Splitting integrators for nonlinear Schrödinger ... - Semantic Scholar
Splitting integrators for nonlinear Schr¨odinger equations over long times Ludwig Gauckler1 and Christian Lubich1 July 7, 2008 1
Mathematisches Institut, Universit¨ at T¨ ubingen, Auf der Morgenstelle 10, D-72076 T¨ ubingen, Germany. email: {gauckler,lubich}@na.uni-tuebingen.de
Abstract Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie-Trotter splitting in time for cubic Schr¨ odinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schr¨ odinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension. Keywords: Nonlinear Schr¨ odinger equation – Splitting integrators – Split-step Fourier method – Long-time behavior – Near-conservation of actions, energy, and momentum – Modulated Fourier expansion MSC (2000): 65P10, 65M70
1
Introduction
We consider the nonlinear Schr¨odinger equation iut = −∆u + V ∗ u + |u|2 u,
(1)
where u = u(x, t), x ∈ Td = Rd /2πZd , t ≥ 0, in dimension d ≥ 1 with periodic boundary conditions. The potential V = V (x) ∈ L2 (Td ) is assumed to be periodic with real Fourier coefficients and acts by convolution on u. This linear potential of convolution type makes the linear part of the equation less unpleasant in arbitrary spatial dimension d, see for example [1], [2], and [6] where such equations are studied. A standard way to discretize this equation is a spectral semi-discretization in space followed by a Lie-Trotter splitting in time. In this paper, we show that the actions of the linear Schr¨odinger equation iut = −∆u + V ∗ u as well as energy and momentum of (1) are approximately conserved along the numerical solution over long times. Energy and momentum are exact invariants of (1), and the actions are approximately conserved along solutions of (1) over long times as is shown in [1] and [7]. In [7] these quantities were studied for spectral semi-discretizations in space. Here, we extend these results to full discretizations using in addition a Lie-Trotter splitting in time. A long time analysis of semi-discretizations in time via a Lie-Trotter splitting for linear Schr¨odinger equations with a multiplicative potential has been performed by Dujardin and Faou [5]. The proof of our results is done by analyzing a modulated Fourier expansion of the numerical solution. Such expansions have been used by Cohen, Hairer, and Lubich [3] to analyze full discretizations of semilinear wave equations with trigonometric integrators and the St¨ ormer-Verlet method as time integrators. In comparison with backward error analysis – the standard tool for proving conservation properties over long times (successfully applied to numerical solutions of ordinary differential equations without high frequencies) – the technique of modulated Fourier expansions does not rely on the smallness of hω where h is the time step size and ω the largest 1
frequency of the linear part of the system, see [9, Chapters IX and XIII]. This is of major importance in the context of partial differential equations presented here where the frequencies tend to infinity as the spatial discretization is refined. The modulated Fourier expansion of the numerical solution is obtained as in [3] by an approximate, time-continuous solution of a formal modulation system. It describes the numerical solution at times nh for natural numbers n up to a small defect. The main conceptual difference in comparison to [3] is that we do not use invariants of the modulation system to get conservation properties. Instead of this, we define an auxiliary function which has almost-invariants close to the actions, energy, and momentum of the numerical solution and describes the approximate solution of the modulation system at times nh up to a small defect. In Section 2 we recall results concerning (almost) invariants of the exact solution of (1). The considered discretization of (1) is presented in Section 3. In the subsequent sections we analyze the (almost) invariants of the exact solution along this numerical approximation. In Section 4 our main theorem for the numerical solution is stated and illustrated by numerical experiments. The proof of this result is given in Sections 5 and 6 where we introduce a modulated Fourier expansion of the numerical solution and analyze its conservation properties.
2
The nonlinear Schr¨ odinger equation over long times
In this section we introduce some quantities which are known to be conserved (exactly or approximately over long times) along solutions of the nonlinear Schr¨odinger equation (1) with small initial data.
2.1
Conservation of energy and momentum
Being a Hamiltonian system, the total energy or Hamiltonian Z 1 1 H(u, u ¯) = |∇u|2 + (V ∗ u)¯ u + |u|4 dx d 2(2π) Td 2
(2)
is exactly conserved along solutions of (1). Here, |·| denotes the Euclidean norm. The same is true for the momentum Z 1 (u∇¯ u−u ¯∇u)dx. (3) K(u, u ¯) = i (2π)d Td
2.2
Near-conservation of actions over long times
P Along any solution u(x, t) = j∈Zd uj (t)ei(j·x) with j·x = j1 x1 +· · ·+jd xd of the linear Schr¨odinger equation iut = −∆u + V ∗ u the actions Ij (u, u ¯) =
1 |uj |2 2
(j ∈ Zd )
(4)
are exactly conserved. Along a solution of our nonlinear equation (1) these actions are approximately conserved over long times under a condition of small initial data and a non-resonance condition, see [1] and [7]. The smallness of the initial data is measured in the Sobolev norm kuks =
X
|ωj |s |uj |2
j∈Zd
21
for s ≥ 0 where ωj = |j|2 + Fj (V ) = (j12 + · · · + jd2 ) + Fj (V ) (j ∈ Zd )
(5)
are the frequencies (eigenvalues) of the linear P part of (1) and Fj (V ) = Vj denotes the j-th Fourier coefficient of the periodic function V = j Vj ei(j·x) (by assumption Vj ∈ R). In the definition of k·ks a frequency ωj = 0 has to be replaced by 1; this has to be done also in the following whenever 2
the absolute value of a frequency ωj = 0 appears. The initial data is assumed to be of size ε ≪ 1 in the norm k·ks . We consider the almost-conservation of actions on time intervals of length ε−N for natural numbers N . Besides the smallness of the initial data we need a non-resonance condition on the frequencies (5). For the precise statement of this non-resonance condition we introduce the following notations of [4] and [7]. For a sequence k = (kl )l∈Zd of integers kl and the sequence ω = (ωl )l∈Zd of frequencies (5) we write Y σ|k | X X X ωl l (6) kl ωl , ω σ|k| = |kl |, k · ω = kl l, kkk = j(k) = l∈Zd
l∈Zd
l∈Zd
l∈Zd
for σ ∈ R. We further define the set of near-resonant indices k 1
Rε = { (j, k) : j = j(k), k 6= hji, |k · ω − ωj | < ε 2 , kkk ≤ 2N + 2 } where hji = (δjl )l∈Zd with Kronecker’s delta. The non-resonance condition then reads sup (j,k)∈Rε
|ωj |s− |ω (s−
d+1 2
d+1 2 )|k|
|
εkkk+1 ≤ C0 ε2N +4
(7)
with a constant C0 (independent of ε), and we have the following theorem (for a similar theorem see [1, Theorem 3.26]). Theorem 1 ([7, Theorem 1]). For given N and s ≥ d+1 there exists ε0 > 0 such that the following holds: Under the conditions of small initial data ku(0)ks ≤ ε ≤ ε0 and of non-resonance (7), the estimate X 3 |Il (u(·, t), u ¯(·, t)) − Il (u(·, 0), u ¯(·, 0))| ≤ Cε 2 for 0 ≤ t ≤ ε−N |ωl |s 2 ε d l∈Z
holds for solutions u(x, t) of (1) with a constant C which depends on C0 , d, N , s, and V but is independent of ε and t. The non-resonance condition is fulfilled for a large set of potentials V as is shown in [7, Proposition 2].
3
Discretization of the nonlinear Schr¨ odinger equation
We describe a standard way to discretize the nonlinear Schr¨odinger equation (1), see for example [10].
3.1
Spectral semi-discretization in space
The semi-discretization in space is done by a spectral collocation method as in [7]. We write M = {−M, . . . , M − 1}d . The trigonometric polynomial X uM (x, t) = qj (t)ei(j·x) j∈M
is chosen as an ansatz for the solution u of the nonlinear Schr¨odinger equation (1), inserted in (1), and evaluated at the collocation points xk =
π k, M
k ∈ M.
Writing uM (xk , t)k∈M = F2M (qj (t))j∈M with the d-dimensional discrete Fourier transform F2M equation (1) reduces to the system of ordinary differential equations i
duM −1 M (xk , t)k∈M = F2M ΩF2M u (xk , t)k∈M + (|uM (xk , t)|2 uM (xk , t))k∈M dt 3
(8)
where Ω = diag((ωl )l∈M ) is the diagonal matrix with the frequencies ωl , l ∈ M, on its diagonal. The equation for the initial value in t = 0 becomes uM (xk , 0)k∈M = u(xk , 0)k∈M . Equation (8) describes a finite dimensional complex Hamiltonian system with Hamiltonian HM (uM (xk , t)k∈M , uM (xk , t)k∈M ) = HM (uM , uM ), T 1 X M 1 −1 M u (xk , t)k∈M + HM (uM , uM ) = uM (xk , t)k∈M F2M ΩF2M |u (xk )|4 2 4 k∈M (9) Z 1 1 M 2 M M 4 M |∇u | + (V ∗ u )u + Q(|u | ) dx, = 2(2π)d Td 2 P where Q(v) denotes the trigonometric interpolation of a periodic function v = j∈Zd vj ei(j·x) in the collocation points, X X vj+2M m ei(j·x) . Q(v) = j∈M m∈Zd
The Hamiltonian HM is exactly conserved along solutions of the semi-discretized system (8) whereas the Hamiltonian H (equation (2)) of the nonlinear Schr¨odinger equation (1) is approximately conserved over long times along solutions of (8), cf. [7, Theorem 2].
3.2
Lie-Trotter splitting in time
The space discretized nonlinear Schr¨odinger equation (8) is discretized in time by a Lie-Trotter splitting with step size h: If un denotes the approximation to the solution uM of (8) after n time steps at time tn = nh, we compute the approximation after n + 1 time steps by n 2
v n (x) = Q(e−ih|u | un )(x), X un+1 (x) = e−iωj h vjn ei(j·x) .
(10)
j∈M
The iterates can be computed efficiently using the fast Fourier transform. As initial value of the iteration we choose u0 = Q(u(·, 0)). (11) v n (xk )k∈M is the exact solution of the differential equation i
duM (xk , t)k∈M = (|uM (xk , t)|2 uM (xk , t))k∈M dt
(12)
at time tn+1 with initial value un (xk )k∈M at tn , and un+1 (xk )k∈M is the exact solution of the differential equation duM −1 M i u (xk , t)k∈M (13) (xk , t)k∈M = F2M ΩF2M dt at time tn+1 with initial value v n (xk )k∈M at tn . A more popular splitting algorithm is the Strang or symmetric Lie-Trotter splitting where half a time step with (13) is performed before and after the time step with (12). Since the numerical solution obtained from that method differs from that of method (10) only by half a time step with (13) at the beginning and the end, the long-time behavior of both methods is the same. We therefore only consider the simpler method (10) in this article.
4
Results and numerical experiments
We now turn to the formulation of the main result of this paper: The (approximately) conserved quantities of the exact solution of (1) introduced in Section 2 are approximately conserved along the numerical solution introduced in Section 3 over long times. We adopt the notations (6), but as in [7, Section 5] k and ω are finite sequences (kl )l∈M and (ωl )l∈M , respectively, and X j(k) = kl l mod 2M ∈ M l∈M
4
where mod 2M denotes the entry-wise reduction modulo 2M with representative chosen in M. The set of near-resonant indices takes the form 1
Rε,M,h = { (j, k) : j = j(k), k 6= hji, |ei(ωj −k·ω)h − 1| < ε 2 h, kkk ≤ 2N + 2 }, and the non-resonance condition reads sup (j,k)∈Rε,M,h
|ωj |s− |ω
d+1 2
(s− d+1 2 )|k|
|
εkkk+1 ≤ C0 ε2N +4 ,
sup (j,k)∈Rε,M,h
s l∈M |kl ||ωl | d+1 d+1 |ω (s− 2 )|k| ||ωj | 2
P
1
ε 2 kkk ≤ C0 εN +1
(14) with a constant C0 independent of ε where the zero frequencies are replaced by 1. The second condition of (14) is not needed in the time-continuous situation (7). However, in Appendix A we will show that it is true in this situation under the non-resonance condition of Bambusi and Gr´ebert [1]. Note that in the limit h → 0 we have h1 |ei(ωj −k·ω)h − 1| → |k · ω − ωj | and hence the non-resonance condition (14) reduces to a non-resonance condition which is fulfilled for a large set of potentials. Theorem 2. For given N and s ≥ d + 1 there exists ε0 > 0 such that the following holds: Under the conditions of small initial data ku0 ks ≤ ε ≤ ε0 and of non-resonance (14), the estimates X
|ωl |s
l∈M
3 |Il (un , un ) − Il (u0 , u0 )| ≤ Cε 2 2 ε
3 |H(un , un ) − H(u0 , u0 )| ≤ Cε 2 ε2 3 |HM (un , un ) − HM (u0 , u0 )| ≤ Cε 2 2 ε d n n X 3 |Kr (u , u ) − Kr (u0 , u0 )| ≤ Cε 2 2 ε r=1
for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N
hold for the numerical solution un described in Section 3 with time step size h ≤ 1 with a constant C which depends on C0 , d, N , s, and V but is independent of ε, M , h, and n.
100
100
10−2
10−2
10−4
10−4
10−6
10−6
10−8
10−8
10−10
10−10
10−12
10−12
10−14
10−14
10−16
10−16
10−18
10−18
10−20 0
10000
10−20 0
20000
10000
20000
Figure 1: Example in one dimension: Actions (black lines), discrete energy (upper grey line), and momentum (lower grey line) for non-resonant time step size h = 0.005+2π/ω−6 (left) and resonant time step size h = 2π/ω−6 (right). 5
.0025 .0020 .0015 .0010 .0005 .01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.02
.03
.04
.05
.06
.07
.08
.09
.10
.02
.03
.04
.05
.06
.07
.08
.09
.10
101 100 10−1 10−2 .01 10−6 10−9 10−12 10−15 .01
Figure 2: Example in one dimension: Maximal deviation in actions (upper image), energy (middle image), and momentum (lower image) for time step sizes h ∈ [0.01, 0.1]. The proof of this theorem will be the subject of Sections 5 and 6. The theorem implies the spatial regularity of the numerical solution over long times: by the conservation of actions we have 7 |kun k2s − ku0 k2s | ≤ Cε 2 and hence kun k ≤ 2ε over long times 0 ≤ tn = nh ≤ ε−N . The same results as in Theorem 2 are true if the Lie-Trotter splitting is replaced by the symmetric Strang splitting since this change does not affect the long-time behavior. We further mention that a generalization of Theorem 2 to nonlinear Schr¨odinger equations with nonlinearities of the form g(|u|2 )|u|2 u where g is a polynomial is straightforward. We conclude this section with some numerical experiments in one and two spatial dimensions. The initial value u(·, 0) in one dimension is chosen as x 3 x 2 3 x 3 x −1 + 1 + i · 0.1 · −1 +1 , u(x, 0) = 0.1 · π π π π
and the potential V is chosen such that
ωj =
q
|j|4 + rj
where rj = 0.5 for j ≥ 0 and rj = 0.8 for j < 0. For the discretization in space we take M = 27 , that is 28 collocation points, and the time step size is chosen as h = 0.005+2π/ω−6 and h = 2π/ω−6 . As explained by Theorem 2, we observe conservation of actions, energy, and momentum for the first time step size in Figure 1 where the eight largest actions Ij (equation (4)) and sixteen other actions, the (discrete) energy HM (equation (9)), and the momentum K (equation (3)) are plotted. For the second time step size we observe numerical resonances in Figure 1. In fact, this time step size does not fulfill the non-resonance condition (14): For k = h−6i + h−(M − 3)i we have j(k) = M − 3 and ei(ωj(k) −k·ω)h = e2πi(ωM −3 −ω−(M −3) )/ω−6 ≈ 1, i. e. (j(k), k) is near-resonant, but this pair does not fulfill (14). 6
10−4
10−4
10−6
10−6
10−8
10−8
10−10
10−10
10−12
10−12
10−14
10−14
10−16
10−16
0
500000
1000000
0
500000
1000000
Figure 3: Example in two dimensions: Actions (black lines), discrete energy (upper grey line), and first component of momentum (lower grey line) for non-resonant frequencies (15) (left) and resonant frequencies (16) (right). In Figure 2 we study the effect of resonant time step sizes in the example in one dimension. We integrate with time step sizes h ∈ [0.01, 0.1] over a time interval of length 5000, and plot the maximal deviation in the actions, in the energy, and in the momentum for these different time steps with a resolution of 0.0001. In two dimensions we choose 2 x x 2 x x 1 2 2 1 −1 +1 −1 +1 u(x1 , x2 , 0) = 0.01 · π π π π x 2 x 2 x 2 x 2 1 1 2 2 + i · 0.01 · −1 +1 −1 +1 π π π π as initial value. For the potential we make the choice q ω(j1 ,j2 ) = ωj = |j|4 + rj ,
(15)
where rj = 0.5 for j1 , j2 ≥ 0, rj = 0.6 for j1 ≥ 0 > j2 , rj = 0.7 for j1 < 0 ≤ j2 , and rj = 0.8 for j1 , j2 < 0. We further study the resonant situation V = 0 where ω(j1 ,j2 ) = ωj = |j|2 .
(16)
For the numerical discretization we choose M = 24 and h = 0.1. In Figure 3 the actions Ij , the (discrete) energy HM , and the first component of the momentum K are plotted. In the case of non-resonant frequencies the good long-time behavior is explained by Theorem 2.
5
Modulated Fourier expansions
The analysis of the numerical solution is done by the method of modulated Fourier expansions. We follow [3], where a full discretization of semilinear wave equations is studied, and [7] where the exact solution of the nonlinear Schr¨odinger equation (1) and its semi-discretization in space (8) are studied. Throughout this section we work under the assumptions of Theorem 2. All appearing constants will be denoted by C. The main point is that all these constants do not depend on ε, M , h, and the time 0 ≤ t ≤ ε−1 ; however, they may depend on C0 and N from the non-resonance condition (14), the dimension d, the regularity parameter s, and the potential V . 7
5.1
The modulation system
As an ansatz for a function u ˜ which approximates the numerical solution un at time tn we choose a modulated Fourier expansion, X X k u ˜(x, t) = zj(k) (εt)ei(j(k)·x) e−i(k·ω)t = z k (x, εt)e−i(k·ω)t (17) kkk≤K
kkk≤K
k with z k (x, εt) = zj(k) (εt)ei(j(k)·x) where zlk (εt) = 0 for l 6= j(k). In the following we tacitly assume kkk ≤ K = 2N + 2 unless stated otherwise. If we insert (17) in the splitting ansatz (10) we get from the conditions un = u ˜(·, tn ) and un+1 = u ˜(·, tn+1 )
X
k zj(k) (ε(t + h))ei(j(k)·x) e−i(k·ω)(t+h) =
XX
e−iωj h
k j∈M
k
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
∞ X (−ih)m m! m=0
1 m+1 z km+2 . . . z k2m+1 ei(j·x) e−i(k·ω)t . Fj Q z k . . . z k
For k = k1 + · · · + km+1 − km+2 − · · · − k2m+1 we have ( 1 1 k km+1 k2m+1 km+2 zj(k 1 ) . . . zj(km+1 ) zj(km+2 ) . . . zj(k2m+1 ) , 2m+1 k km+1 km+2 k z ...z = Fj Q z . . . z 0,
j = j(k), else
since j(k) ≡ j(k1 )+· · ·+j(km+1 )−j(km+2 )−· · ·−j(k2m+1 ) (mod 2M ) and j(k) ∈ M. Comparing the coefficients of ei(j(k)·x) e−i(k·ω)t thus gives the modulation system k Lkj(k) zj(k) =
∞ X (−ih)m m! m=1
1
X
m+1
k +···+k −km+2 −···−k2m+1 =k
1 m+1 Fj(k) Q z k . . . z k z km+2 . . . z k2m+1
(18a)
where (Lkj zjk )(εt) = zjk (ε(t + h))ei(ωj −k·ω)h − zjk (εt) ∞ X (εh)l k (l) (zj ) (εt)ei(ωj −k·ω)h . = (ei(ωj −k·ω)h − 1)zjk (εt) + l! l=1
Here we denote by yields
(l)
the l-th derivative with respect to τ = εt. The initial condition (11) further X
zjk (0) = u0j .
(18b)
k
5.2
Results on the modulation functions
The remaining part of this section is devoted to the construction of an approximate solution of the modulation system (18) for 0 ≤ εt = τ ≤ 1. The properties of this approximate solution are colk lected in the following proposition. For measuring the size of functions z = (z k )k = (zj(k) ei(j(k)·x) )k we use the norm
X
2 21 X X
k|z|ks = {z k } = |ωj |s |zjk | k
s
j∈M
k
from [7] where
{v}(x) =
X
|vj |ei(j·x)
j∈Zd
for a periodic function v(x) =
P
j∈Zd
vj ei(j·x) , and the scaling
ˆ = (ˆ z z k )k = (|ω 8
2s−d−1 |k| 4
|z k )k .
Proposition 1. There exists a function u ˜(x, t) = and 0 ≤ εt ≤ 1 satisfying kun − u ˜(·, tn )ks ≤ CεN +2
P
k i(j(k)·x) −i(k·ω)t e kkk≤2N +2 zj(k) (εt)e
for x ∈ Td
for 0 ≤ tn = nh ≤ ε−1 .
(19a)
Moreover, the following estimates hold: • u ˜ is small, k˜ u(·, t)ks ≤ Cε.
(19b)
• z is small, k zj(k) = 0 for (j(k), k) ∈ Rε,M,h ,
k|ˆ z|k d+1 ≤ Cε, 2
X
hji
|ωj |s |zj |2 ≤ Cε2 ,
j∈Zd
X
|ωj |s
j∈Zd
X
k6=hji
2 |zjk | ≤ Cε5 ,
X
j∈Zd
|ωj |
d+1 2
X
k6=hji
2 |ˆ zjk | ≤ Cε5 .
(19c)
All constants are independent of ε, M , h, and 0 ≤ t ≤ ε−1 but may depend on C0 , d, N , s, and V. The proof of this proposition will cover the remaining part of this section except Subsection 5.9.
5.3
Iterative solution of the modulation system
As in [7] we collect those pairs (j(k), k) with k 6= hj(k)i which are not in Rε,M,h in the set Sε,M,h = { (j, k) : j = j(k), k 6= hji, (j, k) 6∈ Rε,M,h , kkk ≤ K }. The solution of the modulation system (18) is determined up to a small defect by an iterative procedure. We start by setting i0 i0 h h hji k zj = u0j and zj(k) = 0 for k 6= hj(k)i
for 0 ≤ εt = τ ≤ 1. For n ≥ 0 and 0 ≤ εt = τ ≤ 1 we set +2 h in+1 h NX 1 (εh)l k (l) = i(ω −k·ω)h zjk (zj ) (εt)ei(ωj −k·ω)h − l! e j −1 l=1 +
N X (−ih)m m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
in 1 m+1 z km+2 . . . z k2m+1 Fj Q z k . . . z k
for (j, k) ∈ Sε,M,h , +2 in+1 h NX h (εh)l hji (l) hji z˙j = ε−1 h−1 − (zj ) (εt) l! l=2
N X (−ih)m + m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =hji
h
in+1 hX in hji zj (0) = u0j − zjk (0) ,
h
zjk
in 1 m+1 z km+2 . . . z k2m+1 , Fj Q z k . . . z k
k6=hji
in+1
= 0 for (j, k) ∈ Rε,M,h .
The notation [·]n means that the n-th iterates of the variables within the brackets are taken. In this construction we have truncated the expansion of the exponential function and the Taylor expansion of zjk after N and N + 2 terms, respectively. We will show that after L = 2N + 2 iterations we obtain functions z = [z]L with the properties described in Proposition 1. 9
5.4
Abstract formulation of the iteration
Let
( max( 12 (kkk + 1), 2), [[k]] = 1 2 (kkk + 1) = 1,
We scale the variables as in [7] ( ε−[[k]] zjk , k aj = 0,
k = hji, k 6= hji
bkj
and
k 6= hji, k = hji.
( 0, k = hji, = ε−[[k]] zjk , k = 6 hji,
and we write a = (ak )k = (akj(k) ei(j(k)·x) )k , b = (bk )k = (bkj(k) ei(j(k)·x) )k , and c = a + b. We further define A(a)kj = −
N +2 X (εh)l k (l) (εh)l k (l) (aj ) (εt), B(b)kj = − (bj ) (εt)ei(ωj −k·ω)h , l! l! l=1 l=2 ( i(ωj −k·ω)h k (e − 1)cj , (j, k) ∈ Sε,M,h , (Ωc)kj = 1 k 2 else, ε hcj ,
N +2 X
and N X (−ih)m m! m=1 1 X 2m+1 m+1 ]] k1 Fj Q ε[[k ]]+···+[[k c . . . ck ckm+2 . . . ck2m+1 .
F(c)kj = ε− max([[k]],2)
k1 +···+km+1 −km+2 −···−k2m+1 =k
The iteration in the rescaled variables becomes in in h h in+1 h for (j, k) ∈ Sε,M,h , = (Ω−1 B(b))kj + (Ω−1 F(c))kj bkj in h in+1 in h hX in+1 h in h hji hji hji hji , aj (0) = ε−1 u0j − + h−1 F(c)j ε[[k]]−1 bkj (0) . = ε−1 h−1 A(a)j a˙ j k6=hji
We also use a second rescaling of the variables, a ˆkj = |ω ˆ With ˆ=a ˆ + b. c |k| ˆ c)k = ε− max([[k]],2) |ω 2s−d−1 4 F(ˆ | j
|ω −(
N X (−ih)m m! m=1
2s−d−1 )(|k1 |+···+|k2m+1 |) 4
1
2s−d−1 |k| 4
X
k1 +···+km+1 −km+2 −···−k2m+1 =k m+1
|ˆ ck . . . cˆk
1 2m+1 ]] Fj Q ε[[k ]]+···+[[k
cˆkm+2 . . . cˆk2m+1
ˆ becomes the iteration for b in in h h in+1 h −1 ˆ k + (Ω−1 F(ˆ ˆbk ˆ c))k = (Ω B( b)) j j j
5.5
2s−d−1 |akj , ˆbkj = |ω 4 |k| |bkj , and
for (j, k) ∈ Sε,M,h .
Estimating the nonlinearity
The estimation of the nonlinearity is based on the inequality
X
2
X
2 X
2
{Q(ck dl )} ≤ C {ck } {dl } = Ck|c|ks k|d|ks . k,l
s
s
k
l
s
from [7, Lemmas 1 and 6] which is valid for s > d2 . The constant depends on d, s, and V but not on M . 10
Lemma 1. We have 1
k|Ω−1 c|ks ≤ ε− 2 h−1 k|c|ks
(20a)
k|F(c)|ks ≤ εhk|c|ks P1 (k|c|ks ) ˜|ks P2 (max(k|c|ks , k|˜ k|F(c) − F(˜ c)|ks ≤ εhk|c − c c|ks ))
(20b) (20c)
where P1 and P2 are polynomials of degree 2N whose coefficients are nonnegative and bounded ˆ and k| · |k d+1 instead of c, c ˆ˜, F, ˆ, c ˜, independently of ε, M , and h. The same estimates hold for c 2 F, and k| · |ks , respectively. Proof. The first estimate (20a) follows as in [7, Lemma 1], and the second and the third one can be seen similarly as there: We note that for m ≥ 1 and k1 + · · · + km+1 − km+2 − · · · − k2m+1 = k [[k1 ]] + · · · + [[k2m+1 ]] ≥ max([[k]], 2) + m. With this inequality and [7, Lemma 6] each summand of F(c) can be estimated as the nonlinearity F(c) of [7], and the triangle inequality yields (20b) and (20c). For example concerning (20b) we have N X (εh)m 2m 2m+1 k|F(c)|ks ≤ C k|c|ks . m! m=1
5.6
Size of the iterated modulation functions
Estimates of the modulation functions are obtained similarly as in [7, Subsection 3.6] with the help of Lemma 1. We remark that for 0 ≤ εt = τ ≤ 1 k|A(a(l+1) (τ ))|ks ≤ Cε2 h2 k|B(b(l) (τ ))|ks ≤ Cεh
sup k=2,...,N +2
sup k=1,...,N +2
k|a(l+1+k) (τ )|ks ,
k|b(l+k) (τ )|ks
with a constant which depends on N but not on ε, M , h, and τ . With these estimates and Lemma 1 we get for αn = βn = α ˆn = βˆn =
max
sup k|[a(l) (τ )]n |ks ,
max
sup k|[b(l) (τ )]n |ks ,
max
sup k|[ˆ a(l) (τ )]n |k d+1 ,
max
ˆ (l) (τ )]n |k d+1 sup k|[b
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1 l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
2
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
2
as in [7] αn = α ˆ n ≤ C,
1
βn ≤ Cε 2 ,
1 βˆn ≤ Cε 2
(21)
for n = 0, . . . , L with a constant C which depends on d, L, n, s, and V but not on ε, M , and h. With k|[˜ u]n |ks ≤ εk|[c]n |ks (see [7, Subsection 3.6] this yields for z = [z]L the estimates (19b) and (19c) of Proposition 1.
5.7
Defect of the iterated modulation functions
After n steps the defect in the modulation system (18a) is (with j = j(k)) h
Lkj zjk −
∞ X (−ih)m m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 in m+1 Fj Q z k . . . z k z km+2 . . . z k2m+1 .
11
This has to be considered for all indices k, not only for those with kkk ≤ K. We set z k = 0 for kkk > K. In comparison to [7] there are two additional sources of defect due to the truncation of the expansion of the exponential function and of the Taylor expansion in the iterative procedure described in Subsection 5.3. The defect in (18a) after n iteration steps is decomposed as follows (the notation is chosen to be consistent with Proposition 2 below). The defect arising from the truncation of the Taylor expansion of zjk is h
hkj − ekj
in
N +2 h in X (εh)l k (l) i(ωj −k·ω)h = Lkj zjk − (zj ) e − (ei(ωj −k·ω)h − 1)zjk , l! l=1
the defect arising from the truncation of the expansion of the exponential function is ∞ h in h X dkj = −
m=N +1
(−ih)m m!
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
in 1 m+1 , z km+2 . . . z k2m+1 Fj Q z k . . . z k
and the remaining defect is decomposed as N +2 h in h X (εh)l k (l) i(ωj −k·ω)h ekj + fjk + gjk = (ei(ωj −k·ω)h − 1)zjk + (zj ) e l! l=1
−
N X
m
(−ih) m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 in m+1 Fj Q z k . . . z k z km+2 . . . z k2m+1 ,
where [ekj ]n = 0 for (j, k) ∈ 6 Sε,M,h and k 6= hji, [fjk ]n = 0 for (j, k) 6∈ Rε,M,h , and [gjk ]n = 0 for kkk ≤ K (dk and g k can be nonzero for kkk > K). The defect in equation (18b) for the initial condition reads h in hX in hji d˜j = u0j − zjk (0) . k
Now, we estimate these quantities for n = 0, . . . , L. For f we get as in [7, Subsection 3.7] using the first part of the non-resonance condition (14), Lemma 1, and (21) n
k|[f ]
2 |ks
ˆ c)]n |k2d+1 ≤ k|[F(ˆ 2
sup (j,k)∈Rε,M,h
|ω | 2s−d−1 4 j |ω
2s−d−1 |k| 4
|
ε[[k]]
2
≤ (CεN +3 h)2
(22)
for n = 0, . . . , L with a constant which depends on C0 , d, N , n, s, and V but not on ε, M , and h. In addition we have ˆ c)]n |k d+1 ≤ Cεh. k|[(ε−[[k]] fˆk )k ]n |k d+1 ≤ k|[F(ˆ (23) 2
2
With the same arguments as in the proof of Lemma 1 using in addition (21) we obtain
X 1
{[g k ]n } ≤ Cε 2 (K+2) εh = CεN +3 h
K
d 2
5.8.1
Size of the numerical solution
with a constant only depending on d, s, and V , cf. [7, Lemma 1].
We first determine the size of the numerical solution un . For this solution we have with (30) kun+1 ks ≤
∞ X 2 n 2 2 n 2 0 2 hm 2m n 2m+1 C ku ks = kun keC hku ks ≤ ku0 keC h(ku ks +···+ku ks ) . m! m=0
13
Hence, for ku1 ks , . . . , kun ks ≤ 2ε (note ku0 k ≤ ε) the estimate kun+1 k ≤ 2ε is fulfilled if (n + 1)h ≤ ε−1 and ε ≤ log(2)/(4C 2 ). Thus we have shown kun ks ≤ 2ε
for 0 ≤ tn = nh ≤ ε−1
(31)
if ε is small enough. 5.8.2
Error on [0, ε−1 ]
In t = 0 we have by (29)
˜ L |k ≤ CεN +2 ku0 − u ˜(·, 0)ks ≤ k|[d] s
The constant depends on d, N , s, and V but not on ε, M , and h. For ku1 ks , ku2 ks ≤ cε the m difference of the powers um 1 and u2 satisfies with (30) m
X
m kum 2−j (u1 + u2 )j−1 (u1 − u2 )(u1m−j + u2m−j ) 1 − u2 ks = j=1
≤ C m−1
m X
s
2−j (2cε)j−1 ku1 − u2 ks 2(cε)m−j = (Ccε)m−1 mku1 − u2 ks .
j=1
Hence ∞
X X (−ih)m m+1 m
m (u1 u ¯1 − um+1 u ¯ ) e−iωj h Fj
2 2 m! s m=1 j∈M ∞ X
≤
≤
hm m+1 m m+1 m (kum+1 u ¯m u ¯2 ks + kum+1 u ¯m u ¯2 ks ) 1 − u1 2 − u2 1 1 m! m=1
∞ X hm m+1 (C (cε)m+1 (Ccε)m−1 mku1 − u2 ks + C m (cε)m (Ccε)m (m + 1)ku1 − u2 ks ) m! m=1
≤3
∞ X
2 hm (Ccε)2m ku1 − u2 ks = 3h(Ccε)2 e(Ccε) h ku1 − u2 ks . (m − 1)! m=1
Using the size of un (31), the size of u ˜ (19b) from Proposition 1, and the estimates of the defect (22), (24), (25), (26), (28) we get ∞
X X (−ih)m
kun+1 − u ˜(·, tn+1 )ks ≤ e−iωj h Fj ((un )m+1 (un )m − (˜ u(·, tn ))m+1 (˜ u(·, tn ))m ) m! s m=0 j∈M
X
+ [(hk − ek + dk + ek + f k + g k )(εtn+1 )]L e−i(k·ω)tn+1 s
k∈ZM
2
≤ (1 + 3h(Cε)2 e(Cε) h )kun − u ˜(·, tn )ks + CεN +3 h.
We have thus shown the smallness of the error (19a) for (n + 1)h ≤ ε−1 . This concludes the proof of Proposition 1.
5.9
An auxiliary function close to the modulation functions
In [3], conservation properties of the numerical solution are deduced from almost-invariants of the modulation system. In the situation of the numerical solution of the semilinear wave equation with trigonometric integrators considered in that paper, the nonlinearity in the modulation system is, up to a multiplication with real-valued filter functions, the same as the nonlinearity in the modulation system for the exact solution of the wave equation, cf. [3, equation (34)] and [4, equation (15)]. In the case of the nonlinear Schr¨odinger equation and the Lie-Trotter splitting considered in the present paper, we encounter a different situation. While the nonlinearity in the modulation system 14
for the exact solution is cubic (cf. [7, equation (7a)]), it contains terms of arbitrary high order in the modulation system for the time-discrete equation (cf. equation (18a)). The point is, that some of the terms are multiplied by an imaginary factor while others are multiplied by a real factor, which prevents the arguments from [7, Subsection 4.1] for the derivation of almost-invariants of the modulation system to work. For this reason, we now introduce an auxiliary function wn = wn (x, t) = (wk,n (x, t))k on each interval [nh, (n + 1)h], which satisfies a differential equation whose nonlinearity is the nonlinearity in the modulation system for the exact solution [7, equation (7a)]. This allows to deduce almostinvariants of this auxiliary function as in [7]. Moreover, the auxiliary functions wn are constructed in such a way that they approximate the modulation functions z at the points nh and (n + 1)h up to a small defect. To be precise, let z = [z]L , f = [f ]L , and h = [h]L . We set for kkk ≤ (2N + 1)K, (n + 1)h ≤ ε−1 , and 0 ≤ t ≤ h wk,n (x, t) =
N X (−it)m m! m=0
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 m+1 Q zk · · · zk z km2 · · · z k2m+1
(x,εtn )
.
These functions have the properties stated in the following proposition where again only those k with kkk ≤ K are considered. k,n Proposition 2. For (n+1)h ≤ ε−1 the functions wn = wn (x, t) = (wk,n (x, t))k = (wj(k) (t)ei(j(k)·x) )k for x ∈ Td and 0 ≤ t ≤ h have the following properties:
• They are related to z by zjk (εtn ) = wjk,n (0)
zjk (εtn+1 )ei(ωj −k·ω)h = wjk,n (h) + fjk (εtn+1 ) + hkj (εtn+1 ) (32a)
and
with k|h|ks ≤ CεN +3 h,
ˆ d+1 ≤ CεN +3 h, k|h|k 2
k|f |ks ≤ CεN +3 h,
1
k|(ε− 2 (kkk+1) fˆk )k |k d+1 ≤ Cεh
(32b)
2
where fjk = 0 for (j, k) 6∈ Rε,M,h and hkj = 0 for j 6= j(k) or (j, k) ∈ Rε,M,h . • They satisfy i
d k,n w = dt
1
X
Q(wk
,n
2
wk
,n
wk3 ,n ) + rk,n
(32c)
k1 +k2 −k3 =k
with k|rn |ks ≤ CεN +3 ,
k|ˆrn |k d+1 ≤ CεN +3 .
(32d)
2
• They are small, k|wn |ks ≤ Cε,
ˆ n |k d+1 ≤ Cε, k|w 2
1
ˆ k,n )k |k d+1 ≤ C. k|(ε− 2 (kkk+1) w 2
(32e)
All constants are independent of ε, M , h, n, and t but may depend on C0 , d, N , s, and V . Proof. For j 6= j(k) we have wjk,n = 0 on [0, h], and for kkk ≤ K we have zjk (εtn ) = wjk,n (0) and zjk (εtn+1 )ei(ωj −k·ω)h = wjk,n (h) + fjk (εtn+1 ) + hkj (εtn+1 ) with the notations of Subsection (5.7). This proves (32a), and the estimates (22), (23), (26), and (28) yield (32b). The estimates (32e) of w are obtained by estimating the nonlinearity as in Lemma 1. 15
The auxiliary functions wn satisfy the differential equation (32c) as we show next. With rk,n (t) = −
3N +1 X
(−it)k−1 (−1)l m!n!l! k=N +1 m+n+l=k−1 1 X k+1 Q z k · · · z k z kk+2 · · · z k2k+1 X
k1 +···+kk+1 −k −···−k2k+1 =k
(x,εtn )
k+2
1
X
+
Q(wk
,n
2
wk
,n
wk3 ,n ).
k1 +k2 −k3 =k kk1 k>K or kk2 k>K or kk3 k>K
and Q(Q(v)Q(w)) = Q(vw) we have X 1 2 Q(wk ,n wk ,n wk3 ,n ) k1 +k2 −k3 =k
1
X
=−
Q(wk
,n
2
wk
,n
wk3 ,n ) +
k1 +k2 −k3 =k kk1 k>K or kk2 k>K or kk3 k>K
X
k1 +···+km+n+l+2 −···−k2(m+n+l+1)+1 =k
m+n+l+3
−k
= −rk,n (x, t) +
N X (−it)k−1
k=1
X
m+n+l=k−1
(k − 1)!
N X
m,n,l=0
(−it)m+n+l (−1)l m!n!l!
1 m+n+l+2 Q zk · · · zk z km+n+l+3 · · · z k2(m+n+l+1)+1 X
k1 +···+kk+1 −kk+2 −···−k2k+1 =k
1 k+1 Q z k · · · z k z kk+2 · · · z k2k+1
(x,εtn )
(x,εtn )
(−1)l (k − 1)! d = i wk,n (x, t) − rk,n (x, t). m!n!l! dt
Here, the last equality follows from X
m+n+l=k−1
k−1 k−1−n X (k − 1)! (k − 1 − n)! (−1)l (k − 1)! X = =1 (−1)l m!n!l! n!(k − 1 − n)! l!(k − 1 − n − l)! n=0 l=0
by the binomial theorem. The estimates (32d) of r are obtained in the same way as the estimates of the defects dk and k g . This completes the proof the proposition.
6
Conservation properties
Let z and wn be the functions of Propositions 1 and 2. In this section we prove Theorem 2. As mentioned before we do not use invariants of the modulation system (18) to get conservation properties but rather the auxiliary functions wn of Proposition 2 which have almost-invariants and describe z at the discrete points εtn and εtn+1 up to a small defect.
6.1
Almost-invariants
The functions wn satisfy (32c), and this system possesses almost-invariants as we show next. We set Z X 1 2 1 Q(wk wk wk3 wk4 )dx U (w) = d (2π) d T 1 2 3 4 k +k −k −k =0
where we omit the index n of w and where the sum is (as always) over those k with kkk ≤ K. R R 1 1 Q(¯ v )dx = (2π) Q(v)dx and Q(vw) = Q(Q(v)Q(w)) we get for real sequences Using (2π) d d Td Td 16
µ = (µl )l∈M as in [7, Subsection 6.2] X d X d k k rk U (e−i(k·µ)θ wk )k = 2 0= (k · µ)|wj(k) |2 + 4 Re i(k · µ)wj(k) j(k) . dθ θ=0 dt k
Hence
Iµ (wn (·, t)) =
(33)
k
1X k,n (k · µ)|wj(k) (t)|2 2 k
is an almost-invariant of the system (32c) on [0, h]. The following lemma is similar to [7, Lemma 3]. Due to the usage of wn we now also have to control the entries of wn corresponding to near-resonant indices (note that we don’t have wjk = 0 for (j, k) ∈ Rε,M,h ). This is done with the second part of the non-resonance condition (14). 1
Lemma 2. Let wk = xk + ε 2 (kkk+1) y k and rk = pk + q k with xkj = pkj = 0 for j 6= j(k) or (j, k) ∈ Rε,M,h and yjk = qjk = 0 for (j, k) 6∈ Rε,M,h . The following estimate holds for s ≥ d + 1 with a constant C which depends only on C0 , d, K, s, and V . X X 3 k k y|k d+1 k|ˆ p|k d+1 + CεN + 2 k|ˆ q|k d+1 . |ωl |s |kl ||wj(k) ||rj(k) | ≤ Ck|ˆ x|k d+1 k|ˆ l∈M
2
2
2
2
k
Proof. We have with Sε,M,h = { (j, k) : j = j(k), k 6= hji, (j, k) 6∈ Rε,M,h , kkk ≤ K } P s X X X d+1 l∈M |kl ||ωl | k k xkj ||ˆ pkj | |ωl |s |kl ||wj(k) ||rj(k) |= |ωj | 2 d+1 d+1 |ˆ (s− )|k| 2 2 |ω ||ωj | l∈M k k=hji or (j,k)∈Sε,M,h P s X d+1 1 l∈M |kl ||ωl | 2 (kkk+1) |ˆ yjk ||ˆ qjk |. + |ωj | 2 d+1 d+1 ε (s− )|k| 2 2 |ω ||ω | j (j,k)∈Rε,M,h The non-resonance condition (14) implies P s 3 1 l∈M |kl ||ωl | 2 (kkk+1) ≤ C εN + 2 . sup 0 d+1 d+1 ε (s− )|k| 2 2 (j,k)∈Rε,M,h |ω ||ωj | The proof of sup k=hji or (j,k)∈Sε,M,h
s l∈M |kl ||ωl | d+1 d+1 |ω (s− 2 )|k| ||ωj | 2
P
≤ C.
for a constant C that depends on d, K, s, and V is given in [7, Lemma 3 and Section 6.2]. The Cauchy-Schwarz inequality now yields the first estimate of the lemma. Lemma 3. We have for 0 ≤ tn = nh ≤ ε−1 X |ωl |s Ihli (z(·, εtn )) − Ihli (z(·, 0)) ≤ CεN +3 l∈M
with a constant C which depends on C0 , d, N , s, and V but not on ε, M , h, and n. k Proof. For (n + 1)h ≤ ε−1 we have with (32a) and fj(k) and hkj(k) evaluated at εtn+1
2Iµ (z(·, εtn+1 )) − 2Iµ (z(·, εtn )) =
X
k,n k,n k (k · µ)(|wj(k) (h) + fj(k) + hkj(k) |2 − |wj(k) (0)|2 )
k
h
=
Z
h
=
Z
0
0
X d k,n k k Iµ (wn (t))dt + (k · µ)(|fj(k) + hkj(k) |2 + 2 Re(wj(k) (h)(fj(k) + hkj(k) ))) dt k
X d k k k k (εt Iµ (wn (t))dt + (k · µ)(−|fj(k) + hkj(k) |2 + 2 Re(zj(k) n+1 )(fj(k) + hj(k) ))) dt k
17
and with (33) Z
0
h
X d k,n k,n Iµ (·, wn (t))dt ≤ h sup |k · µ||wj(k) (t)||rj(k) (t)|. dt 0≤t≤h k
Using the first estimate of Lemma 2 we thus get X |ωl |s Ihli (z(·, εtn+1 )) − Ihli (z(·, εtn )) l∈M
3
1
ˆ n |k d+1 k|ˆrn |k d+1 + CεN + 2 hk|(ε− 2 (kkk+1) w ˆ k,n )k |k d+1 k|ˆrn |k d+1 ≤ Chk|w 2
2
2
2
ˆ d+1 k|ˆ ˆ 2d+1 + CεN + 32 k|(ε− 21 (kkk+1) fˆk )k |k d+1 k|ˆf |k d+1 + Ck|h|k z|k d+1 . + Ck|h|k 2
2
2
2
2
where the quantities are evaluated in εtn+1 or the supremum over 0 ≤ t ≤ h is taken. With Propositions 1 and (2) this can be estimated by CεN +4 h where C is independent of ε, M , h, and n. The claimed estimate follows with the triangle inequality by patching n ≤ (εh)−1 intervals of length h together.
6.2
Relationship between almost-invariants and actions
With the same proof as [7, Lemma 5] we get. Lemma 4. We have for 0 ≤ tn = nh ≤ ε−1 X 7 |ωl |s Ihli (z(·, εtn )) − Il (un , un ) ≤ Cε 2 l∈M
with a constant C which depends on C0 , d, N , s, and V but not on ε, M , h, and n.
6.3
From short to long time intervals
By now we have proven Theorem 1 on the short time interval [0, ε−1 ]. The extension to long time intervals [0, ε−N ] as stated in the theorem can be done as in [4, Section 4.5] (see also [7, Section 4.5]). Let nε be the largest integer with nε h ≤ ε−1 . We consider the approximate solution of the modulation system given by Proposition 1 on intervals [mnε h, (m + 1)eε h] for integers m where the numerical solution after mnε time steps is taken as initial value. The conservation of Ihli on these intervals is controlled by Lemma 3 and the difference at the boundaries of these intervals can be controlled as in [7, Lemma 3 and Proposition 4]. Patching together 2ε−N +1 intervals of length nε h (note nε h ≥ 21 ε−1 for h ≤ 1 ≤ ε−1 ) and using the estimates of the difference to the actions Il of Lemma 4 this proves the almost-conservation of actions in Theorem 1.
6.4
Near-conservation of energy and momentum
We now turn to the proof of the long-time near-conservation of the energy H (equation (2)), the discrete energy HM (equation (9)), and the momentum K (equation (3)) which read in terms of the Fourier coefficients Z 1 X 1 n 2 n n ωj |uj | + H(u , u ) = |un |4 dx, d 2 4(2π) d T j∈Zd Z X 1 1 Q(|un |4 )dx, ωj |unj |2 + HM (un , un ) = 2 4(2π)d Td j∈M X n n Kr (u , u ) = 2 jr |unj |2 for r = 1, . . . , d. j∈M
The conservation of actions as stated in Theorem 2 implies as in [7, Subsection 6.4] X 7 |Kr (un , un ) − Kr (u0 , u0 )| ≤ 2 |jr ||Ij (un , un ) − Ij (u0 , u0 )| ≤ Cε 2 j∈M
18
for 0 ≤ tn = nh ≤ ε−N and r = 1, . . . , d since |jr | ≤ C|ωj |s . This proves the near-conservation of the momentum K. For the conservation of energy we note first Z 1 |un |4 dx ≤ k|un |4 k0 ≤ Ckun k4s (2π)d Td and
Z 1 Q(|un |4 )dx ≤ kQ(|un |4 )k0 ≤ Ckun k4s (2π)d Td by the Cauchy-Schwarz inequality and [7, Lemma 6]. The near-conservation of actions implies kun ks ≤ Cε and X X X 7 |ωj ||Ij (un , un ) − Ij (u0 , u0 )| ≤ Cε 2 ωj |unj |2 − ωj |u0j |2 ≤ j∈M
j∈M
j∈M
for 0 ≤ tn = nh ≤ ε−N . This yields the long-time near-conservation of H and HM and completes the proof of Theorem 2.
A
On the non-resonance condition
For the analysis of the numerical solution of (1) we used the non-resonance condition (14). The second inequality of (14) is not necessary for the analysis of the exact solution of (1) and the solution of the spatial semi-discretization of (1), cf. (7) and [7]. We recall that the first inequality of (14) is true in the continuous situation for a large set of potentials V as is shown in [7, Proposition 2]. In this appendix we discuss the validity of the additional non-resonance condition in the continuous situation. For this purpose let, as in Section 2, 1
Rε = { (j, k) : j = j(k), k 6= hji, |k · ω − ωj | < ε 2 , kkk ≤ 2N + 2 = K }. Combining the techniques of [7, Proposition 2] and [7, Lemma 3] we obtain the following Proposition. Proposition 3. Fix N . For sufficiently large s the non-resonance condition P s 1 l∈Zd |kl ||ωl | 2 kkk ≤ C εN +1 sup 0 d+1 ε d+1 )|k| (s− 2 2 (j,k)∈Rε |ω ||ωj | holds for all V ∈ S where S is the set of [7, Proposition 1] with a constant C0 which only depends on N , s, and V . Proof. P Let V ∈ S and (j, k) ∈ Rε . As in the proof of [7, Proposition 2] we write k · ω = kl ωl + |j|≤L,j6=l kj ωj with |l| ≥ L and L ≥ 1 minimal. We then have
1 γ ≤ |k · ω − ωj | < ε 2 α L where we use the notations of [7, Proposition 1] and the argument of [7, Proposition 2]. Now, we proceed as in the proof of [7, Lemma 3] by separating the indices k. For |kl | > 1 or d+1 d+1 1 kj 6= 0 for j 6= l with |j| > 2K |l| we have |ω (s− 2 )|k| | ≥ C|ωl |2(s− 2 ) ≥ C|ωl |s L2(s−d−1) with a constant which depends on d, K, s, and V . Hence P s 2(s−d−1) s−d−1 K|ωl |s l∈Zd |kl ||ωl | ≤C ≤ CKγ − α ε α d+1 s L2(s−d−1) (s− )|k| d+1 |ω | 2 l |ω ||ωj |
and we have to choose s ≥ α(N + 1) + d + 1. d+1 d+1 d+1 d+1 For the other k we have as in [7, Lemma 3] |ω (s− 2 )|k| ||ωj | 2 ≥ C|ωl |s− 2 L2s−d−1 |ωl | 2 and the choice s ≥ α(N + 1) + d+1 2 yields the desired result. Proposition 3 and [7, Proposition 2] show that the numerical non-resonance condition (14) reduces in the limit h → 0 to a non-resonance condition which is fulfilled for a large set of potentials V. 19
Acknowledgement This work was supported by DFG, Project LU 532/4-1.
References [1] D. Bambusi and B. Gr´ebert. Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J., 135(3):507–567, 2006. [2] J. Bourgain. Quasi-periodic solutions of Hamiltonian perturbations of 2D linear Schr¨odinger equations. Ann. of Math. (2), 148(2):363–439, 1998. [3] D. Cohen, E. Hairer, and C. Lubich. Conservation of energy, momentum and actions in numerical discretizations of nonlinear wave equations. To appear in Numer. Math., 2008. [4] D. Cohen, E. Hairer, and C. Lubich. Long-time analysis of nonlinearly perturbed wave equations via modulated Fourier expansions. Arch. Ration. Mech. Anal., 187(2):341–368, 2008. [5] G. Dujardin and E. Faou. Normal form and long time analysis of splitting schemes for the linear Schr¨ odinger equation with small potential. Numer. Math., 108(2):223–262, 2007. [6] L. H. Eliasson and S. B. Kuksin. KAM for the non-linear Schr¨odinger equation. To appear in Ann. of Math., 2008. [7] L. Gauckler and C. Lubich. Nonlinear Schr¨odinger equations and their spectral semidiscretizations over long times. Preprint, http://na.uni-tuebingen.de/preprints.shtml, 2008. [8] E. Hairer and C. Lubich. Spectral semi-discretisations of weakly nonlinear wave equations over long times. To appear in Found. Comput. Math., 2008. [9] E. Hairer, C. Lubich, and G. Wanner. Geometric numerical integration. Structure-preserving algorithms for ordinary differential equations, volume 31 of Springer Series in Computational Mathematics. Springer-Verlag, Berlin, second edition, 2006. [10] J. A. C. Weideman and B. M. Herbst. Split-step methods for the solution of the nonlinear Schr¨odinger equation. SIAM J. Numer. Anal., 23(3):485–507, 1986.
20
Mathematisches Institut, Universit¨ at T¨ ubingen, Auf der Morgenstelle 10, D-72076 T¨ ubingen, Germany. email: {gauckler,lubich}@na.uni-tuebingen.de
Abstract Conservation properties of a full discretization via a spectral semi-discretization in space and a Lie-Trotter splitting in time for cubic Schr¨ odinger equations with small initial data (or small nonlinearity) are studied. The approximate conservation of the actions of the linear Schr¨ odinger equation, energy, and momentum over long times is shown using modulated Fourier expansions. The results are valid in arbitrary spatial dimension. Keywords: Nonlinear Schr¨ odinger equation – Splitting integrators – Split-step Fourier method – Long-time behavior – Near-conservation of actions, energy, and momentum – Modulated Fourier expansion MSC (2000): 65P10, 65M70
1
Introduction
We consider the nonlinear Schr¨odinger equation iut = −∆u + V ∗ u + |u|2 u,
(1)
where u = u(x, t), x ∈ Td = Rd /2πZd , t ≥ 0, in dimension d ≥ 1 with periodic boundary conditions. The potential V = V (x) ∈ L2 (Td ) is assumed to be periodic with real Fourier coefficients and acts by convolution on u. This linear potential of convolution type makes the linear part of the equation less unpleasant in arbitrary spatial dimension d, see for example [1], [2], and [6] where such equations are studied. A standard way to discretize this equation is a spectral semi-discretization in space followed by a Lie-Trotter splitting in time. In this paper, we show that the actions of the linear Schr¨odinger equation iut = −∆u + V ∗ u as well as energy and momentum of (1) are approximately conserved along the numerical solution over long times. Energy and momentum are exact invariants of (1), and the actions are approximately conserved along solutions of (1) over long times as is shown in [1] and [7]. In [7] these quantities were studied for spectral semi-discretizations in space. Here, we extend these results to full discretizations using in addition a Lie-Trotter splitting in time. A long time analysis of semi-discretizations in time via a Lie-Trotter splitting for linear Schr¨odinger equations with a multiplicative potential has been performed by Dujardin and Faou [5]. The proof of our results is done by analyzing a modulated Fourier expansion of the numerical solution. Such expansions have been used by Cohen, Hairer, and Lubich [3] to analyze full discretizations of semilinear wave equations with trigonometric integrators and the St¨ ormer-Verlet method as time integrators. In comparison with backward error analysis – the standard tool for proving conservation properties over long times (successfully applied to numerical solutions of ordinary differential equations without high frequencies) – the technique of modulated Fourier expansions does not rely on the smallness of hω where h is the time step size and ω the largest 1
frequency of the linear part of the system, see [9, Chapters IX and XIII]. This is of major importance in the context of partial differential equations presented here where the frequencies tend to infinity as the spatial discretization is refined. The modulated Fourier expansion of the numerical solution is obtained as in [3] by an approximate, time-continuous solution of a formal modulation system. It describes the numerical solution at times nh for natural numbers n up to a small defect. The main conceptual difference in comparison to [3] is that we do not use invariants of the modulation system to get conservation properties. Instead of this, we define an auxiliary function which has almost-invariants close to the actions, energy, and momentum of the numerical solution and describes the approximate solution of the modulation system at times nh up to a small defect. In Section 2 we recall results concerning (almost) invariants of the exact solution of (1). The considered discretization of (1) is presented in Section 3. In the subsequent sections we analyze the (almost) invariants of the exact solution along this numerical approximation. In Section 4 our main theorem for the numerical solution is stated and illustrated by numerical experiments. The proof of this result is given in Sections 5 and 6 where we introduce a modulated Fourier expansion of the numerical solution and analyze its conservation properties.
2
The nonlinear Schr¨ odinger equation over long times
In this section we introduce some quantities which are known to be conserved (exactly or approximately over long times) along solutions of the nonlinear Schr¨odinger equation (1) with small initial data.
2.1
Conservation of energy and momentum
Being a Hamiltonian system, the total energy or Hamiltonian Z 1 1 H(u, u ¯) = |∇u|2 + (V ∗ u)¯ u + |u|4 dx d 2(2π) Td 2
(2)
is exactly conserved along solutions of (1). Here, |·| denotes the Euclidean norm. The same is true for the momentum Z 1 (u∇¯ u−u ¯∇u)dx. (3) K(u, u ¯) = i (2π)d Td
2.2
Near-conservation of actions over long times
P Along any solution u(x, t) = j∈Zd uj (t)ei(j·x) with j·x = j1 x1 +· · ·+jd xd of the linear Schr¨odinger equation iut = −∆u + V ∗ u the actions Ij (u, u ¯) =
1 |uj |2 2
(j ∈ Zd )
(4)
are exactly conserved. Along a solution of our nonlinear equation (1) these actions are approximately conserved over long times under a condition of small initial data and a non-resonance condition, see [1] and [7]. The smallness of the initial data is measured in the Sobolev norm kuks =
X
|ωj |s |uj |2
j∈Zd
21
for s ≥ 0 where ωj = |j|2 + Fj (V ) = (j12 + · · · + jd2 ) + Fj (V ) (j ∈ Zd )
(5)
are the frequencies (eigenvalues) of the linear P part of (1) and Fj (V ) = Vj denotes the j-th Fourier coefficient of the periodic function V = j Vj ei(j·x) (by assumption Vj ∈ R). In the definition of k·ks a frequency ωj = 0 has to be replaced by 1; this has to be done also in the following whenever 2
the absolute value of a frequency ωj = 0 appears. The initial data is assumed to be of size ε ≪ 1 in the norm k·ks . We consider the almost-conservation of actions on time intervals of length ε−N for natural numbers N . Besides the smallness of the initial data we need a non-resonance condition on the frequencies (5). For the precise statement of this non-resonance condition we introduce the following notations of [4] and [7]. For a sequence k = (kl )l∈Zd of integers kl and the sequence ω = (ωl )l∈Zd of frequencies (5) we write Y σ|k | X X X ωl l (6) kl ωl , ω σ|k| = |kl |, k · ω = kl l, kkk = j(k) = l∈Zd
l∈Zd
l∈Zd
l∈Zd
for σ ∈ R. We further define the set of near-resonant indices k 1
Rε = { (j, k) : j = j(k), k 6= hji, |k · ω − ωj | < ε 2 , kkk ≤ 2N + 2 } where hji = (δjl )l∈Zd with Kronecker’s delta. The non-resonance condition then reads sup (j,k)∈Rε
|ωj |s− |ω (s−
d+1 2
d+1 2 )|k|
|
εkkk+1 ≤ C0 ε2N +4
(7)
with a constant C0 (independent of ε), and we have the following theorem (for a similar theorem see [1, Theorem 3.26]). Theorem 1 ([7, Theorem 1]). For given N and s ≥ d+1 there exists ε0 > 0 such that the following holds: Under the conditions of small initial data ku(0)ks ≤ ε ≤ ε0 and of non-resonance (7), the estimate X 3 |Il (u(·, t), u ¯(·, t)) − Il (u(·, 0), u ¯(·, 0))| ≤ Cε 2 for 0 ≤ t ≤ ε−N |ωl |s 2 ε d l∈Z
holds for solutions u(x, t) of (1) with a constant C which depends on C0 , d, N , s, and V but is independent of ε and t. The non-resonance condition is fulfilled for a large set of potentials V as is shown in [7, Proposition 2].
3
Discretization of the nonlinear Schr¨ odinger equation
We describe a standard way to discretize the nonlinear Schr¨odinger equation (1), see for example [10].
3.1
Spectral semi-discretization in space
The semi-discretization in space is done by a spectral collocation method as in [7]. We write M = {−M, . . . , M − 1}d . The trigonometric polynomial X uM (x, t) = qj (t)ei(j·x) j∈M
is chosen as an ansatz for the solution u of the nonlinear Schr¨odinger equation (1), inserted in (1), and evaluated at the collocation points xk =
π k, M
k ∈ M.
Writing uM (xk , t)k∈M = F2M (qj (t))j∈M with the d-dimensional discrete Fourier transform F2M equation (1) reduces to the system of ordinary differential equations i
duM −1 M (xk , t)k∈M = F2M ΩF2M u (xk , t)k∈M + (|uM (xk , t)|2 uM (xk , t))k∈M dt 3
(8)
where Ω = diag((ωl )l∈M ) is the diagonal matrix with the frequencies ωl , l ∈ M, on its diagonal. The equation for the initial value in t = 0 becomes uM (xk , 0)k∈M = u(xk , 0)k∈M . Equation (8) describes a finite dimensional complex Hamiltonian system with Hamiltonian HM (uM (xk , t)k∈M , uM (xk , t)k∈M ) = HM (uM , uM ), T 1 X M 1 −1 M u (xk , t)k∈M + HM (uM , uM ) = uM (xk , t)k∈M F2M ΩF2M |u (xk )|4 2 4 k∈M (9) Z 1 1 M 2 M M 4 M |∇u | + (V ∗ u )u + Q(|u | ) dx, = 2(2π)d Td 2 P where Q(v) denotes the trigonometric interpolation of a periodic function v = j∈Zd vj ei(j·x) in the collocation points, X X vj+2M m ei(j·x) . Q(v) = j∈M m∈Zd
The Hamiltonian HM is exactly conserved along solutions of the semi-discretized system (8) whereas the Hamiltonian H (equation (2)) of the nonlinear Schr¨odinger equation (1) is approximately conserved over long times along solutions of (8), cf. [7, Theorem 2].
3.2
Lie-Trotter splitting in time
The space discretized nonlinear Schr¨odinger equation (8) is discretized in time by a Lie-Trotter splitting with step size h: If un denotes the approximation to the solution uM of (8) after n time steps at time tn = nh, we compute the approximation after n + 1 time steps by n 2
v n (x) = Q(e−ih|u | un )(x), X un+1 (x) = e−iωj h vjn ei(j·x) .
(10)
j∈M
The iterates can be computed efficiently using the fast Fourier transform. As initial value of the iteration we choose u0 = Q(u(·, 0)). (11) v n (xk )k∈M is the exact solution of the differential equation i
duM (xk , t)k∈M = (|uM (xk , t)|2 uM (xk , t))k∈M dt
(12)
at time tn+1 with initial value un (xk )k∈M at tn , and un+1 (xk )k∈M is the exact solution of the differential equation duM −1 M i u (xk , t)k∈M (13) (xk , t)k∈M = F2M ΩF2M dt at time tn+1 with initial value v n (xk )k∈M at tn . A more popular splitting algorithm is the Strang or symmetric Lie-Trotter splitting where half a time step with (13) is performed before and after the time step with (12). Since the numerical solution obtained from that method differs from that of method (10) only by half a time step with (13) at the beginning and the end, the long-time behavior of both methods is the same. We therefore only consider the simpler method (10) in this article.
4
Results and numerical experiments
We now turn to the formulation of the main result of this paper: The (approximately) conserved quantities of the exact solution of (1) introduced in Section 2 are approximately conserved along the numerical solution introduced in Section 3 over long times. We adopt the notations (6), but as in [7, Section 5] k and ω are finite sequences (kl )l∈M and (ωl )l∈M , respectively, and X j(k) = kl l mod 2M ∈ M l∈M
4
where mod 2M denotes the entry-wise reduction modulo 2M with representative chosen in M. The set of near-resonant indices takes the form 1
Rε,M,h = { (j, k) : j = j(k), k 6= hji, |ei(ωj −k·ω)h − 1| < ε 2 h, kkk ≤ 2N + 2 }, and the non-resonance condition reads sup (j,k)∈Rε,M,h
|ωj |s− |ω
d+1 2
(s− d+1 2 )|k|
|
εkkk+1 ≤ C0 ε2N +4 ,
sup (j,k)∈Rε,M,h
s l∈M |kl ||ωl | d+1 d+1 |ω (s− 2 )|k| ||ωj | 2
P
1
ε 2 kkk ≤ C0 εN +1
(14) with a constant C0 independent of ε where the zero frequencies are replaced by 1. The second condition of (14) is not needed in the time-continuous situation (7). However, in Appendix A we will show that it is true in this situation under the non-resonance condition of Bambusi and Gr´ebert [1]. Note that in the limit h → 0 we have h1 |ei(ωj −k·ω)h − 1| → |k · ω − ωj | and hence the non-resonance condition (14) reduces to a non-resonance condition which is fulfilled for a large set of potentials. Theorem 2. For given N and s ≥ d + 1 there exists ε0 > 0 such that the following holds: Under the conditions of small initial data ku0 ks ≤ ε ≤ ε0 and of non-resonance (14), the estimates X
|ωl |s
l∈M
3 |Il (un , un ) − Il (u0 , u0 )| ≤ Cε 2 2 ε
3 |H(un , un ) − H(u0 , u0 )| ≤ Cε 2 ε2 3 |HM (un , un ) − HM (u0 , u0 )| ≤ Cε 2 2 ε d n n X 3 |Kr (u , u ) − Kr (u0 , u0 )| ≤ Cε 2 2 ε r=1
for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N , for 0 ≤ tn = nh ≤ ε−N
hold for the numerical solution un described in Section 3 with time step size h ≤ 1 with a constant C which depends on C0 , d, N , s, and V but is independent of ε, M , h, and n.
100
100
10−2
10−2
10−4
10−4
10−6
10−6
10−8
10−8
10−10
10−10
10−12
10−12
10−14
10−14
10−16
10−16
10−18
10−18
10−20 0
10000
10−20 0
20000
10000
20000
Figure 1: Example in one dimension: Actions (black lines), discrete energy (upper grey line), and momentum (lower grey line) for non-resonant time step size h = 0.005+2π/ω−6 (left) and resonant time step size h = 2π/ω−6 (right). 5
.0025 .0020 .0015 .0010 .0005 .01
.02
.03
.04
.05
.06
.07
.08
.09
.10
.02
.03
.04
.05
.06
.07
.08
.09
.10
.02
.03
.04
.05
.06
.07
.08
.09
.10
101 100 10−1 10−2 .01 10−6 10−9 10−12 10−15 .01
Figure 2: Example in one dimension: Maximal deviation in actions (upper image), energy (middle image), and momentum (lower image) for time step sizes h ∈ [0.01, 0.1]. The proof of this theorem will be the subject of Sections 5 and 6. The theorem implies the spatial regularity of the numerical solution over long times: by the conservation of actions we have 7 |kun k2s − ku0 k2s | ≤ Cε 2 and hence kun k ≤ 2ε over long times 0 ≤ tn = nh ≤ ε−N . The same results as in Theorem 2 are true if the Lie-Trotter splitting is replaced by the symmetric Strang splitting since this change does not affect the long-time behavior. We further mention that a generalization of Theorem 2 to nonlinear Schr¨odinger equations with nonlinearities of the form g(|u|2 )|u|2 u where g is a polynomial is straightforward. We conclude this section with some numerical experiments in one and two spatial dimensions. The initial value u(·, 0) in one dimension is chosen as x 3 x 2 3 x 3 x −1 + 1 + i · 0.1 · −1 +1 , u(x, 0) = 0.1 · π π π π
and the potential V is chosen such that
ωj =
q
|j|4 + rj
where rj = 0.5 for j ≥ 0 and rj = 0.8 for j < 0. For the discretization in space we take M = 27 , that is 28 collocation points, and the time step size is chosen as h = 0.005+2π/ω−6 and h = 2π/ω−6 . As explained by Theorem 2, we observe conservation of actions, energy, and momentum for the first time step size in Figure 1 where the eight largest actions Ij (equation (4)) and sixteen other actions, the (discrete) energy HM (equation (9)), and the momentum K (equation (3)) are plotted. For the second time step size we observe numerical resonances in Figure 1. In fact, this time step size does not fulfill the non-resonance condition (14): For k = h−6i + h−(M − 3)i we have j(k) = M − 3 and ei(ωj(k) −k·ω)h = e2πi(ωM −3 −ω−(M −3) )/ω−6 ≈ 1, i. e. (j(k), k) is near-resonant, but this pair does not fulfill (14). 6
10−4
10−4
10−6
10−6
10−8
10−8
10−10
10−10
10−12
10−12
10−14
10−14
10−16
10−16
0
500000
1000000
0
500000
1000000
Figure 3: Example in two dimensions: Actions (black lines), discrete energy (upper grey line), and first component of momentum (lower grey line) for non-resonant frequencies (15) (left) and resonant frequencies (16) (right). In Figure 2 we study the effect of resonant time step sizes in the example in one dimension. We integrate with time step sizes h ∈ [0.01, 0.1] over a time interval of length 5000, and plot the maximal deviation in the actions, in the energy, and in the momentum for these different time steps with a resolution of 0.0001. In two dimensions we choose 2 x x 2 x x 1 2 2 1 −1 +1 −1 +1 u(x1 , x2 , 0) = 0.01 · π π π π x 2 x 2 x 2 x 2 1 1 2 2 + i · 0.01 · −1 +1 −1 +1 π π π π as initial value. For the potential we make the choice q ω(j1 ,j2 ) = ωj = |j|4 + rj ,
(15)
where rj = 0.5 for j1 , j2 ≥ 0, rj = 0.6 for j1 ≥ 0 > j2 , rj = 0.7 for j1 < 0 ≤ j2 , and rj = 0.8 for j1 , j2 < 0. We further study the resonant situation V = 0 where ω(j1 ,j2 ) = ωj = |j|2 .
(16)
For the numerical discretization we choose M = 24 and h = 0.1. In Figure 3 the actions Ij , the (discrete) energy HM , and the first component of the momentum K are plotted. In the case of non-resonant frequencies the good long-time behavior is explained by Theorem 2.
5
Modulated Fourier expansions
The analysis of the numerical solution is done by the method of modulated Fourier expansions. We follow [3], where a full discretization of semilinear wave equations is studied, and [7] where the exact solution of the nonlinear Schr¨odinger equation (1) and its semi-discretization in space (8) are studied. Throughout this section we work under the assumptions of Theorem 2. All appearing constants will be denoted by C. The main point is that all these constants do not depend on ε, M , h, and the time 0 ≤ t ≤ ε−1 ; however, they may depend on C0 and N from the non-resonance condition (14), the dimension d, the regularity parameter s, and the potential V . 7
5.1
The modulation system
As an ansatz for a function u ˜ which approximates the numerical solution un at time tn we choose a modulated Fourier expansion, X X k u ˜(x, t) = zj(k) (εt)ei(j(k)·x) e−i(k·ω)t = z k (x, εt)e−i(k·ω)t (17) kkk≤K
kkk≤K
k with z k (x, εt) = zj(k) (εt)ei(j(k)·x) where zlk (εt) = 0 for l 6= j(k). In the following we tacitly assume kkk ≤ K = 2N + 2 unless stated otherwise. If we insert (17) in the splitting ansatz (10) we get from the conditions un = u ˜(·, tn ) and un+1 = u ˜(·, tn+1 )
X
k zj(k) (ε(t + h))ei(j(k)·x) e−i(k·ω)(t+h) =
XX
e−iωj h
k j∈M
k
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
∞ X (−ih)m m! m=0
1 m+1 z km+2 . . . z k2m+1 ei(j·x) e−i(k·ω)t . Fj Q z k . . . z k
For k = k1 + · · · + km+1 − km+2 − · · · − k2m+1 we have ( 1 1 k km+1 k2m+1 km+2 zj(k 1 ) . . . zj(km+1 ) zj(km+2 ) . . . zj(k2m+1 ) , 2m+1 k km+1 km+2 k z ...z = Fj Q z . . . z 0,
j = j(k), else
since j(k) ≡ j(k1 )+· · ·+j(km+1 )−j(km+2 )−· · ·−j(k2m+1 ) (mod 2M ) and j(k) ∈ M. Comparing the coefficients of ei(j(k)·x) e−i(k·ω)t thus gives the modulation system k Lkj(k) zj(k) =
∞ X (−ih)m m! m=1
1
X
m+1
k +···+k −km+2 −···−k2m+1 =k
1 m+1 Fj(k) Q z k . . . z k z km+2 . . . z k2m+1
(18a)
where (Lkj zjk )(εt) = zjk (ε(t + h))ei(ωj −k·ω)h − zjk (εt) ∞ X (εh)l k (l) (zj ) (εt)ei(ωj −k·ω)h . = (ei(ωj −k·ω)h − 1)zjk (εt) + l! l=1
Here we denote by yields
(l)
the l-th derivative with respect to τ = εt. The initial condition (11) further X
zjk (0) = u0j .
(18b)
k
5.2
Results on the modulation functions
The remaining part of this section is devoted to the construction of an approximate solution of the modulation system (18) for 0 ≤ εt = τ ≤ 1. The properties of this approximate solution are colk lected in the following proposition. For measuring the size of functions z = (z k )k = (zj(k) ei(j(k)·x) )k we use the norm
X
2 21 X X
k|z|ks = {z k } = |ωj |s |zjk | k
s
j∈M
k
from [7] where
{v}(x) =
X
|vj |ei(j·x)
j∈Zd
for a periodic function v(x) =
P
j∈Zd
vj ei(j·x) , and the scaling
ˆ = (ˆ z z k )k = (|ω 8
2s−d−1 |k| 4
|z k )k .
Proposition 1. There exists a function u ˜(x, t) = and 0 ≤ εt ≤ 1 satisfying kun − u ˜(·, tn )ks ≤ CεN +2
P
k i(j(k)·x) −i(k·ω)t e kkk≤2N +2 zj(k) (εt)e
for x ∈ Td
for 0 ≤ tn = nh ≤ ε−1 .
(19a)
Moreover, the following estimates hold: • u ˜ is small, k˜ u(·, t)ks ≤ Cε.
(19b)
• z is small, k zj(k) = 0 for (j(k), k) ∈ Rε,M,h ,
k|ˆ z|k d+1 ≤ Cε, 2
X
hji
|ωj |s |zj |2 ≤ Cε2 ,
j∈Zd
X
|ωj |s
j∈Zd
X
k6=hji
2 |zjk | ≤ Cε5 ,
X
j∈Zd
|ωj |
d+1 2
X
k6=hji
2 |ˆ zjk | ≤ Cε5 .
(19c)
All constants are independent of ε, M , h, and 0 ≤ t ≤ ε−1 but may depend on C0 , d, N , s, and V. The proof of this proposition will cover the remaining part of this section except Subsection 5.9.
5.3
Iterative solution of the modulation system
As in [7] we collect those pairs (j(k), k) with k 6= hj(k)i which are not in Rε,M,h in the set Sε,M,h = { (j, k) : j = j(k), k 6= hji, (j, k) 6∈ Rε,M,h , kkk ≤ K }. The solution of the modulation system (18) is determined up to a small defect by an iterative procedure. We start by setting i0 i0 h h hji k zj = u0j and zj(k) = 0 for k 6= hj(k)i
for 0 ≤ εt = τ ≤ 1. For n ≥ 0 and 0 ≤ εt = τ ≤ 1 we set +2 h in+1 h NX 1 (εh)l k (l) = i(ω −k·ω)h zjk (zj ) (εt)ei(ωj −k·ω)h − l! e j −1 l=1 +
N X (−ih)m m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
in 1 m+1 z km+2 . . . z k2m+1 Fj Q z k . . . z k
for (j, k) ∈ Sε,M,h , +2 in+1 h NX h (εh)l hji (l) hji z˙j = ε−1 h−1 − (zj ) (εt) l! l=2
N X (−ih)m + m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =hji
h
in+1 hX in hji zj (0) = u0j − zjk (0) ,
h
zjk
in 1 m+1 z km+2 . . . z k2m+1 , Fj Q z k . . . z k
k6=hji
in+1
= 0 for (j, k) ∈ Rε,M,h .
The notation [·]n means that the n-th iterates of the variables within the brackets are taken. In this construction we have truncated the expansion of the exponential function and the Taylor expansion of zjk after N and N + 2 terms, respectively. We will show that after L = 2N + 2 iterations we obtain functions z = [z]L with the properties described in Proposition 1. 9
5.4
Abstract formulation of the iteration
Let
( max( 12 (kkk + 1), 2), [[k]] = 1 2 (kkk + 1) = 1,
We scale the variables as in [7] ( ε−[[k]] zjk , k aj = 0,
k = hji, k 6= hji
bkj
and
k 6= hji, k = hji.
( 0, k = hji, = ε−[[k]] zjk , k = 6 hji,
and we write a = (ak )k = (akj(k) ei(j(k)·x) )k , b = (bk )k = (bkj(k) ei(j(k)·x) )k , and c = a + b. We further define A(a)kj = −
N +2 X (εh)l k (l) (εh)l k (l) (aj ) (εt), B(b)kj = − (bj ) (εt)ei(ωj −k·ω)h , l! l! l=1 l=2 ( i(ωj −k·ω)h k (e − 1)cj , (j, k) ∈ Sε,M,h , (Ωc)kj = 1 k 2 else, ε hcj ,
N +2 X
and N X (−ih)m m! m=1 1 X 2m+1 m+1 ]] k1 Fj Q ε[[k ]]+···+[[k c . . . ck ckm+2 . . . ck2m+1 .
F(c)kj = ε− max([[k]],2)
k1 +···+km+1 −km+2 −···−k2m+1 =k
The iteration in the rescaled variables becomes in in h h in+1 h for (j, k) ∈ Sε,M,h , = (Ω−1 B(b))kj + (Ω−1 F(c))kj bkj in h in+1 in h hX in+1 h in h hji hji hji hji , aj (0) = ε−1 u0j − + h−1 F(c)j ε[[k]]−1 bkj (0) . = ε−1 h−1 A(a)j a˙ j k6=hji
We also use a second rescaling of the variables, a ˆkj = |ω ˆ With ˆ=a ˆ + b. c |k| ˆ c)k = ε− max([[k]],2) |ω 2s−d−1 4 F(ˆ | j
|ω −(
N X (−ih)m m! m=1
2s−d−1 )(|k1 |+···+|k2m+1 |) 4
1
2s−d−1 |k| 4
X
k1 +···+km+1 −km+2 −···−k2m+1 =k m+1
|ˆ ck . . . cˆk
1 2m+1 ]] Fj Q ε[[k ]]+···+[[k
cˆkm+2 . . . cˆk2m+1
ˆ becomes the iteration for b in in h h in+1 h −1 ˆ k + (Ω−1 F(ˆ ˆbk ˆ c))k = (Ω B( b)) j j j
5.5
2s−d−1 |akj , ˆbkj = |ω 4 |k| |bkj , and
for (j, k) ∈ Sε,M,h .
Estimating the nonlinearity
The estimation of the nonlinearity is based on the inequality
X
2
X
2 X
2
{Q(ck dl )} ≤ C {ck } {dl } = Ck|c|ks k|d|ks . k,l
s
s
k
l
s
from [7, Lemmas 1 and 6] which is valid for s > d2 . The constant depends on d, s, and V but not on M . 10
Lemma 1. We have 1
k|Ω−1 c|ks ≤ ε− 2 h−1 k|c|ks
(20a)
k|F(c)|ks ≤ εhk|c|ks P1 (k|c|ks ) ˜|ks P2 (max(k|c|ks , k|˜ k|F(c) − F(˜ c)|ks ≤ εhk|c − c c|ks ))
(20b) (20c)
where P1 and P2 are polynomials of degree 2N whose coefficients are nonnegative and bounded ˆ and k| · |k d+1 instead of c, c ˆ˜, F, ˆ, c ˜, independently of ε, M , and h. The same estimates hold for c 2 F, and k| · |ks , respectively. Proof. The first estimate (20a) follows as in [7, Lemma 1], and the second and the third one can be seen similarly as there: We note that for m ≥ 1 and k1 + · · · + km+1 − km+2 − · · · − k2m+1 = k [[k1 ]] + · · · + [[k2m+1 ]] ≥ max([[k]], 2) + m. With this inequality and [7, Lemma 6] each summand of F(c) can be estimated as the nonlinearity F(c) of [7], and the triangle inequality yields (20b) and (20c). For example concerning (20b) we have N X (εh)m 2m 2m+1 k|F(c)|ks ≤ C k|c|ks . m! m=1
5.6
Size of the iterated modulation functions
Estimates of the modulation functions are obtained similarly as in [7, Subsection 3.6] with the help of Lemma 1. We remark that for 0 ≤ εt = τ ≤ 1 k|A(a(l+1) (τ ))|ks ≤ Cε2 h2 k|B(b(l) (τ ))|ks ≤ Cεh
sup k=2,...,N +2
sup k=1,...,N +2
k|a(l+1+k) (τ )|ks ,
k|b(l+k) (τ )|ks
with a constant which depends on N but not on ε, M , h, and τ . With these estimates and Lemma 1 we get for αn = βn = α ˆn = βˆn =
max
sup k|[a(l) (τ )]n |ks ,
max
sup k|[b(l) (τ )]n |ks ,
max
sup k|[ˆ a(l) (τ )]n |k d+1 ,
max
ˆ (l) (τ )]n |k d+1 sup k|[b
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1 l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
2
l=0,...,1+(N +2)(2L−n) 0≤τ ≤1
2
as in [7] αn = α ˆ n ≤ C,
1
βn ≤ Cε 2 ,
1 βˆn ≤ Cε 2
(21)
for n = 0, . . . , L with a constant C which depends on d, L, n, s, and V but not on ε, M , and h. With k|[˜ u]n |ks ≤ εk|[c]n |ks (see [7, Subsection 3.6] this yields for z = [z]L the estimates (19b) and (19c) of Proposition 1.
5.7
Defect of the iterated modulation functions
After n steps the defect in the modulation system (18a) is (with j = j(k)) h
Lkj zjk −
∞ X (−ih)m m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 in m+1 Fj Q z k . . . z k z km+2 . . . z k2m+1 .
11
This has to be considered for all indices k, not only for those with kkk ≤ K. We set z k = 0 for kkk > K. In comparison to [7] there are two additional sources of defect due to the truncation of the expansion of the exponential function and of the Taylor expansion in the iterative procedure described in Subsection 5.3. The defect in (18a) after n iteration steps is decomposed as follows (the notation is chosen to be consistent with Proposition 2 below). The defect arising from the truncation of the Taylor expansion of zjk is h
hkj − ekj
in
N +2 h in X (εh)l k (l) i(ωj −k·ω)h = Lkj zjk − (zj ) e − (ei(ωj −k·ω)h − 1)zjk , l! l=1
the defect arising from the truncation of the expansion of the exponential function is ∞ h in h X dkj = −
m=N +1
(−ih)m m!
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
in 1 m+1 , z km+2 . . . z k2m+1 Fj Q z k . . . z k
and the remaining defect is decomposed as N +2 h in h X (εh)l k (l) i(ωj −k·ω)h ekj + fjk + gjk = (ei(ωj −k·ω)h − 1)zjk + (zj ) e l! l=1
−
N X
m
(−ih) m! m=1
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 in m+1 Fj Q z k . . . z k z km+2 . . . z k2m+1 ,
where [ekj ]n = 0 for (j, k) ∈ 6 Sε,M,h and k 6= hji, [fjk ]n = 0 for (j, k) 6∈ Rε,M,h , and [gjk ]n = 0 for kkk ≤ K (dk and g k can be nonzero for kkk > K). The defect in equation (18b) for the initial condition reads h in hX in hji d˜j = u0j − zjk (0) . k
Now, we estimate these quantities for n = 0, . . . , L. For f we get as in [7, Subsection 3.7] using the first part of the non-resonance condition (14), Lemma 1, and (21) n
k|[f ]
2 |ks
ˆ c)]n |k2d+1 ≤ k|[F(ˆ 2
sup (j,k)∈Rε,M,h
|ω | 2s−d−1 4 j |ω
2s−d−1 |k| 4
|
ε[[k]]
2
≤ (CεN +3 h)2
(22)
for n = 0, . . . , L with a constant which depends on C0 , d, N , n, s, and V but not on ε, M , and h. In addition we have ˆ c)]n |k d+1 ≤ Cεh. k|[(ε−[[k]] fˆk )k ]n |k d+1 ≤ k|[F(ˆ (23) 2
2
With the same arguments as in the proof of Lemma 1 using in addition (21) we obtain
X 1
{[g k ]n } ≤ Cε 2 (K+2) εh = CεN +3 h
K
d 2
5.8.1
Size of the numerical solution
with a constant only depending on d, s, and V , cf. [7, Lemma 1].
We first determine the size of the numerical solution un . For this solution we have with (30) kun+1 ks ≤
∞ X 2 n 2 2 n 2 0 2 hm 2m n 2m+1 C ku ks = kun keC hku ks ≤ ku0 keC h(ku ks +···+ku ks ) . m! m=0
13
Hence, for ku1 ks , . . . , kun ks ≤ 2ε (note ku0 k ≤ ε) the estimate kun+1 k ≤ 2ε is fulfilled if (n + 1)h ≤ ε−1 and ε ≤ log(2)/(4C 2 ). Thus we have shown kun ks ≤ 2ε
for 0 ≤ tn = nh ≤ ε−1
(31)
if ε is small enough. 5.8.2
Error on [0, ε−1 ]
In t = 0 we have by (29)
˜ L |k ≤ CεN +2 ku0 − u ˜(·, 0)ks ≤ k|[d] s
The constant depends on d, N , s, and V but not on ε, M , and h. For ku1 ks , ku2 ks ≤ cε the m difference of the powers um 1 and u2 satisfies with (30) m
X
m kum 2−j (u1 + u2 )j−1 (u1 − u2 )(u1m−j + u2m−j ) 1 − u2 ks = j=1
≤ C m−1
m X
s
2−j (2cε)j−1 ku1 − u2 ks 2(cε)m−j = (Ccε)m−1 mku1 − u2 ks .
j=1
Hence ∞
X X (−ih)m m+1 m
m (u1 u ¯1 − um+1 u ¯ ) e−iωj h Fj
2 2 m! s m=1 j∈M ∞ X
≤
≤
hm m+1 m m+1 m (kum+1 u ¯m u ¯2 ks + kum+1 u ¯m u ¯2 ks ) 1 − u1 2 − u2 1 1 m! m=1
∞ X hm m+1 (C (cε)m+1 (Ccε)m−1 mku1 − u2 ks + C m (cε)m (Ccε)m (m + 1)ku1 − u2 ks ) m! m=1
≤3
∞ X
2 hm (Ccε)2m ku1 − u2 ks = 3h(Ccε)2 e(Ccε) h ku1 − u2 ks . (m − 1)! m=1
Using the size of un (31), the size of u ˜ (19b) from Proposition 1, and the estimates of the defect (22), (24), (25), (26), (28) we get ∞
X X (−ih)m
kun+1 − u ˜(·, tn+1 )ks ≤ e−iωj h Fj ((un )m+1 (un )m − (˜ u(·, tn ))m+1 (˜ u(·, tn ))m ) m! s m=0 j∈M
X
+ [(hk − ek + dk + ek + f k + g k )(εtn+1 )]L e−i(k·ω)tn+1 s
k∈ZM
2
≤ (1 + 3h(Cε)2 e(Cε) h )kun − u ˜(·, tn )ks + CεN +3 h.
We have thus shown the smallness of the error (19a) for (n + 1)h ≤ ε−1 . This concludes the proof of Proposition 1.
5.9
An auxiliary function close to the modulation functions
In [3], conservation properties of the numerical solution are deduced from almost-invariants of the modulation system. In the situation of the numerical solution of the semilinear wave equation with trigonometric integrators considered in that paper, the nonlinearity in the modulation system is, up to a multiplication with real-valued filter functions, the same as the nonlinearity in the modulation system for the exact solution of the wave equation, cf. [3, equation (34)] and [4, equation (15)]. In the case of the nonlinear Schr¨odinger equation and the Lie-Trotter splitting considered in the present paper, we encounter a different situation. While the nonlinearity in the modulation system 14
for the exact solution is cubic (cf. [7, equation (7a)]), it contains terms of arbitrary high order in the modulation system for the time-discrete equation (cf. equation (18a)). The point is, that some of the terms are multiplied by an imaginary factor while others are multiplied by a real factor, which prevents the arguments from [7, Subsection 4.1] for the derivation of almost-invariants of the modulation system to work. For this reason, we now introduce an auxiliary function wn = wn (x, t) = (wk,n (x, t))k on each interval [nh, (n + 1)h], which satisfies a differential equation whose nonlinearity is the nonlinearity in the modulation system for the exact solution [7, equation (7a)]. This allows to deduce almostinvariants of this auxiliary function as in [7]. Moreover, the auxiliary functions wn are constructed in such a way that they approximate the modulation functions z at the points nh and (n + 1)h up to a small defect. To be precise, let z = [z]L , f = [f ]L , and h = [h]L . We set for kkk ≤ (2N + 1)K, (n + 1)h ≤ ε−1 , and 0 ≤ t ≤ h wk,n (x, t) =
N X (−it)m m! m=0
X
k1 +···+km+1 −km+2 −···−k2m+1 =k
1 m+1 Q zk · · · zk z km2 · · · z k2m+1
(x,εtn )
.
These functions have the properties stated in the following proposition where again only those k with kkk ≤ K are considered. k,n Proposition 2. For (n+1)h ≤ ε−1 the functions wn = wn (x, t) = (wk,n (x, t))k = (wj(k) (t)ei(j(k)·x) )k for x ∈ Td and 0 ≤ t ≤ h have the following properties:
• They are related to z by zjk (εtn ) = wjk,n (0)
zjk (εtn+1 )ei(ωj −k·ω)h = wjk,n (h) + fjk (εtn+1 ) + hkj (εtn+1 ) (32a)
and
with k|h|ks ≤ CεN +3 h,
ˆ d+1 ≤ CεN +3 h, k|h|k 2
k|f |ks ≤ CεN +3 h,
1
k|(ε− 2 (kkk+1) fˆk )k |k d+1 ≤ Cεh
(32b)
2
where fjk = 0 for (j, k) 6∈ Rε,M,h and hkj = 0 for j 6= j(k) or (j, k) ∈ Rε,M,h . • They satisfy i
d k,n w = dt
1
X
Q(wk
,n
2
wk
,n
wk3 ,n ) + rk,n
(32c)
k1 +k2 −k3 =k
with k|rn |ks ≤ CεN +3 ,
k|ˆrn |k d+1 ≤ CεN +3 .
(32d)
2
• They are small, k|wn |ks ≤ Cε,
ˆ n |k d+1 ≤ Cε, k|w 2
1
ˆ k,n )k |k d+1 ≤ C. k|(ε− 2 (kkk+1) w 2
(32e)
All constants are independent of ε, M , h, n, and t but may depend on C0 , d, N , s, and V . Proof. For j 6= j(k) we have wjk,n = 0 on [0, h], and for kkk ≤ K we have zjk (εtn ) = wjk,n (0) and zjk (εtn+1 )ei(ωj −k·ω)h = wjk,n (h) + fjk (εtn+1 ) + hkj (εtn+1 ) with the notations of Subsection (5.7). This proves (32a), and the estimates (22), (23), (26), and (28) yield (32b). The estimates (32e) of w are obtained by estimating the nonlinearity as in Lemma 1. 15
The auxiliary functions wn satisfy the differential equation (32c) as we show next. With rk,n (t) = −
3N +1 X
(−it)k−1 (−1)l m!n!l! k=N +1 m+n+l=k−1 1 X k+1 Q z k · · · z k z kk+2 · · · z k2k+1 X
k1 +···+kk+1 −k −···−k2k+1 =k
(x,εtn )
k+2
1
X
+
Q(wk
,n
2
wk
,n
wk3 ,n ).
k1 +k2 −k3 =k kk1 k>K or kk2 k>K or kk3 k>K
and Q(Q(v)Q(w)) = Q(vw) we have X 1 2 Q(wk ,n wk ,n wk3 ,n ) k1 +k2 −k3 =k
1
X
=−
Q(wk
,n
2
wk
,n
wk3 ,n ) +
k1 +k2 −k3 =k kk1 k>K or kk2 k>K or kk3 k>K
X
k1 +···+km+n+l+2 −···−k2(m+n+l+1)+1 =k
m+n+l+3
−k
= −rk,n (x, t) +
N X (−it)k−1
k=1
X
m+n+l=k−1
(k − 1)!
N X
m,n,l=0
(−it)m+n+l (−1)l m!n!l!
1 m+n+l+2 Q zk · · · zk z km+n+l+3 · · · z k2(m+n+l+1)+1 X
k1 +···+kk+1 −kk+2 −···−k2k+1 =k
1 k+1 Q z k · · · z k z kk+2 · · · z k2k+1
(x,εtn )
(x,εtn )
(−1)l (k − 1)! d = i wk,n (x, t) − rk,n (x, t). m!n!l! dt
Here, the last equality follows from X
m+n+l=k−1
k−1 k−1−n X (k − 1)! (k − 1 − n)! (−1)l (k − 1)! X = =1 (−1)l m!n!l! n!(k − 1 − n)! l!(k − 1 − n − l)! n=0 l=0
by the binomial theorem. The estimates (32d) of r are obtained in the same way as the estimates of the defects dk and k g . This completes the proof the proposition.
6
Conservation properties
Let z and wn be the functions of Propositions 1 and 2. In this section we prove Theorem 2. As mentioned before we do not use invariants of the modulation system (18) to get conservation properties but rather the auxiliary functions wn of Proposition 2 which have almost-invariants and describe z at the discrete points εtn and εtn+1 up to a small defect.
6.1
Almost-invariants
The functions wn satisfy (32c), and this system possesses almost-invariants as we show next. We set Z X 1 2 1 Q(wk wk wk3 wk4 )dx U (w) = d (2π) d T 1 2 3 4 k +k −k −k =0
where we omit the index n of w and where the sum is (as always) over those k with kkk ≤ K. R R 1 1 Q(¯ v )dx = (2π) Q(v)dx and Q(vw) = Q(Q(v)Q(w)) we get for real sequences Using (2π) d d Td Td 16
µ = (µl )l∈M as in [7, Subsection 6.2] X d X d k k rk U (e−i(k·µ)θ wk )k = 2 0= (k · µ)|wj(k) |2 + 4 Re i(k · µ)wj(k) j(k) . dθ θ=0 dt k
Hence
Iµ (wn (·, t)) =
(33)
k
1X k,n (k · µ)|wj(k) (t)|2 2 k
is an almost-invariant of the system (32c) on [0, h]. The following lemma is similar to [7, Lemma 3]. Due to the usage of wn we now also have to control the entries of wn corresponding to near-resonant indices (note that we don’t have wjk = 0 for (j, k) ∈ Rε,M,h ). This is done with the second part of the non-resonance condition (14). 1
Lemma 2. Let wk = xk + ε 2 (kkk+1) y k and rk = pk + q k with xkj = pkj = 0 for j 6= j(k) or (j, k) ∈ Rε,M,h and yjk = qjk = 0 for (j, k) 6∈ Rε,M,h . The following estimate holds for s ≥ d + 1 with a constant C which depends only on C0 , d, K, s, and V . X X 3 k k y|k d+1 k|ˆ p|k d+1 + CεN + 2 k|ˆ q|k d+1 . |ωl |s |kl ||wj(k) ||rj(k) | ≤ Ck|ˆ x|k d+1 k|ˆ l∈M
2
2
2
2
k
Proof. We have with Sε,M,h = { (j, k) : j = j(k), k 6= hji, (j, k) 6∈ Rε,M,h , kkk ≤ K } P s X X X d+1 l∈M |kl ||ωl | k k xkj ||ˆ pkj | |ωl |s |kl ||wj(k) ||rj(k) |= |ωj | 2 d+1 d+1 |ˆ (s− )|k| 2 2 |ω ||ωj | l∈M k k=hji or (j,k)∈Sε,M,h P s X d+1 1 l∈M |kl ||ωl | 2 (kkk+1) |ˆ yjk ||ˆ qjk |. + |ωj | 2 d+1 d+1 ε (s− )|k| 2 2 |ω ||ω | j (j,k)∈Rε,M,h The non-resonance condition (14) implies P s 3 1 l∈M |kl ||ωl | 2 (kkk+1) ≤ C εN + 2 . sup 0 d+1 d+1 ε (s− )|k| 2 2 (j,k)∈Rε,M,h |ω ||ωj | The proof of sup k=hji or (j,k)∈Sε,M,h
s l∈M |kl ||ωl | d+1 d+1 |ω (s− 2 )|k| ||ωj | 2
P
≤ C.
for a constant C that depends on d, K, s, and V is given in [7, Lemma 3 and Section 6.2]. The Cauchy-Schwarz inequality now yields the first estimate of the lemma. Lemma 3. We have for 0 ≤ tn = nh ≤ ε−1 X |ωl |s Ihli (z(·, εtn )) − Ihli (z(·, 0)) ≤ CεN +3 l∈M
with a constant C which depends on C0 , d, N , s, and V but not on ε, M , h, and n. k Proof. For (n + 1)h ≤ ε−1 we have with (32a) and fj(k) and hkj(k) evaluated at εtn+1
2Iµ (z(·, εtn+1 )) − 2Iµ (z(·, εtn )) =
X
k,n k,n k (k · µ)(|wj(k) (h) + fj(k) + hkj(k) |2 − |wj(k) (0)|2 )
k
h
=
Z
h
=
Z
0
0
X d k,n k k Iµ (wn (t))dt + (k · µ)(|fj(k) + hkj(k) |2 + 2 Re(wj(k) (h)(fj(k) + hkj(k) ))) dt k
X d k k k k (εt Iµ (wn (t))dt + (k · µ)(−|fj(k) + hkj(k) |2 + 2 Re(zj(k) n+1 )(fj(k) + hj(k) ))) dt k
17
and with (33) Z
0
h
X d k,n k,n Iµ (·, wn (t))dt ≤ h sup |k · µ||wj(k) (t)||rj(k) (t)|. dt 0≤t≤h k
Using the first estimate of Lemma 2 we thus get X |ωl |s Ihli (z(·, εtn+1 )) − Ihli (z(·, εtn )) l∈M
3
1
ˆ n |k d+1 k|ˆrn |k d+1 + CεN + 2 hk|(ε− 2 (kkk+1) w ˆ k,n )k |k d+1 k|ˆrn |k d+1 ≤ Chk|w 2
2
2
2
ˆ d+1 k|ˆ ˆ 2d+1 + CεN + 32 k|(ε− 21 (kkk+1) fˆk )k |k d+1 k|ˆf |k d+1 + Ck|h|k z|k d+1 . + Ck|h|k 2
2
2
2
2
where the quantities are evaluated in εtn+1 or the supremum over 0 ≤ t ≤ h is taken. With Propositions 1 and (2) this can be estimated by CεN +4 h where C is independent of ε, M , h, and n. The claimed estimate follows with the triangle inequality by patching n ≤ (εh)−1 intervals of length h together.
6.2
Relationship between almost-invariants and actions
With the same proof as [7, Lemma 5] we get. Lemma 4. We have for 0 ≤ tn = nh ≤ ε−1 X 7 |ωl |s Ihli (z(·, εtn )) − Il (un , un ) ≤ Cε 2 l∈M
with a constant C which depends on C0 , d, N , s, and V but not on ε, M , h, and n.
6.3
From short to long time intervals
By now we have proven Theorem 1 on the short time interval [0, ε−1 ]. The extension to long time intervals [0, ε−N ] as stated in the theorem can be done as in [4, Section 4.5] (see also [7, Section 4.5]). Let nε be the largest integer with nε h ≤ ε−1 . We consider the approximate solution of the modulation system given by Proposition 1 on intervals [mnε h, (m + 1)eε h] for integers m where the numerical solution after mnε time steps is taken as initial value. The conservation of Ihli on these intervals is controlled by Lemma 3 and the difference at the boundaries of these intervals can be controlled as in [7, Lemma 3 and Proposition 4]. Patching together 2ε−N +1 intervals of length nε h (note nε h ≥ 21 ε−1 for h ≤ 1 ≤ ε−1 ) and using the estimates of the difference to the actions Il of Lemma 4 this proves the almost-conservation of actions in Theorem 1.
6.4
Near-conservation of energy and momentum
We now turn to the proof of the long-time near-conservation of the energy H (equation (2)), the discrete energy HM (equation (9)), and the momentum K (equation (3)) which read in terms of the Fourier coefficients Z 1 X 1 n 2 n n ωj |uj | + H(u , u ) = |un |4 dx, d 2 4(2π) d T j∈Zd Z X 1 1 Q(|un |4 )dx, ωj |unj |2 + HM (un , un ) = 2 4(2π)d Td j∈M X n n Kr (u , u ) = 2 jr |unj |2 for r = 1, . . . , d. j∈M
The conservation of actions as stated in Theorem 2 implies as in [7, Subsection 6.4] X 7 |Kr (un , un ) − Kr (u0 , u0 )| ≤ 2 |jr ||Ij (un , un ) − Ij (u0 , u0 )| ≤ Cε 2 j∈M
18
for 0 ≤ tn = nh ≤ ε−N and r = 1, . . . , d since |jr | ≤ C|ωj |s . This proves the near-conservation of the momentum K. For the conservation of energy we note first Z 1 |un |4 dx ≤ k|un |4 k0 ≤ Ckun k4s (2π)d Td and
Z 1 Q(|un |4 )dx ≤ kQ(|un |4 )k0 ≤ Ckun k4s (2π)d Td by the Cauchy-Schwarz inequality and [7, Lemma 6]. The near-conservation of actions implies kun ks ≤ Cε and X X X 7 |ωj ||Ij (un , un ) − Ij (u0 , u0 )| ≤ Cε 2 ωj |unj |2 − ωj |u0j |2 ≤ j∈M
j∈M
j∈M
for 0 ≤ tn = nh ≤ ε−N . This yields the long-time near-conservation of H and HM and completes the proof of Theorem 2.
A
On the non-resonance condition
For the analysis of the numerical solution of (1) we used the non-resonance condition (14). The second inequality of (14) is not necessary for the analysis of the exact solution of (1) and the solution of the spatial semi-discretization of (1), cf. (7) and [7]. We recall that the first inequality of (14) is true in the continuous situation for a large set of potentials V as is shown in [7, Proposition 2]. In this appendix we discuss the validity of the additional non-resonance condition in the continuous situation. For this purpose let, as in Section 2, 1
Rε = { (j, k) : j = j(k), k 6= hji, |k · ω − ωj | < ε 2 , kkk ≤ 2N + 2 = K }. Combining the techniques of [7, Proposition 2] and [7, Lemma 3] we obtain the following Proposition. Proposition 3. Fix N . For sufficiently large s the non-resonance condition P s 1 l∈Zd |kl ||ωl | 2 kkk ≤ C εN +1 sup 0 d+1 ε d+1 )|k| (s− 2 2 (j,k)∈Rε |ω ||ωj | holds for all V ∈ S where S is the set of [7, Proposition 1] with a constant C0 which only depends on N , s, and V . Proof. P Let V ∈ S and (j, k) ∈ Rε . As in the proof of [7, Proposition 2] we write k · ω = kl ωl + |j|≤L,j6=l kj ωj with |l| ≥ L and L ≥ 1 minimal. We then have
1 γ ≤ |k · ω − ωj | < ε 2 α L where we use the notations of [7, Proposition 1] and the argument of [7, Proposition 2]. Now, we proceed as in the proof of [7, Lemma 3] by separating the indices k. For |kl | > 1 or d+1 d+1 1 kj 6= 0 for j 6= l with |j| > 2K |l| we have |ω (s− 2 )|k| | ≥ C|ωl |2(s− 2 ) ≥ C|ωl |s L2(s−d−1) with a constant which depends on d, K, s, and V . Hence P s 2(s−d−1) s−d−1 K|ωl |s l∈Zd |kl ||ωl | ≤C ≤ CKγ − α ε α d+1 s L2(s−d−1) (s− )|k| d+1 |ω | 2 l |ω ||ωj |
and we have to choose s ≥ α(N + 1) + d + 1. d+1 d+1 d+1 d+1 For the other k we have as in [7, Lemma 3] |ω (s− 2 )|k| ||ωj | 2 ≥ C|ωl |s− 2 L2s−d−1 |ωl | 2 and the choice s ≥ α(N + 1) + d+1 2 yields the desired result. Proposition 3 and [7, Proposition 2] show that the numerical non-resonance condition (14) reduces in the limit h → 0 to a non-resonance condition which is fulfilled for a large set of potentials V. 19
Acknowledgement This work was supported by DFG, Project LU 532/4-1.
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