THE GENERATION AND EVOLUTION OF LUMP SOLITARY WAVES ...
c 2000 Society for Industrial and Applied Mathematics
SIAM J. APPL. MATH. Vol. 61, No. 3, pp. 731–750
THE GENERATION AND EVOLUTION OF LUMP SOLITARY WAVES IN SURFACE-TENSION-DOMINATED FLOWS∗ KURT M. BERGER† AND PAUL A. MILEWSKI† Abstract. Three-dimensional solitary waves or lump solitons are known to be solutions to the Kadomtsev–Petviashvili I equation, which models small-amplitude shallow-water waves when the Bond number is greater than 13 . Recently, Pego and Quintero presented a proof of the existence of such waves for the Benney–Luke equation with surface tension. Here we establish an explicit connection between the lump solitons of these two equations and numerically compute the Benney– Luke lump solitons and their speed-amplitude relation. Furthermore, we numerically collide two Benney–Luke lump solitons to illustrate their soliton wave character. Finally, we study the flow over an obstacle near the linear shallow-water speed and show that three-dimensional lump solitons are periodically generated. Key words. solitary waves, capillary-gravity waves, flow over topography AMS subject classifications. 76B15, 76B25, 73D20, 35Q51 PII. S0036139999356971
1. Introduction. The Benney–Luke equation [3] describes the evolution of threedimensional, weakly nonlinear water waves whose horizontal length scale is long compared to the water depth. Recently Milewski [10] generalized this equation to include the effects of surface tension and topographical forcing: 1 ∆2 Φ + Φt ∆Φ + (∇Φ)2t = F hx (x + F t, y). Φtt − ∆Φ + B − (1.1) 3 In this equation, referred to hereafter as the generalized Benney–Luke (gBL) equation, 1 is the ratio of amplitude to depth and Φ(x, y, t) is the leading order velocity potential, which is independent of the vertical coordinate z and the undisturbed water depth H. The water surface is given by H+η(x, y, t), where, to leading order, η = −Φt . Note that to be physically valid, all derivative terms in (1.1) must be O(1). The right-hand side of the gBL equation arises from a uniform flow of normalized speed or Froude number F = √U over bottom topography h(x, y). Here z = −H − gH
h(x, y) denotes the bottom of the channel so that a positive obstacle corresponds to T h < 0. The Bond number, B = ρgH 2 , is a dimensionless quantity indicating the relative importance of surface tension and gravity and appears in the gBL equation as a parameter. Here T ≥ 0 is the surface tension coefficient of the liquid and ρ is its density. Initially, we will be interested in the unforced form of this equation, for which the term on the right-hand side is zero. The gBL equation reduces to the Kadomtsev–Petviashvili (KP) equation (I or II), which describes weakly three-dimensional waves, and the Korteweg–de Vries (KdV) equation describing two-dimensional waves, in the appropriate limits (see [11]). All of these water wave equations are easier to analyze and to compute solutions to than the full Euler equations. Because of its more general scaling, the spatially isotropic ∗ Received
by the editors June 4, 1999; accepted for publication (in revised form) March 24, 2000; published electronically August 29, 2000. http://www.siam.org/journals/siap/61-3/35697.html † Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706 ([email protected], [email protected]). The research of the second author was partially supported by NSF-DMS and a Sloan Research Fellowship. 731
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gBL equation is more versatile than the forced or unforced KP equation, modeling a wider range of initial conditions and topographical forcing, including noncritical mean flows. All of these equations admit two-dimensional sech2 soliton solutions for B = 13 . For B < 13 , the soliton is elevated above the surface, otherwise it is a wave of depression. Both Fokas and Ablowitz [5] and Manakov et al. [9] used inverse scattering techniques to show that for B > 13 , the KP-I equation admits three-dimensional lump solitons, whose amplitude decays to zero in every direction. Recently, Pego and Quintero [15] presented a proof of the existence of such lump solitons for the gBL equation using the concentration-compactness method. Here we derive an explicit formula for a class of symmetric lump solitons that approximate solutions to the gBL equation for B > 13 by modifying the KP-I lump solitons. Using these solutions as a starting point, we compute lump soliton solutions to the gBL equation and compute the speedamplitude relation for these solitons, comparing them to the KP-I solitons. Next, we verify the soliton nature of these waves by numerically colliding them. Note that although there is an inverse scattering transform for KP-I, there are no closed form multilump interaction solutions of the form we are seeking (see section 4). Solitary waves can be generated when √ the uniform flow over an obstacle is near the critical shallow water speed C0 = gH, i.e., Froude number F ≈ 1. Solutions to this problem when the free surface is one-dimensional were first sought by Wu and Wu [18] and later by Akylas [2], Cole [4], and Wu [17], among others. Grimshaw and Smyth [6] investigated a similar problem, but for stratified flows. In most of these studies, the free surface evolution is described by a forced KdV equation when F ≈ 1, and the solution involves the periodic shedding of solitary waves upstream. Recently, the one-dimensional free surface case with surface tension was considered by Milewski and Vanden-Broeck [13]. They derived a forced fifth-order KdV equation for B ≈ 13 , F ≈ 1 and found unsteady capillary-gravity flows past an obstacle to be significantly different from those in the pure gravity case. They found many complicated behaviors including the shedding of solitary waves upstream and downstream, Wilton’s ripples, and depression solitary waves with oscillating tails. For three-dimensional gravity-wave flow, Katsis and Akylas [7] studied a moving pressure distribution on the free surface, an equivalent problem to uniform flow over an obstacle. They found that for flow near the linear shallow water speed, the response was governed by a forced KP equation. They observed the upstream generation of three-dimensional unsteady disturbances due to nonlinear effects. Milewski and Tabak [12] computed time-dependent solutions using a forced version of the Benney– Luke equation. They generated three-dimensional nonlinear bow waves upstream from a localized three-dimensional obstacle. Moreover, using two such obstacles separated in the y-direction, but at the same x location, they were able to see hexagonal-like patterns upstream from the obstacles. In this study we examine the uniform flow over an obstacle when the effect of surface tension is significant (B > 13 ) using the forced gBL equation.1 For F ≈ 1 we observe, in the frame of the obstacle, the periodic shedding of three-dimensional lump solitons at an angle to the downstream flow. The remainder of this paper is organized as follows. In section 2, we examine the KP-I equation and its lump soliton solutions and derive lump solutions to the gBL equation from those of KP-I. Section 3 is devoted to the calculation of the speed1 We note that a Bond number of 1 corresponds to a water layer of depth approximately 0.5 cm. 3 Thus in this regime, viscous effects may become important. We do not, however, consider viscosity in this paper.
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amplitude relation for the gBL lump solitons, while section 4 shows the results of colliding two of these solitons at an angle. Finally, in the last section, we investigate time-dependent three-dimensional uniform flow over an obstacle when surface tension is significant. 2. KP-I and gBL lump solitons. Using inverse scattering techniques it has been shown (see Ablowitz and Clarkson [1]) that the KP-I equation 3 1 1 1 ηt + ηx + ηηx + (2.1) + ηyy = 0 − B ηxxx 2 2 2 3 x with B > 13 admits a family of three-dimensional lump solitons that decay to zero radially in every direction. In this form of the equation, x, y, and t are dimensionless coordinates in the “laboratory” frame and η is the dimensionless free surface displacement. The KP-II equation, corresponding to B < 13 , does not admit lump soliton solutions. For (2.1), a class of symmetric lump soliton solutions is given by
(2.2)
η(X, Y ) =
A 3A2 2 X 2 + 64(B− +1 1 Y 8(B− 13 ) 3) , A 2 A 3A 2 2 + 1]2 [− 8(B− 1 X + 1 Y 64(B− 3 ) 3)
3 A. 16 Here is a small amplitude parameter and A < 0 is the minimum amplitude of the depression soliton. Equation (2.1) is given in the laboratory frame, but x, y, t, and η are dimensionless. If H is the depth of the undisturbed water and the surface elevation η(x, y, t) is measured from z = H, we use H for the vertical length scale, L (a typical wavelength) for the horizontal length scale, and a (a typical amplitude) for the scale of the surface 1 elevation. To complete our scaling, we use the linear shallow water speed C0 = (gH) 2 L as the scale for wave speed and C0 as a time scale. These scales define two ratios a commonly used in parameterizing water wave equations: = H and µ = H L . In the derivation of both the KP and Benney–Luke equations, we set = µ2 to balance nonlinear and dispersive effects (see [3, 11]). Using these length and time scales and the definitions of and µ, we can rewrite the KP-I lump soliton solution in dimensional coordinates in the laboratory frame: (A) 3(A) 3 2 2 2 (x − C (1 + )t) + (A) y + 1 0 16 64H 2 (B− 13 ) 8H 2 (B− 13 ) η(x, y, t) = AH
2 . −(A) 3(A) 3 2 2 2 (x − C0 (1 + 16 )t) + 64H 2 (B− 1 ) (A) y + 1 8H 2 (B− 1 ) 1
X = x − ct, Y = 2 y, c = 1 +
3
3
Note that the minimum amplitude of the soliton is AH and that the solution, for a given Bond number, depends only on the parameter A. The dimensional speed, which also has this property, is c = C0 (1 + 3(A) 16 ) in the x direction. Solutions to the gBL or BL equation also have this property as will be discussed in section 3. Figure 1 shows a typical KP-I lump soliton solution in the laboratory frame. Unlike the KP equation, the gBL equation is written in terms of the velocity potential Φ. Since to leading order Φtt = ∆Φ the unforced form of (1.1) can be also written, preserving the order of approximation, in the more general form
Φtt − ∆Φ + a∆2 Φ − b∆Φtt + Φt ∆Φ + (∇Φ)2t = 0 (2.3)
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KURT M. BERGER AND PAUL A. MILEWSKI
0 0.01 0.02 0.03 0.04
50 y
A
100
0 50
100 0 y
20 100
100
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x
10
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10
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Fig. 1. Two views of a lump soliton solution of the KP-I equation in the laboratory frame. Here we use A = −0.05, B = 23 , with = 0.1. Note that the scale in y is much larger than the scale in x.
in which a − b = B − 13 and a and b are O(1). In this form, for a given Bond number, the choice of the parameters a and b primarily affects the linear dispersion relation of the equation for high wave numbers, which is a useful modification for computing numerical solutions (see section 5). In the derivation of the Benney–Luke equation [11], the free surface displacement η(x, y, t) is related to the velocity potential Φ(x, y, t) by η = −Φt + O(). Thus if 1 1 1 η = η(x−ct, 2 y), we can write Φ = Φ(x−ct, 2 y) and also η = −Φt +η1 (x−ct, 2 y). To leading order then, η = −Φt = cΦx . Using these approximations in the laboratory form of KP-I (2.1) leads to the following KP-I equation in terms of Φ, correct to O(2 ): 2 1 2 Φtt − 2Φxx − Φyy + B − Φxxxx + Φt Φxx + (Φx )t = 0. (2.4) c 3 Note that the substitution of η1 into the O(1) terms of (2.1) yields O(2 ) terms due to the assumed form of η1 and the fact that 1 − c = O(). A solution to (2.4) can be calculated directly by integrating the exact solution η (2.2) with respect to x: X A . ΦL (X, Y ) = c − A 1 X 2 + 3A2 1 Y 2 + 1 8(B− ) 64(B− ) 3
3
3 It is easy to show that by slightly altering the speed of the soliton from c = 1+ 16 A to c = 2c − 1, ΦL also exactly satisfies the equation 1 2 Φxxxx + Φt Φxx + (Φx )t = 0, Φtt − ∆Φ + B − (2.5) 3 ∗
which is identical to the gBL equation, but it excludes the following dispersive and nonlinear terms involving y derivatives: 1 2 B− (2Φxxyy + Φyyyy ) + Φt Φyy + (Φy )t . 3 1
However, for the above lump solitons, Φ = 1c F (x − ct, 2 y), and this reduces to 1 2 B− (2FXXY Y + FY Y Y Y ) − FX FY Y − 2FY FXY = O(2 ). c 3
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As long as the partial derivatives of F (X, Y ) remain order one, these terms are of O(2 ) and can be neglected, since terms of this order are implicitly ignored in gBL. Thus, the exact KP-I lump soliton solution ΦL (X, Y ) with a modified speed is, to leading order, an approximate solution to the isotropic gBL equation. An expansion of the form Φ(X, Y ) = ΦL (X, Y ) + f (X, Y ) in the gBL equation leads to a linear, nonconstant coefficient, inhomogeneous, elliptic partial differential equation for f . The analysis of this equation is beyond the scope of this work. In the next section, however, we numerically compute lump soliton solutions to the gBL equation, along with their corresponding speeds. 3. Speed-amplitude relations. In the previous section, we discussed two models of the three-dimensional evolution of the water surface when the effects of surface tension are significant. The KP-I model is known to have exact lump soliton solutions but is valid only for flows that are quasi-one-dimensional. Lump solutions to 3 KP-I satisfy the speed-amplitude relation c = 1 + 16 A, in which A is the minimum amplitude of free surface displacement η normalized by the depth H. Although the form of the lump solitons depends on two parameters, amplitude and Bond number, the speed-amplitude relation is Bond number independent. The above family of solitons are also approximate solutions to the gBL equation, with a slightly modified speed-amplitude relation. We use these KP-I solutions to compute solutions to the gBL equation which do not necessarily obey the KP scaling, δy δx , and thus are better modeled by the gBL equation. We compute the speedamplitude curve for lump solutions to the gBL equation numerically by using the exact KP-I lump soliton with the modified speed for a given amplitude and fixed Bond number as an initial guess to look for a traveling-wave solution to the gBL equation using the Fourier pseudospectral method described below. (This method was used by Milewski and Keller [11] to compute periodic solutions to the Benney– Luke equation.) After converging to a lump solution to the gBL equation, we choose another point on the KP-I speed-amplitude curve and repeat the process. Before applying the pseudospectral method, we change variables in the unforced form of (1.1) to Φ(x, y, t) = f (θ1 , θ2 ) with θ1 = kx − ωt, θ2 = ly. The resulting equation, given below, is parameterized by the wavenumbers k and l and the frequency ω. 1 ω 2 ∂θ1 θ1 f − (k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )f + B − (k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )2 f 3 (3.1)
−[(ω∂θ1 )f ][(k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )f ] − (ω∂θ1 )[(k∂θ1 f )2 + (l∂θ2 f )2 ] = 0.
By assuming that the two-phase traveling-wave solution f (θ1 , θ2 ) is 2π periodic in each phase, we restrict our computational domain to a period rectangle in the xy plane. The solution travels with velocity
ω c(k, l, ω) = ,0 . k Next we seek solutions of the form f (θ1 , θ2 ) =
N
N
m=−N n=−N
amn e(imθ1 +inθ2 )
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KURT M. BERGER AND PAUL A. MILEWSKI Speed vs. Amplitude for KP I and gBL Models 1
C/C0
0.95 0.9
0.85
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0.1
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0.5 | A|
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1
Fig. 2. Speed-amplitude curves for the KP-I and gBL lump solitons. The star indicates the speed and minimum amplitude of a lump soliton generated through flow over topography as described in section 5.
imposing am,n = a∗−m,−n to obtain a real solution. Note that a00 is arbitrary since it does not affect the displacement of the free surface. Substitution of this form into (3.1) results in a set of 4N 2 + 4N real nonlinear algebraic equations for the 4N 2 + 4N + 1 unknowns, i.e., the real and complex parts of amn and ω. To equate the number of unknowns and equations and to remove a degree of freedom, we add a constraint on the amplitude of the wave: Aˆ2t =
N
N
|amn |2 .
m=−N n=−N
Here, Aˆ is the target L2 norm of the wave. The resulting system of 4N 2 + 4N + 1 equations is solved for the amn and ω using Newton’s method. Using the exact KP-I lump soliton we choose the lengths Lx and Ly so that the solution has effectively decayed to zero within a box of dimensions 2Lx by 2Ly . Obviously, Lx and Ly depend strongly on the given amplitude, since the amplitude is related to the horizontal and vertical extent of the wave. We then choose the wavenumbers to be k = Lπx , l = Lπy so that we have a 2π-periodic solution in θ1 and θ2 . The Fourier transform of this solution, with some necessary augmentation for the highest modes, is the initial amn guess for Newton’s method. Initially, ω is set to kc∗ and the fixed target amplitude Aˆt is computed from the exact KP-I solution, i.e., from the initial amn . After Newton’s method converges, we find the minimum amplitude of the solution and compute the speed as ωk , thus obtaining a point on the speed-amplitude curve of the gBL equation. Figure 2 shows the speed-amplitude relationship of lump soliton solutions to KP-I and gBL. Note that the speed for gBL solitons increasingly deviates from that of KPI as the amplitude increases. It can be shown that as A becomes O(1), derivatives in gBL that are not present in KP, such as FXXY Y and FX FY Y , cease to be only O(1) and are much larger, so that all terms in the gBL equation are comparable. This can be seen from the values of Lx and Ly used at different points on this curve. The left-most point of the curve corresponds to a solution with spatial extent of approximately Lx = 68, Ly = 3025, while the right-most point on the gBL curve L has Lx = 4, Ly = 10.5. The ratio Lxy has decreased from 44 to 2.6, indicating that
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the typical KP scaling is no longer present and that the ignored derivatives are no longer O(1). As a result, the solutions to gBL and KP-I deviate, as shown in the speed-amplitude graph. The speed-amplitude curves shown in Figure 2 are unaffected by the choice of the small parameter . All physically meaningful solutions, however, must satisfy A 1. We showed in section 2 that the exact dimensional lump soliton solution to the KP-I equation depends only on the combination A. That lump solutions to the gBL equation also only depend on the combination A is less obvious since appears explicitly as a parameter in that equation. However, suppose we have a solution Φ(x, y, t) (with corresponding η(x, y, t)) to the gBL equation and we have assumed some value for the parameter . Let the minimum value of η be A. It is easily shown ˆ x , y , t ) is a solution to gBL with ˆ = 2 , and that ηˆ = α2 η so that ˆ times that αΦ( α α α α min(ˆ η ) remains A. Theoretically, then, we could choose = 1 and effectively remove from the gBL equation or equivalently scale it out of the equation. In this case, however, physically meaningful solutions would be restricted to have small derivatives so that the O(2 ) terms, which were ignored in (1.1), remain small. The gBL curve in Figure 2 was produced using = 0.1. 4. Collisions of gBL lump solitons. Although the gBL lump solitons derived in section 3 travel in the x direction only, the gBL equation is isotropic and thus solutions can travel in any direction. To test the “soliton” properties of these solutions, we collide two lump solitary waves. These solitons start at the same x location, but are separated in the y direction, one above the other. The top soliton travels at an angle of −α with respect to the x-axis, while the other travels at an angle of +α. The collision of lump soliton solutions of the KP-I equation was first considered by Manakov et al. [9], in which they claim the solitons do not interact (i.e., zero phase shift). More recently, Villarroel and Ablowitz [16] find, using a “small norm” expansion, nontrivial interactions between solitons. These results, however, are not comparable directly to those presented in this paper since the √ nonisotropic scaling of KP forces the solutions to separate at a rate proportional to t and not t, as is the case here. The solutions of Villarroel and Ablowitz are probably an “inner solution” near the interaction region of our gBL computations. However, the matching of the KP solution to the gBL solution is not considered further here. We use the pseudospectral algorithm proposed by Milewski and Tabak [12] with the initial function being the superposition of two soliton solutions of gBL. We choose the computational domain, assumed periodic, and the initial location of the solitons such that they have effectively decayed to zero within a reasonable distance of each other and the boundaries. Figure 3 shows contour plots of the solution at four different times. Here α = 5◦ , the domain in Lx = 30π by Ly = 150π, and we use Nx = 512 and Ny = 512 points in the x and y directions, respectively. With a time step of ∆t = 0.05, we are able to see a fairly smooth nonlinear interaction with the solitons, traveling from left to right, emerging intact after the collision. Although our computations are not resolved enough to study precisely whether the collisions are completely elastic, they do yield the soliton-like phase shift and do confirm that the lumps are stable to within the numerical error. This is in contrast to recent calculations of one-dimensional free surface soliton interactions for B ≈ 13 (see Malomed and Vanden-Broeck [8]), in which the solitons may break up following collisions. Both solitons, which are identical to the soliton shown in Figure 1, have A = −0.04992 and a speed in the direction of propagation of approximately c = 0.9901.
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KURT M. BERGER AND PAUL A. MILEWSKI T=0
T = 1200
T = 1400
T = 2400
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Fig. 3. The collision of two gBL lump solitons of equal amplitude traveling at ±5◦ to the horizontal. The domain is 30π by 150π and the mesh is 512 by 512. We used ∆t = 0.05 and = 0.1. The dotted lines show the direction of travel before and after the collision.
An angle-dependent phase shift due to the nonlinear interaction was observed and is shown in Figure 4. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision, which is approximately ∆x = 3.4 and ∆y = ±10.7. In other words, we observe a phase lead in x and a phase lag in y for both solitons. In relation to the grid resolution for this computation, which is hx = 0.184, hy = 0.920, the measured phase shifts are significant. The results of a similar 5◦ collision using a soliton of amplitude A = −0.4992 (as in Figure 4) and 2A are shown in Figure 5. Here the phase shifts are approximately ∆x = 4.0 (lead) and ∆y = 8.9 (lag) for the soliton of amplitude A and ∆x = 7.4 (lead) and ∆y = −1.2 (lag) for the 2A soliton. Figure 6 shows the results for the collision of two lump solitons of equal amplitude at an angle of ±45◦ to the horizontal. At this large angle, the solitons pass quickly through one another and produce no measurable phase shift. 5. Generation of lump solitons by resonant flow over a bump. Given that the gBL equation supports lump solitons and that these appear to be stable under collision with other solitons, the next logical step is to study whether they can be generated in more realistic flow conditions. The forced KdV and KP equations have
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Fig. 4. The path of the center of two lump solitons of equal amplitude traveling from left to right at ±5◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision. Tracking the center of each soliton is impossible in the interaction region.
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Fig. 5. The path of the center of two lump solitons of amplitude A (starting from upper left) and 2A traveling from left to right at ±5◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision. Tracking the center of each soliton is impossible in the interaction region.
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KURT M. BERGER AND PAUL A. MILEWSKI
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Fig. 6. The path of the center of two lump solitons of amplitude A traveling from left to right at ±45◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton based on the soliton’s known speed. The x’s mark the actual computed location and illustrate a negligible phase shift.
been used to study free surface flow over topography (see Wu and Wu [18], Akylas [2], Cole [4], Grimshaw and Smyth [6], and Katsis and Akylas [7], among others). Under certain conditions the forced KdV equation will produce solitary waves over the forcing topography. Recently, Nadiga, Margolin, and Smolarkiewicz [14] compared the generation of solitary wave solutions to various shallow-water equations, including forced versions of the KdV, Green–Naghdi, and Boussinesq equations, to those of the full Euler equations for flow over variable topography. They showed that while details of the flow differ, the generation of solitary waves remains qualitatively correct. We study here whether the (forced) gBL equation could, under the right conditions, produce three-dimensional lump solitons. Milewski and Tabak [12] previously computed time-dependent solutions to the forced gBL equation for zero surface tension, outside the regime of lump solitons. We consider the free surface flow past a submerged obstacle in a channel of finite
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z
0.99
0.995
1 50
y
0
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10 x
20
30
Fig. 7. Bottom topography used in numerical experiments for flow over a bump. The bottom is given by z = −1 − 2 h(x, y) in which h(x, y) = −sech2 .302 (x − x0 )2 + .152 (y − y0 )2 . Here we use = 0.1.
depth. Upstream from the obstacle, the flow is uniform with a constant velocity U in the positive x direction and of constant depth H. We let z = −H − h(x, y) denote the bottom of the channel so that a positive obstacle corresponds to h < 0. Milewski [10] derived the forced version of the gBL equation, repeated here in its more general form, to model the three-dimensional evolution of the free surface for this problem:
(5.1) Φtt − ∆Φ + a∆2 Φ − b∆Φtt + Φt ∆Φ + (∇Φ)2t ) = F hx (x + F t, y). Recall that a − b = B − 13 and F is the Froude number defined as F = CU0 . Note that this equation applies in a frame moving with the mean velocity of the flow and is therefore still isotropic. In this frame, the obstacle is moving with a speed −U . Milewski and Tabak [12] computed time-dependent solutions to this equation for B = 0, F = 1 for a three-dimensional bump of the form h(x, y) = sech2 (x2 + y 2 ). In the derivation of (1.1), the length scale of the obstacle h(x, y) is 2 H so that 2 the bottom of the channel is given by For our purposes we use z 2= −1 − 2 h(x, y). 2 the positive bump h(x, y) = −sech .30 (x − x0 ) + .152 (y − y0 )2 , which is shown in Figure 7. Our interest is in the creation of three-dimensional lump solitons through uniform flow over a bump. Lump solitons are solutions to the unforced gBL equation only when both surface tension and nonlinear effects are significant. Since flow over topography near the critical speed leads to large free-surface displacement and thus nonlinear effects, we expect the possibility of lump solitons only for B > 13 , F ≈ 1. Before considering solutions to the forced gBL equation at the critical speed, however, we first examine the linear solution at a range of flow speeds. 5.1. Linear solution. Ignoring the two nonlinear terms in (5.1), we can rewrite this equation in operator form as (∂tt + L2 )Φ = G(h), in which L2 = (1 − b∆)−1 (−∆ + a∆2 )
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KURT M. BERGER AND PAUL A. MILEWSKI
and G(h) = (1 − b∆)−1 (F hx (x + F t, y)) . We factor the left side by introducing u = (∂t − iL)Φ and recast the equation in terms of u as ut + iLu = G(h).
(5.2)
Given u(x, y, t), the free surface in a frame moving with the mean velocity of the flow is, to leading order, η = −Φt = −Re(u). Equation (5.2) can be solved as an initial value problem if we assume that the bump is introduced at t = 0 and η(x, y, 0) = Φ(x, y, 0) = 0, which implies u(x, y, 0) = 0. Taking the two-dimensional Fourier transform of (5.2) gives an equation for u ˆ(k) in which k = (k, l): ˆ ˆ u = G( ˆ h) u ˆt + iLˆ
(5.3) with ˆ= L
|k|2 + a|k|4 1 + b|k|2
12
,
|k|2 = k 2 + l2 ,
and ˆ = ˆ h) G(
ikF ˆ h(k)eikF t . 1 + b|k|2
Equation (5.3) is easily solved in a frame fixed with respect to the bump. Letting v(x + F t, y, t) = u(x, y, t) be the solution in the frame of the bump, the solution is ˆ h(k) kF ˆ (5.4) [1 − e−i(L(k)+F k)t ]. vˆ(k) = 2 ˆ 1 + b|k| F k + L(k) ˆ The linear dispersion relation ω(k) = L(k) for the gBL equation, given above, shows that the linear problem is well-posed only if the parameters a and b are nonnegative. Figure 8 shows the phase speed in the x direction as a function of k for B = 0 and B = 23 for the unmodified dispersion relation (b = 0), and for B = 23 with b = 12 . For B < 13 the free surface evolution is dominated by gravity and the wave speed increases with wavelength. For B > 13 the free surface evolution is dominated by surface tension and the wave speed decreases with wavelength. The b = 0 curve is close to the b = 0 curve when k is O(1), but they differ as k becomes large, a regime that is outside the physical range of validity of the gBL equation. Thus the effect of b > 0 is to bound the speed of the high wavenumbers, which are not well approximated by the gBL equation, to give more manageable numerical simulations. The solution of (5.4) has two regimes, subcritical (F > 1) and supercritical (F < 1). Note that these definitions are the opposite of those used for gravity flows due to the difference in the linear dispersion relations. In both cases the subcritical regime consists of Froude numbers for which there exist linear wave speeds, i.e., there exist steady waves in the frame of the bump. The linear solution to (5.4) for subcritical flows (F > 1) contains a steady wave pattern given by the family of wavenumbers (K cos θ, K sin θ) for −1 −1 −1 −1 cos < θ < 2π − cos F F
743
LUMP SOLITARY WAVES 1.5 B = 2/3, b = 0
cx
B = 2/3, b = 1/2 1 B = 0, b = 0
0.5
0
0.5
1
1.5
2
2.5 k
3
3.5
4
4.5
Fig. 8. Linear dispersion relation cx (k, 0) for the gBL equation for B = 0 and B = b = 0. Also shown is the curve for B = 23 and b = 12 .
5 2 3
with
TIME = 300 250
200
y
150
100
50
0
30
40
50
60
70 x
80
90
100
110
Fig. 9. Uniform linear flow past a positive bump for (F = 1.1, B = 23 ) as viewed in the frame of the obstacle. The obstacle is centered at the intersection of the dotted lines.
and
K=
F 2 cos2 θ − 1 . (a − bF 2 cos2 θ) Cg (K)
These waves travel in the xy plane along xy = Cgy (K) lines, with the limiting lines x ). These two lines have an angle with the x-axis given by K = 0 and θ = cos−1 ( −1 F of ±ξ ∗ , with ξ ∗ = tan−1 ( √F12 −1 ). Thus there exist waves upstream of the obstacle outside of the wedge of half-angle ξ ∗ . Figure 9 shows the linear response in the frame of the obstacle for (F = 1.1, B = 23 ) at t = 300 for the positive bump described
744
KURT M. BERGER AND PAUL A. MILEWSKI TIME = 300
250
200
y
150
100
50
0
0
20
40
60
80
100
x
Fig. 10. Uniform linear flow past a positive bump for (F = 0.9, B = 23 ). Viewed in the frame of the obstacle, the transient waves are moving to the left leaving a steady depression wave above the obstacle. The obstacle is centered at the intersection of the dotted lines.
above. The half-angle of the wedge in this figure is very close to the predicted value for F = 1.1, which is ξ ∗ = 65.4◦ . In the supercritical regime (F < 1), the steady solution, obtained from the first term in (5.4), is a localized disturbance over the bump as t → ∞. The asymptotic form of the transient waves generated by the bump can be obtained using stationary phase arguments on the second term in (5.4). Figure 10 shows the linear response for a supercritical flow (F = 0.9, B = 23 ) at t = 300 for the positive bump described above. As expected, we observe a stationary depression above the obstacle and transient waves moving off to the left. Nonlinear simulations of these cases produced virtually identical results. In the critical F = 1 case the linear solution will grow unbounded in time. Figure 11 shows the linear response in the frame of the obstacle for (F = 1.0, B = 23 ) at t = 300. The response is localized above the bump with an amplitude roughly an order of magnitude larger than the subcritical and supercritical cases at the same time. 5.2. Nonlinear resonant solutions. Since the linear solution at the critical flow speed F = 1 is unbounded in time and large for F ≈ 1, we expect nonlinear terms to become important in the evolution of the free surface in this regime. Thus we examine the behavior of the forced gBL equation using the pseudospectral method described earlier for soliton collisions. Using a 512 by 512 mesh and the same bottom topography, we observe the symmetric pairwise periodic generation of lump solitons downstream of the obstacle for B > 13 and F ≈ 1. The lump solitons are generated along the lines which form angles ±ξs with respect to the positive x-axis, and their amplitude and period of generation depend on the speed of the flow. Figures 12
745
LUMP SOLITARY WAVES TIME = 300 250
200
y
150
100
50
0
30
40
50
60
70 x
80
90
Fig. 11. Uniform linear flow past a positive bump for (F = 1.0, B = of the obstacle. The amplitude of the response grows with time.
2 ) 3
100
110
as viewed in the frame
and 13 show the free surface evolution at four different times for (F = 1, B = 23 ) for the positive sech2 obstacle. Lump solitons emerge at an angle above and below the downstream direction in the frame of the obstacle. In this frame of reference, the angle of travel with respect to the positive x-axis is approximately ±65◦ and the speed is c ≈ 0.168. In a frame traveling with the mean velocity of the water (F = 1), the angle is α ≈ ±9.2◦ with respect to the upstream direction (negative x-axis) and the speed is c ≈ 0.937. With a minimum amplitude of A ≈ −2.95, these solitons are represented by the point (|A|, c) = (.295, .937), which is indicated on the speed-amplitude curve for the gBL equation given in Figure 2. Lump solitons are also generated for close to critical flow speeds F ≈ 1. Figure 14 shows the development of solitons for F = 0.98, using the same positive topography, at T = 500 and T = 900. In comparison to the F = 1 case at T = 500 (see Figure 13), the F = 0.98 solitons develop at a slower rate. Figure 15 shows the free surface evolution for F = 1.02. Note that the solitons emerge sooner and travel somewhat faster in the frame of the bump due to the increased mean flow. In this case the angle ξs is approximately 65◦ and lies inside the “shadow” wedge given by ξ ∗ = 79◦ for F = 1.02. 6. Conclusion. Three-dimensional solitary waves or lump solitons are shown to be solutions to the isotropic Benney–Luke equation when surface tension effects are significant. We have established an explicit connection between the lump solitons of the KP-I equation and approximate lump solitons of the gBL equation. Moreover, we have numerically computed exact gBL lump solitons and their speed-amplitude relation. We then numerically collide two gBL lump solitons in time to illustrate their solitary wave behavior. Finally, we show that three-dimensional lump solitons are generated in the uniform flow over an obstacle near the critical speed when surface tension is significant.
746
KURT M. BERGER AND PAUL A. MILEWSKI
TIME = 100.0
200
y
150
100
50
0
30
40
50
60
70 x
80
90
100
110
80
90
100
110
TIME = 300.0
200
y
150
100
50
0
30
40
50
60
70 x
Fig. 12. Uniform nonlinear flow past a positive obstacle for (F = 1, B = 23 ) at t = 100 and t = 300. Solitons emerge at a large angle to the downstream direction as viewed in the frame of the obstacle. The obstacle is centered at the intersection of the dotted lines.
747
LUMP SOLITARY WAVES
TIME = 500.0
200
y
150
100
50
0
30
40
50
60
70 x
80
90
100
110
80
90
100
110
TIME = 700.0
200
y
150
100
50
0
30
40
50
60
70 x
Fig. 13. Uniform nonlinear flow past a positive obstacle for (F = 1, B = 23 ) at t = 500 and t = 700. The obstacle is centered at the intersection of the dotted lines. The minimum amplitude of the right-most solitons at t = 700 is approximately A = −2.95 and their speed in this frame is 0.168.
748
KURT M. BERGER AND PAUL A. MILEWSKI
TIME = 500.0
200
y
150
100
50
0
30
40
50
60
70 x
80
90
100
110
80
90
100
110
TIME = 900.0
200
y
150
100
50
0
30
40
50
60
70 x
Fig. 14. Uniform nonlinear flow past a positive obstacle for (F = 0.98, B = t = 900. The obstacle is centered at the intersection of the dotted lines.
2 ) 3
at t = 500 and
749
LUMP SOLITARY WAVES
TIME = 300.0
200
y
150
100
50
0
30
40
50
60
70 x
80
90
100
110
80
90
100
110
TIME = 500.0
200
y
150
100
50
0
30
40
50
60
70 x
Fig. 15. Uniform nonlinear flow past a positive obstacle for (F = 1.02, B = t = 500. The obstacle is centered at the intersection of the dotted lines.
2 ) 3
at t = 300 and
750
KURT M. BERGER AND PAUL A. MILEWSKI REFERENCES
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, UK, 1991. [2] T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech., 141 (1984), pp. 455–466. [3] D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), pp. 309–313. [4] S. L. Cole, Transient waves produced by flow past a bump, Wave Motion, 7 (1985), pp. 579–587. [5] A. S. Fokas and M. J. Ablowitz, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation, Phys. Lett., 94 (1983), pp. 67–70. [6] R. H. Grimshaw and N. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech., 169 (1986), pp. 429–464. [7] C. Katsis and T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects, J. Fluid Mech., 177 (1987), pp. 49– 65. [8] B. Malomed and J.-M. Vanden-Broeck, Solitary wave interactions for the fifth-order KdV equation, Mathematical Problems in the Theory of Water Waves, Contemp. Math. 200, AMS, Providence, RI, 1996, pp. 133–143. [9] S. V. Manakov, V. E. Zakhorov, L. A. Bordag, A. R. Its, and V. B. Matveev, Twodimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett., 63 (1977), pp. 205–206. [10] P. A. Milewski, A formulation for water waves over topography, Stud. Appl. Math., 100 (1998), pp. 95–106. [11] P. A. Milewski and J. B. Keller, Three-dimensional water waves, Stud. Appl. Math., 37 (1996), pp. 149–166. [12] P. A. Milewski and E. G. Tabak, A pseudospectral procedure for the solution of nonlinear wave equations with example from free-surface flows, SIAM J. Sci. Comput., 21 (2000), pp. 1102–1114. [13] P. A. Milewski and J.-M. Vanden-Broeck, Time dependent gravity-capillary flows past an obstacle, Wave Motion, 29 (1999), pp. 63–79. [14] B. T. Nadiga, L. G. Margolin, and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8 (1996), pp. 2066–2077. [15] R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation, Phys. D, 132 (1999), pp. 476–496. [16] J. Villarroel and M. J. Ablowitz, Solutions to the time-dependent Schrodinger and the Kadomtsev-Petviashvili equations, Phys. Rev. Lett., 78 (1997), pp. 570–573. [17] T. Y. Wu, Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech., 184 (1987), pp. 75–99. [18] T. Y. Wu and D. M. Wu, Generation of upstream advancing solitons by moving disturbances, in Proceedings of the 14th Symposium on Naval Hydrodynamics, National Academy of Sciences, Washington, D.C., 1982, pp. 103–125.
SIAM J. APPL. MATH. Vol. 61, No. 3, pp. 731–750
THE GENERATION AND EVOLUTION OF LUMP SOLITARY WAVES IN SURFACE-TENSION-DOMINATED FLOWS∗ KURT M. BERGER† AND PAUL A. MILEWSKI† Abstract. Three-dimensional solitary waves or lump solitons are known to be solutions to the Kadomtsev–Petviashvili I equation, which models small-amplitude shallow-water waves when the Bond number is greater than 13 . Recently, Pego and Quintero presented a proof of the existence of such waves for the Benney–Luke equation with surface tension. Here we establish an explicit connection between the lump solitons of these two equations and numerically compute the Benney– Luke lump solitons and their speed-amplitude relation. Furthermore, we numerically collide two Benney–Luke lump solitons to illustrate their soliton wave character. Finally, we study the flow over an obstacle near the linear shallow-water speed and show that three-dimensional lump solitons are periodically generated. Key words. solitary waves, capillary-gravity waves, flow over topography AMS subject classifications. 76B15, 76B25, 73D20, 35Q51 PII. S0036139999356971
1. Introduction. The Benney–Luke equation [3] describes the evolution of threedimensional, weakly nonlinear water waves whose horizontal length scale is long compared to the water depth. Recently Milewski [10] generalized this equation to include the effects of surface tension and topographical forcing: 1 ∆2 Φ + Φt ∆Φ + (∇Φ)2t = F hx (x + F t, y). Φtt − ∆Φ + B − (1.1) 3 In this equation, referred to hereafter as the generalized Benney–Luke (gBL) equation, 1 is the ratio of amplitude to depth and Φ(x, y, t) is the leading order velocity potential, which is independent of the vertical coordinate z and the undisturbed water depth H. The water surface is given by H+η(x, y, t), where, to leading order, η = −Φt . Note that to be physically valid, all derivative terms in (1.1) must be O(1). The right-hand side of the gBL equation arises from a uniform flow of normalized speed or Froude number F = √U over bottom topography h(x, y). Here z = −H − gH
h(x, y) denotes the bottom of the channel so that a positive obstacle corresponds to T h < 0. The Bond number, B = ρgH 2 , is a dimensionless quantity indicating the relative importance of surface tension and gravity and appears in the gBL equation as a parameter. Here T ≥ 0 is the surface tension coefficient of the liquid and ρ is its density. Initially, we will be interested in the unforced form of this equation, for which the term on the right-hand side is zero. The gBL equation reduces to the Kadomtsev–Petviashvili (KP) equation (I or II), which describes weakly three-dimensional waves, and the Korteweg–de Vries (KdV) equation describing two-dimensional waves, in the appropriate limits (see [11]). All of these water wave equations are easier to analyze and to compute solutions to than the full Euler equations. Because of its more general scaling, the spatially isotropic ∗ Received
by the editors June 4, 1999; accepted for publication (in revised form) March 24, 2000; published electronically August 29, 2000. http://www.siam.org/journals/siap/61-3/35697.html † Department of Mathematics, University of Wisconsin, 480 Lincoln Drive, Madison, WI 53706 ([email protected], [email protected]). The research of the second author was partially supported by NSF-DMS and a Sloan Research Fellowship. 731
732
KURT M. BERGER AND PAUL A. MILEWSKI
gBL equation is more versatile than the forced or unforced KP equation, modeling a wider range of initial conditions and topographical forcing, including noncritical mean flows. All of these equations admit two-dimensional sech2 soliton solutions for B = 13 . For B < 13 , the soliton is elevated above the surface, otherwise it is a wave of depression. Both Fokas and Ablowitz [5] and Manakov et al. [9] used inverse scattering techniques to show that for B > 13 , the KP-I equation admits three-dimensional lump solitons, whose amplitude decays to zero in every direction. Recently, Pego and Quintero [15] presented a proof of the existence of such lump solitons for the gBL equation using the concentration-compactness method. Here we derive an explicit formula for a class of symmetric lump solitons that approximate solutions to the gBL equation for B > 13 by modifying the KP-I lump solitons. Using these solutions as a starting point, we compute lump soliton solutions to the gBL equation and compute the speedamplitude relation for these solitons, comparing them to the KP-I solitons. Next, we verify the soliton nature of these waves by numerically colliding them. Note that although there is an inverse scattering transform for KP-I, there are no closed form multilump interaction solutions of the form we are seeking (see section 4). Solitary waves can be generated when √ the uniform flow over an obstacle is near the critical shallow water speed C0 = gH, i.e., Froude number F ≈ 1. Solutions to this problem when the free surface is one-dimensional were first sought by Wu and Wu [18] and later by Akylas [2], Cole [4], and Wu [17], among others. Grimshaw and Smyth [6] investigated a similar problem, but for stratified flows. In most of these studies, the free surface evolution is described by a forced KdV equation when F ≈ 1, and the solution involves the periodic shedding of solitary waves upstream. Recently, the one-dimensional free surface case with surface tension was considered by Milewski and Vanden-Broeck [13]. They derived a forced fifth-order KdV equation for B ≈ 13 , F ≈ 1 and found unsteady capillary-gravity flows past an obstacle to be significantly different from those in the pure gravity case. They found many complicated behaviors including the shedding of solitary waves upstream and downstream, Wilton’s ripples, and depression solitary waves with oscillating tails. For three-dimensional gravity-wave flow, Katsis and Akylas [7] studied a moving pressure distribution on the free surface, an equivalent problem to uniform flow over an obstacle. They found that for flow near the linear shallow water speed, the response was governed by a forced KP equation. They observed the upstream generation of three-dimensional unsteady disturbances due to nonlinear effects. Milewski and Tabak [12] computed time-dependent solutions using a forced version of the Benney– Luke equation. They generated three-dimensional nonlinear bow waves upstream from a localized three-dimensional obstacle. Moreover, using two such obstacles separated in the y-direction, but at the same x location, they were able to see hexagonal-like patterns upstream from the obstacles. In this study we examine the uniform flow over an obstacle when the effect of surface tension is significant (B > 13 ) using the forced gBL equation.1 For F ≈ 1 we observe, in the frame of the obstacle, the periodic shedding of three-dimensional lump solitons at an angle to the downstream flow. The remainder of this paper is organized as follows. In section 2, we examine the KP-I equation and its lump soliton solutions and derive lump solutions to the gBL equation from those of KP-I. Section 3 is devoted to the calculation of the speed1 We note that a Bond number of 1 corresponds to a water layer of depth approximately 0.5 cm. 3 Thus in this regime, viscous effects may become important. We do not, however, consider viscosity in this paper.
733
LUMP SOLITARY WAVES
amplitude relation for the gBL lump solitons, while section 4 shows the results of colliding two of these solitons at an angle. Finally, in the last section, we investigate time-dependent three-dimensional uniform flow over an obstacle when surface tension is significant. 2. KP-I and gBL lump solitons. Using inverse scattering techniques it has been shown (see Ablowitz and Clarkson [1]) that the KP-I equation 3 1 1 1 ηt + ηx + ηηx + (2.1) + ηyy = 0 − B ηxxx 2 2 2 3 x with B > 13 admits a family of three-dimensional lump solitons that decay to zero radially in every direction. In this form of the equation, x, y, and t are dimensionless coordinates in the “laboratory” frame and η is the dimensionless free surface displacement. The KP-II equation, corresponding to B < 13 , does not admit lump soliton solutions. For (2.1), a class of symmetric lump soliton solutions is given by
(2.2)
η(X, Y ) =
A 3A2 2 X 2 + 64(B− +1 1 Y 8(B− 13 ) 3) , A 2 A 3A 2 2 + 1]2 [− 8(B− 1 X + 1 Y 64(B− 3 ) 3)
3 A. 16 Here is a small amplitude parameter and A < 0 is the minimum amplitude of the depression soliton. Equation (2.1) is given in the laboratory frame, but x, y, t, and η are dimensionless. If H is the depth of the undisturbed water and the surface elevation η(x, y, t) is measured from z = H, we use H for the vertical length scale, L (a typical wavelength) for the horizontal length scale, and a (a typical amplitude) for the scale of the surface 1 elevation. To complete our scaling, we use the linear shallow water speed C0 = (gH) 2 L as the scale for wave speed and C0 as a time scale. These scales define two ratios a commonly used in parameterizing water wave equations: = H and µ = H L . In the derivation of both the KP and Benney–Luke equations, we set = µ2 to balance nonlinear and dispersive effects (see [3, 11]). Using these length and time scales and the definitions of and µ, we can rewrite the KP-I lump soliton solution in dimensional coordinates in the laboratory frame: (A) 3(A) 3 2 2 2 (x − C (1 + )t) + (A) y + 1 0 16 64H 2 (B− 13 ) 8H 2 (B− 13 ) η(x, y, t) = AH
2 . −(A) 3(A) 3 2 2 2 (x − C0 (1 + 16 )t) + 64H 2 (B− 1 ) (A) y + 1 8H 2 (B− 1 ) 1
X = x − ct, Y = 2 y, c = 1 +
3
3
Note that the minimum amplitude of the soliton is AH and that the solution, for a given Bond number, depends only on the parameter A. The dimensional speed, which also has this property, is c = C0 (1 + 3(A) 16 ) in the x direction. Solutions to the gBL or BL equation also have this property as will be discussed in section 3. Figure 1 shows a typical KP-I lump soliton solution in the laboratory frame. Unlike the KP equation, the gBL equation is written in terms of the velocity potential Φ. Since to leading order Φtt = ∆Φ the unforced form of (1.1) can be also written, preserving the order of approximation, in the more general form
Φtt − ∆Φ + a∆2 Φ − b∆Φtt + Φt ∆Φ + (∇Φ)2t = 0 (2.3)
734
KURT M. BERGER AND PAUL A. MILEWSKI
0 0.01 0.02 0.03 0.04
50 y
A
100
0 50
100 0 y
20 100
100
0 20
20
x
10
0 x
10
20
Fig. 1. Two views of a lump soliton solution of the KP-I equation in the laboratory frame. Here we use A = −0.05, B = 23 , with = 0.1. Note that the scale in y is much larger than the scale in x.
in which a − b = B − 13 and a and b are O(1). In this form, for a given Bond number, the choice of the parameters a and b primarily affects the linear dispersion relation of the equation for high wave numbers, which is a useful modification for computing numerical solutions (see section 5). In the derivation of the Benney–Luke equation [11], the free surface displacement η(x, y, t) is related to the velocity potential Φ(x, y, t) by η = −Φt + O(). Thus if 1 1 1 η = η(x−ct, 2 y), we can write Φ = Φ(x−ct, 2 y) and also η = −Φt +η1 (x−ct, 2 y). To leading order then, η = −Φt = cΦx . Using these approximations in the laboratory form of KP-I (2.1) leads to the following KP-I equation in terms of Φ, correct to O(2 ): 2 1 2 Φtt − 2Φxx − Φyy + B − Φxxxx + Φt Φxx + (Φx )t = 0. (2.4) c 3 Note that the substitution of η1 into the O(1) terms of (2.1) yields O(2 ) terms due to the assumed form of η1 and the fact that 1 − c = O(). A solution to (2.4) can be calculated directly by integrating the exact solution η (2.2) with respect to x: X A . ΦL (X, Y ) = c − A 1 X 2 + 3A2 1 Y 2 + 1 8(B− ) 64(B− ) 3
3
3 It is easy to show that by slightly altering the speed of the soliton from c = 1+ 16 A to c = 2c − 1, ΦL also exactly satisfies the equation 1 2 Φxxxx + Φt Φxx + (Φx )t = 0, Φtt − ∆Φ + B − (2.5) 3 ∗
which is identical to the gBL equation, but it excludes the following dispersive and nonlinear terms involving y derivatives: 1 2 B− (2Φxxyy + Φyyyy ) + Φt Φyy + (Φy )t . 3 1
However, for the above lump solitons, Φ = 1c F (x − ct, 2 y), and this reduces to 1 2 B− (2FXXY Y + FY Y Y Y ) − FX FY Y − 2FY FXY = O(2 ). c 3
LUMP SOLITARY WAVES
735
As long as the partial derivatives of F (X, Y ) remain order one, these terms are of O(2 ) and can be neglected, since terms of this order are implicitly ignored in gBL. Thus, the exact KP-I lump soliton solution ΦL (X, Y ) with a modified speed is, to leading order, an approximate solution to the isotropic gBL equation. An expansion of the form Φ(X, Y ) = ΦL (X, Y ) + f (X, Y ) in the gBL equation leads to a linear, nonconstant coefficient, inhomogeneous, elliptic partial differential equation for f . The analysis of this equation is beyond the scope of this work. In the next section, however, we numerically compute lump soliton solutions to the gBL equation, along with their corresponding speeds. 3. Speed-amplitude relations. In the previous section, we discussed two models of the three-dimensional evolution of the water surface when the effects of surface tension are significant. The KP-I model is known to have exact lump soliton solutions but is valid only for flows that are quasi-one-dimensional. Lump solutions to 3 KP-I satisfy the speed-amplitude relation c = 1 + 16 A, in which A is the minimum amplitude of free surface displacement η normalized by the depth H. Although the form of the lump solitons depends on two parameters, amplitude and Bond number, the speed-amplitude relation is Bond number independent. The above family of solitons are also approximate solutions to the gBL equation, with a slightly modified speed-amplitude relation. We use these KP-I solutions to compute solutions to the gBL equation which do not necessarily obey the KP scaling, δy δx , and thus are better modeled by the gBL equation. We compute the speedamplitude curve for lump solutions to the gBL equation numerically by using the exact KP-I lump soliton with the modified speed for a given amplitude and fixed Bond number as an initial guess to look for a traveling-wave solution to the gBL equation using the Fourier pseudospectral method described below. (This method was used by Milewski and Keller [11] to compute periodic solutions to the Benney– Luke equation.) After converging to a lump solution to the gBL equation, we choose another point on the KP-I speed-amplitude curve and repeat the process. Before applying the pseudospectral method, we change variables in the unforced form of (1.1) to Φ(x, y, t) = f (θ1 , θ2 ) with θ1 = kx − ωt, θ2 = ly. The resulting equation, given below, is parameterized by the wavenumbers k and l and the frequency ω. 1 ω 2 ∂θ1 θ1 f − (k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )f + B − (k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )2 f 3 (3.1)
−[(ω∂θ1 )f ][(k 2 ∂θ1 θ1 + l2 ∂θ2 θ2 )f ] − (ω∂θ1 )[(k∂θ1 f )2 + (l∂θ2 f )2 ] = 0.
By assuming that the two-phase traveling-wave solution f (θ1 , θ2 ) is 2π periodic in each phase, we restrict our computational domain to a period rectangle in the xy plane. The solution travels with velocity
ω c(k, l, ω) = ,0 . k Next we seek solutions of the form f (θ1 , θ2 ) =
N
N
m=−N n=−N
amn e(imθ1 +inθ2 )
736
KURT M. BERGER AND PAUL A. MILEWSKI Speed vs. Amplitude for KP I and gBL Models 1
C/C0
0.95 0.9
0.85
KP I
0.8 0.75
gBL 0
0.1
0.2
0.3
0.4
0.5 | A|
0.6
0.7
0.8
0.9
1
Fig. 2. Speed-amplitude curves for the KP-I and gBL lump solitons. The star indicates the speed and minimum amplitude of a lump soliton generated through flow over topography as described in section 5.
imposing am,n = a∗−m,−n to obtain a real solution. Note that a00 is arbitrary since it does not affect the displacement of the free surface. Substitution of this form into (3.1) results in a set of 4N 2 + 4N real nonlinear algebraic equations for the 4N 2 + 4N + 1 unknowns, i.e., the real and complex parts of amn and ω. To equate the number of unknowns and equations and to remove a degree of freedom, we add a constraint on the amplitude of the wave: Aˆ2t =
N
N
|amn |2 .
m=−N n=−N
Here, Aˆ is the target L2 norm of the wave. The resulting system of 4N 2 + 4N + 1 equations is solved for the amn and ω using Newton’s method. Using the exact KP-I lump soliton we choose the lengths Lx and Ly so that the solution has effectively decayed to zero within a box of dimensions 2Lx by 2Ly . Obviously, Lx and Ly depend strongly on the given amplitude, since the amplitude is related to the horizontal and vertical extent of the wave. We then choose the wavenumbers to be k = Lπx , l = Lπy so that we have a 2π-periodic solution in θ1 and θ2 . The Fourier transform of this solution, with some necessary augmentation for the highest modes, is the initial amn guess for Newton’s method. Initially, ω is set to kc∗ and the fixed target amplitude Aˆt is computed from the exact KP-I solution, i.e., from the initial amn . After Newton’s method converges, we find the minimum amplitude of the solution and compute the speed as ωk , thus obtaining a point on the speed-amplitude curve of the gBL equation. Figure 2 shows the speed-amplitude relationship of lump soliton solutions to KP-I and gBL. Note that the speed for gBL solitons increasingly deviates from that of KPI as the amplitude increases. It can be shown that as A becomes O(1), derivatives in gBL that are not present in KP, such as FXXY Y and FX FY Y , cease to be only O(1) and are much larger, so that all terms in the gBL equation are comparable. This can be seen from the values of Lx and Ly used at different points on this curve. The left-most point of the curve corresponds to a solution with spatial extent of approximately Lx = 68, Ly = 3025, while the right-most point on the gBL curve L has Lx = 4, Ly = 10.5. The ratio Lxy has decreased from 44 to 2.6, indicating that
LUMP SOLITARY WAVES
737
the typical KP scaling is no longer present and that the ignored derivatives are no longer O(1). As a result, the solutions to gBL and KP-I deviate, as shown in the speed-amplitude graph. The speed-amplitude curves shown in Figure 2 are unaffected by the choice of the small parameter . All physically meaningful solutions, however, must satisfy A 1. We showed in section 2 that the exact dimensional lump soliton solution to the KP-I equation depends only on the combination A. That lump solutions to the gBL equation also only depend on the combination A is less obvious since appears explicitly as a parameter in that equation. However, suppose we have a solution Φ(x, y, t) (with corresponding η(x, y, t)) to the gBL equation and we have assumed some value for the parameter . Let the minimum value of η be A. It is easily shown ˆ x , y , t ) is a solution to gBL with ˆ = 2 , and that ηˆ = α2 η so that ˆ times that αΦ( α α α α min(ˆ η ) remains A. Theoretically, then, we could choose = 1 and effectively remove from the gBL equation or equivalently scale it out of the equation. In this case, however, physically meaningful solutions would be restricted to have small derivatives so that the O(2 ) terms, which were ignored in (1.1), remain small. The gBL curve in Figure 2 was produced using = 0.1. 4. Collisions of gBL lump solitons. Although the gBL lump solitons derived in section 3 travel in the x direction only, the gBL equation is isotropic and thus solutions can travel in any direction. To test the “soliton” properties of these solutions, we collide two lump solitary waves. These solitons start at the same x location, but are separated in the y direction, one above the other. The top soliton travels at an angle of −α with respect to the x-axis, while the other travels at an angle of +α. The collision of lump soliton solutions of the KP-I equation was first considered by Manakov et al. [9], in which they claim the solitons do not interact (i.e., zero phase shift). More recently, Villarroel and Ablowitz [16] find, using a “small norm” expansion, nontrivial interactions between solitons. These results, however, are not comparable directly to those presented in this paper since the √ nonisotropic scaling of KP forces the solutions to separate at a rate proportional to t and not t, as is the case here. The solutions of Villarroel and Ablowitz are probably an “inner solution” near the interaction region of our gBL computations. However, the matching of the KP solution to the gBL solution is not considered further here. We use the pseudospectral algorithm proposed by Milewski and Tabak [12] with the initial function being the superposition of two soliton solutions of gBL. We choose the computational domain, assumed periodic, and the initial location of the solitons such that they have effectively decayed to zero within a reasonable distance of each other and the boundaries. Figure 3 shows contour plots of the solution at four different times. Here α = 5◦ , the domain in Lx = 30π by Ly = 150π, and we use Nx = 512 and Ny = 512 points in the x and y directions, respectively. With a time step of ∆t = 0.05, we are able to see a fairly smooth nonlinear interaction with the solitons, traveling from left to right, emerging intact after the collision. Although our computations are not resolved enough to study precisely whether the collisions are completely elastic, they do yield the soliton-like phase shift and do confirm that the lumps are stable to within the numerical error. This is in contrast to recent calculations of one-dimensional free surface soliton interactions for B ≈ 13 (see Malomed and Vanden-Broeck [8]), in which the solitons may break up following collisions. Both solitons, which are identical to the soliton shown in Figure 1, have A = −0.04992 and a speed in the direction of propagation of approximately c = 0.9901.
738
KURT M. BERGER AND PAUL A. MILEWSKI T=0
T = 1200
T = 1400
T = 2400
400
400
400
400
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250
250 y
450
y
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50
50
50
0
0
50 x
0
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Fig. 3. The collision of two gBL lump solitons of equal amplitude traveling at ±5◦ to the horizontal. The domain is 30π by 150π and the mesh is 512 by 512. We used ∆t = 0.05 and = 0.1. The dotted lines show the direction of travel before and after the collision.
An angle-dependent phase shift due to the nonlinear interaction was observed and is shown in Figure 4. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision, which is approximately ∆x = 3.4 and ∆y = ±10.7. In other words, we observe a phase lead in x and a phase lag in y for both solitons. In relation to the grid resolution for this computation, which is hx = 0.184, hy = 0.920, the measured phase shifts are significant. The results of a similar 5◦ collision using a soliton of amplitude A = −0.4992 (as in Figure 4) and 2A are shown in Figure 5. Here the phase shifts are approximately ∆x = 4.0 (lead) and ∆y = 8.9 (lag) for the soliton of amplitude A and ∆x = 7.4 (lead) and ∆y = −1.2 (lag) for the 2A soliton. Figure 6 shows the results for the collision of two lump solitons of equal amplitude at an angle of ±45◦ to the horizontal. At this large angle, the solitons pass quickly through one another and produce no measurable phase shift. 5. Generation of lump solitons by resonant flow over a bump. Given that the gBL equation supports lump solitons and that these appear to be stable under collision with other solitons, the next logical step is to study whether they can be generated in more realistic flow conditions. The forced KdV and KP equations have
739
LUMP SOLITARY WAVES
350
300
y
250
200
INTERACTION REGION
150
0
500
1000
1500
2000
x
Fig. 4. The path of the center of two lump solitons of equal amplitude traveling from left to right at ±5◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision. Tracking the center of each soliton is impossible in the interaction region.
350
300
y
250
INTERACTION REGION
200
150
0
500
1000
1500
2000
x
Fig. 5. The path of the center of two lump solitons of amplitude A (starting from upper left) and 2A traveling from left to right at ±5◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton at T = 100 intervals from T = 0 to T = 2400 based on the soliton’s known speed. The x’s mark the actual computed location and illustrate the phase shift due to the collision. Tracking the center of each soliton is impossible in the interaction region.
740
KURT M. BERGER AND PAUL A. MILEWSKI
300
280
260
240
220
y
200
180
INTERACTION REGION
160
140
120
100
80 180
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240
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320
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360
x
Fig. 6. The path of the center of two lump solitons of amplitude A traveling from left to right at ±45◦ to the positive x-axis. The circles indicate the predicted location of the center of each lump soliton based on the soliton’s known speed. The x’s mark the actual computed location and illustrate a negligible phase shift.
been used to study free surface flow over topography (see Wu and Wu [18], Akylas [2], Cole [4], Grimshaw and Smyth [6], and Katsis and Akylas [7], among others). Under certain conditions the forced KdV equation will produce solitary waves over the forcing topography. Recently, Nadiga, Margolin, and Smolarkiewicz [14] compared the generation of solitary wave solutions to various shallow-water equations, including forced versions of the KdV, Green–Naghdi, and Boussinesq equations, to those of the full Euler equations for flow over variable topography. They showed that while details of the flow differ, the generation of solitary waves remains qualitatively correct. We study here whether the (forced) gBL equation could, under the right conditions, produce three-dimensional lump solitons. Milewski and Tabak [12] previously computed time-dependent solutions to the forced gBL equation for zero surface tension, outside the regime of lump solitons. We consider the free surface flow past a submerged obstacle in a channel of finite
741
LUMP SOLITARY WAVES
z
0.99
0.995
1 50
y
0
0
10 x
20
30
Fig. 7. Bottom topography used in numerical experiments for flow over a bump. The bottom is given by z = −1 − 2 h(x, y) in which h(x, y) = −sech2 .302 (x − x0 )2 + .152 (y − y0 )2 . Here we use = 0.1.
depth. Upstream from the obstacle, the flow is uniform with a constant velocity U in the positive x direction and of constant depth H. We let z = −H − h(x, y) denote the bottom of the channel so that a positive obstacle corresponds to h < 0. Milewski [10] derived the forced version of the gBL equation, repeated here in its more general form, to model the three-dimensional evolution of the free surface for this problem:
(5.1) Φtt − ∆Φ + a∆2 Φ − b∆Φtt + Φt ∆Φ + (∇Φ)2t ) = F hx (x + F t, y). Recall that a − b = B − 13 and F is the Froude number defined as F = CU0 . Note that this equation applies in a frame moving with the mean velocity of the flow and is therefore still isotropic. In this frame, the obstacle is moving with a speed −U . Milewski and Tabak [12] computed time-dependent solutions to this equation for B = 0, F = 1 for a three-dimensional bump of the form h(x, y) = sech2 (x2 + y 2 ). In the derivation of (1.1), the length scale of the obstacle h(x, y) is 2 H so that 2 the bottom of the channel is given by For our purposes we use z 2= −1 − 2 h(x, y). 2 the positive bump h(x, y) = −sech .30 (x − x0 ) + .152 (y − y0 )2 , which is shown in Figure 7. Our interest is in the creation of three-dimensional lump solitons through uniform flow over a bump. Lump solitons are solutions to the unforced gBL equation only when both surface tension and nonlinear effects are significant. Since flow over topography near the critical speed leads to large free-surface displacement and thus nonlinear effects, we expect the possibility of lump solitons only for B > 13 , F ≈ 1. Before considering solutions to the forced gBL equation at the critical speed, however, we first examine the linear solution at a range of flow speeds. 5.1. Linear solution. Ignoring the two nonlinear terms in (5.1), we can rewrite this equation in operator form as (∂tt + L2 )Φ = G(h), in which L2 = (1 − b∆)−1 (−∆ + a∆2 )
742
KURT M. BERGER AND PAUL A. MILEWSKI
and G(h) = (1 − b∆)−1 (F hx (x + F t, y)) . We factor the left side by introducing u = (∂t − iL)Φ and recast the equation in terms of u as ut + iLu = G(h).
(5.2)
Given u(x, y, t), the free surface in a frame moving with the mean velocity of the flow is, to leading order, η = −Φt = −Re(u). Equation (5.2) can be solved as an initial value problem if we assume that the bump is introduced at t = 0 and η(x, y, 0) = Φ(x, y, 0) = 0, which implies u(x, y, 0) = 0. Taking the two-dimensional Fourier transform of (5.2) gives an equation for u ˆ(k) in which k = (k, l): ˆ ˆ u = G( ˆ h) u ˆt + iLˆ
(5.3) with ˆ= L
|k|2 + a|k|4 1 + b|k|2
12
,
|k|2 = k 2 + l2 ,
and ˆ = ˆ h) G(
ikF ˆ h(k)eikF t . 1 + b|k|2
Equation (5.3) is easily solved in a frame fixed with respect to the bump. Letting v(x + F t, y, t) = u(x, y, t) be the solution in the frame of the bump, the solution is ˆ h(k) kF ˆ (5.4) [1 − e−i(L(k)+F k)t ]. vˆ(k) = 2 ˆ 1 + b|k| F k + L(k) ˆ The linear dispersion relation ω(k) = L(k) for the gBL equation, given above, shows that the linear problem is well-posed only if the parameters a and b are nonnegative. Figure 8 shows the phase speed in the x direction as a function of k for B = 0 and B = 23 for the unmodified dispersion relation (b = 0), and for B = 23 with b = 12 . For B < 13 the free surface evolution is dominated by gravity and the wave speed increases with wavelength. For B > 13 the free surface evolution is dominated by surface tension and the wave speed decreases with wavelength. The b = 0 curve is close to the b = 0 curve when k is O(1), but they differ as k becomes large, a regime that is outside the physical range of validity of the gBL equation. Thus the effect of b > 0 is to bound the speed of the high wavenumbers, which are not well approximated by the gBL equation, to give more manageable numerical simulations. The solution of (5.4) has two regimes, subcritical (F > 1) and supercritical (F < 1). Note that these definitions are the opposite of those used for gravity flows due to the difference in the linear dispersion relations. In both cases the subcritical regime consists of Froude numbers for which there exist linear wave speeds, i.e., there exist steady waves in the frame of the bump. The linear solution to (5.4) for subcritical flows (F > 1) contains a steady wave pattern given by the family of wavenumbers (K cos θ, K sin θ) for −1 −1 −1 −1 cos < θ < 2π − cos F F
743
LUMP SOLITARY WAVES 1.5 B = 2/3, b = 0
cx
B = 2/3, b = 1/2 1 B = 0, b = 0
0.5
0
0.5
1
1.5
2
2.5 k
3
3.5
4
4.5
Fig. 8. Linear dispersion relation cx (k, 0) for the gBL equation for B = 0 and B = b = 0. Also shown is the curve for B = 23 and b = 12 .
5 2 3
with
TIME = 300 250
200
y
150
100
50
0
30
40
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60
70 x
80
90
100
110
Fig. 9. Uniform linear flow past a positive bump for (F = 1.1, B = 23 ) as viewed in the frame of the obstacle. The obstacle is centered at the intersection of the dotted lines.
and
K=
F 2 cos2 θ − 1 . (a − bF 2 cos2 θ) Cg (K)
These waves travel in the xy plane along xy = Cgy (K) lines, with the limiting lines x ). These two lines have an angle with the x-axis given by K = 0 and θ = cos−1 ( −1 F of ±ξ ∗ , with ξ ∗ = tan−1 ( √F12 −1 ). Thus there exist waves upstream of the obstacle outside of the wedge of half-angle ξ ∗ . Figure 9 shows the linear response in the frame of the obstacle for (F = 1.1, B = 23 ) at t = 300 for the positive bump described
744
KURT M. BERGER AND PAUL A. MILEWSKI TIME = 300
250
200
y
150
100
50
0
0
20
40
60
80
100
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Fig. 10. Uniform linear flow past a positive bump for (F = 0.9, B = 23 ). Viewed in the frame of the obstacle, the transient waves are moving to the left leaving a steady depression wave above the obstacle. The obstacle is centered at the intersection of the dotted lines.
above. The half-angle of the wedge in this figure is very close to the predicted value for F = 1.1, which is ξ ∗ = 65.4◦ . In the supercritical regime (F < 1), the steady solution, obtained from the first term in (5.4), is a localized disturbance over the bump as t → ∞. The asymptotic form of the transient waves generated by the bump can be obtained using stationary phase arguments on the second term in (5.4). Figure 10 shows the linear response for a supercritical flow (F = 0.9, B = 23 ) at t = 300 for the positive bump described above. As expected, we observe a stationary depression above the obstacle and transient waves moving off to the left. Nonlinear simulations of these cases produced virtually identical results. In the critical F = 1 case the linear solution will grow unbounded in time. Figure 11 shows the linear response in the frame of the obstacle for (F = 1.0, B = 23 ) at t = 300. The response is localized above the bump with an amplitude roughly an order of magnitude larger than the subcritical and supercritical cases at the same time. 5.2. Nonlinear resonant solutions. Since the linear solution at the critical flow speed F = 1 is unbounded in time and large for F ≈ 1, we expect nonlinear terms to become important in the evolution of the free surface in this regime. Thus we examine the behavior of the forced gBL equation using the pseudospectral method described earlier for soliton collisions. Using a 512 by 512 mesh and the same bottom topography, we observe the symmetric pairwise periodic generation of lump solitons downstream of the obstacle for B > 13 and F ≈ 1. The lump solitons are generated along the lines which form angles ±ξs with respect to the positive x-axis, and their amplitude and period of generation depend on the speed of the flow. Figures 12
745
LUMP SOLITARY WAVES TIME = 300 250
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0
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80
90
Fig. 11. Uniform linear flow past a positive bump for (F = 1.0, B = of the obstacle. The amplitude of the response grows with time.
2 ) 3
100
110
as viewed in the frame
and 13 show the free surface evolution at four different times for (F = 1, B = 23 ) for the positive sech2 obstacle. Lump solitons emerge at an angle above and below the downstream direction in the frame of the obstacle. In this frame of reference, the angle of travel with respect to the positive x-axis is approximately ±65◦ and the speed is c ≈ 0.168. In a frame traveling with the mean velocity of the water (F = 1), the angle is α ≈ ±9.2◦ with respect to the upstream direction (negative x-axis) and the speed is c ≈ 0.937. With a minimum amplitude of A ≈ −2.95, these solitons are represented by the point (|A|, c) = (.295, .937), which is indicated on the speed-amplitude curve for the gBL equation given in Figure 2. Lump solitons are also generated for close to critical flow speeds F ≈ 1. Figure 14 shows the development of solitons for F = 0.98, using the same positive topography, at T = 500 and T = 900. In comparison to the F = 1 case at T = 500 (see Figure 13), the F = 0.98 solitons develop at a slower rate. Figure 15 shows the free surface evolution for F = 1.02. Note that the solitons emerge sooner and travel somewhat faster in the frame of the bump due to the increased mean flow. In this case the angle ξs is approximately 65◦ and lies inside the “shadow” wedge given by ξ ∗ = 79◦ for F = 1.02. 6. Conclusion. Three-dimensional solitary waves or lump solitons are shown to be solutions to the isotropic Benney–Luke equation when surface tension effects are significant. We have established an explicit connection between the lump solitons of the KP-I equation and approximate lump solitons of the gBL equation. Moreover, we have numerically computed exact gBL lump solitons and their speed-amplitude relation. We then numerically collide two gBL lump solitons in time to illustrate their solitary wave behavior. Finally, we show that three-dimensional lump solitons are generated in the uniform flow over an obstacle near the critical speed when surface tension is significant.
746
KURT M. BERGER AND PAUL A. MILEWSKI
TIME = 100.0
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Fig. 12. Uniform nonlinear flow past a positive obstacle for (F = 1, B = 23 ) at t = 100 and t = 300. Solitons emerge at a large angle to the downstream direction as viewed in the frame of the obstacle. The obstacle is centered at the intersection of the dotted lines.
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LUMP SOLITARY WAVES
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TIME = 700.0
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Fig. 13. Uniform nonlinear flow past a positive obstacle for (F = 1, B = 23 ) at t = 500 and t = 700. The obstacle is centered at the intersection of the dotted lines. The minimum amplitude of the right-most solitons at t = 700 is approximately A = −2.95 and their speed in this frame is 0.168.
748
KURT M. BERGER AND PAUL A. MILEWSKI
TIME = 500.0
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Fig. 14. Uniform nonlinear flow past a positive obstacle for (F = 0.98, B = t = 900. The obstacle is centered at the intersection of the dotted lines.
2 ) 3
at t = 500 and
749
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TIME = 300.0
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Fig. 15. Uniform nonlinear flow past a positive obstacle for (F = 1.02, B = t = 500. The obstacle is centered at the intersection of the dotted lines.
2 ) 3
at t = 300 and
750
KURT M. BERGER AND PAUL A. MILEWSKI REFERENCES
[1] M. J. Ablowitz and P. A. Clarkson, Solitons, nonlinear evolution equations and inverse scattering, Cambridge University Press, Cambridge, UK, 1991. [2] T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution, J. Fluid Mech., 141 (1984), pp. 455–466. [3] D. J. Benney and J. C. Luke, Interactions of permanent waves of finite amplitude, J. Math. Phys., 43 (1964), pp. 309–313. [4] S. L. Cole, Transient waves produced by flow past a bump, Wave Motion, 7 (1985), pp. 579–587. [5] A. S. Fokas and M. J. Ablowitz, On the inverse scattering and direct linearizing transforms for the Kadomtsev-Petviashvili equation, Phys. Lett., 94 (1983), pp. 67–70. [6] R. H. Grimshaw and N. Smyth, Resonant flow of a stratified fluid over topography, J. Fluid Mech., 169 (1986), pp. 429–464. [7] C. Katsis and T. R. Akylas, On the excitation of long nonlinear water waves by a moving pressure distribution. Part 2. Three-dimensional effects, J. Fluid Mech., 177 (1987), pp. 49– 65. [8] B. Malomed and J.-M. Vanden-Broeck, Solitary wave interactions for the fifth-order KdV equation, Mathematical Problems in the Theory of Water Waves, Contemp. Math. 200, AMS, Providence, RI, 1996, pp. 133–143. [9] S. V. Manakov, V. E. Zakhorov, L. A. Bordag, A. R. Its, and V. B. Matveev, Twodimensional solitons of the Kadomtsev-Petviashvili equation and their interaction, Phys. Lett., 63 (1977), pp. 205–206. [10] P. A. Milewski, A formulation for water waves over topography, Stud. Appl. Math., 100 (1998), pp. 95–106. [11] P. A. Milewski and J. B. Keller, Three-dimensional water waves, Stud. Appl. Math., 37 (1996), pp. 149–166. [12] P. A. Milewski and E. G. Tabak, A pseudospectral procedure for the solution of nonlinear wave equations with example from free-surface flows, SIAM J. Sci. Comput., 21 (2000), pp. 1102–1114. [13] P. A. Milewski and J.-M. Vanden-Broeck, Time dependent gravity-capillary flows past an obstacle, Wave Motion, 29 (1999), pp. 63–79. [14] B. T. Nadiga, L. G. Margolin, and P. K. Smolarkiewicz, Different approximations of shallow fluid flow over an obstacle, Phys. Fluids, 8 (1996), pp. 2066–2077. [15] R. L. Pego and J. R. Quintero, Two-dimensional solitary waves for a Benney-Luke equation, Phys. D, 132 (1999), pp. 476–496. [16] J. Villarroel and M. J. Ablowitz, Solutions to the time-dependent Schrodinger and the Kadomtsev-Petviashvili equations, Phys. Rev. Lett., 78 (1997), pp. 570–573. [17] T. Y. Wu, Generation of upstream advancing solitons by moving disturbances, J. Fluid Mech., 184 (1987), pp. 75–99. [18] T. Y. Wu and D. M. Wu, Generation of upstream advancing solitons by moving disturbances, in Proceedings of the 14th Symposium on Naval Hydrodynamics, National Academy of Sciences, Washington, D.C., 1982, pp. 103–125.
