STUDIES IN RECURRENCE IN NONLINEAR DISPERSIVE WAVE EQUATIONS

Transcription

1 STUDIES IN RECURRENCE IN NONLINEAR DISPERSIVE WAVE EQUATIONS T. Arbogast, J. L. Bona, and J.-M. Yuan December 25, 29 Abstract This paper is concerned with the study of recurrence phenomena in nonlinear dispersive wave equations using numerical simulation. Two classes of equations are investigated, namely, the generalized KdV-equations, u t + u p u x + ɛu xxx =, and the generalized RLW- or BBM-equations, u t + u x + u p u x ɛu xxt =. The numerical evidence presented here shows that even non-integrable equations exhibit recurrence phenomena. We also determine an approximate relationship between recurrence time and amplitude of the initial data. 1 Introduction In the last few decades, a new paradigm has arisen in mathematics. In this conception, numerical simulation and experimentation have been added to the arsenal of techniques available to aid our understanding of mathematical issues. Such a paradigm is also common in physics, and more recently the other sciences and even some of the social sciences. It is distinct from the use of computation in engineering and also in science and social science, where the goal is to make direct, numerically-valued predictions which are used in forecasting, design, control and the like. Early examples of the use of computers to uncover mathematical information include the famous Birch and Swinnerton-Dyer conjectures in number theory and the pioneering work on what came to be known as solitons by Fermi, Pasta, and Ulam. Despite the relative lack of computational power in the 195 s, these works and others like them have proved to be of lasting importance. In the present study, we continue in the line pioneered by Fermi, Pasta and Ulam [16] and later by Zabusky and Kruskal [38]. In [16], numerical simulations indicated that a mass and spring model with non-hooksian springs (modelled as an an-harmonic lattice) features near recurrence of an initial perturbation. This model, which had small cubic and quartic additions to the usual harmonic potential governing the springs, was initially conceived as a possible model for heat conduction. With a sinusoidal initial disturbance, one expected a kind of thermalization, or equipartition of energy into all the modes of oscillation. Thus 1

2 the near recurrence of the initial state was unexpected. Upon reflection, this outcome might not seem surprising in the light of Poincaré s famous observations of chaotic patterns in the dynamics of planetary interactions and his associated recurrence theorem. However, the recurrence observed by Fermi, Pasta and Ulam was different from Poincaré recurrence. For one thing, the recurrence they observed happens on a much shorter time scale. Zabusky and Kruskal [38] revisited the Fermi-Pasta-Ulam system (henceforth the FPU system) in They took a continuum limit of the FPU lattice and came to the Korteweg-de Vries equation (KdV-equation) u t + uu x + ɛu xxx =. (1) Following Fermi et al., they posed this equation as an initial-value problem with periodic boundary conditions and simulated the problem numerically with ɛ =.22 and u(x, ) = cos(πx), x 2. They also found recurrence of initial states for (1). Later, Zabusky [37] pursued in more detail the recurrence phenomenon for the KdV-equation. Fixing the initial data as u(x, ) = cos(2πx), he denoted by t B, the time until shock formulation for (1) with ɛ =, and by t R = t R ( ɛ) the recurrence time for (1) with the given value of ɛ. On the basis of numerical studies of Tappert [33] using a split-step Fourier method, Zabusky suggested the relationships and t R t B =.7 ɛ (2) the number of emergent solitons =.2 ɛ. (3) Indeed, this was another finding of Zabusky and Kruskal s numerical study [38], that general classes of initial data resolved into the special travelling waves called solitary waves by Scott Russell [3], first written analytically by Boussinesq [13], and later termed solitons by Zabusky and Kruskal. Theoretical support for these relationships was later put forward by Lax [21], Novikov [27], Toda [34], McKean & Trubowitz [22], McKean [23], and Bourgain [12]. The recurrence phenomenon for the periodic initial-value problem for the KdV-equation (1) has been linked to the existence of quasi-periodic motion of completely integrable Hamiltonian systems. It is natural to inquire as to whether or not other infinite-dimensional Hamiltonian systems might have the same property. This is the principal perspective of the present study, which is made in the context of the generalized KdVequation (GKdV equation henceforth), u t + u p u x + u xxx =, (4) 2

3 and its counterpart, the generalized BBM-equation (GBBM-equation) u t + u x + u p u x u xxt =. (5) These equations are all Hamiltonian. However, except for (4) with p = 1 or 2, they do not appear to be integrable. In particular, their solitary-wave solutions do not interact exactly as they do for (4) with p = 1 or 2 (see e.g. [8]). However, the GKdV equation for p = 3 and the GBBM-equations for all p seem to have the property of resolution into solitary waves. More precisely, and following Lax [2], for suitably restricted initial values u(, ) = ψ, the solution u emanating therefrom has the property that u(x, t) = k φ ci (x c i t + θ i (t)) + r(x, t) i=1 where φ ci is a solitary-wave solution of speed c i, θ i (t) θ i ( ) as t for 1 i k, and lim t sup x R r(x, t) =. (Thus, while the L 2 -norm of r may remain positive as t, its L -norm tends algebraically to zero, as is characteristic of dispersive phenomena.) It is thought that the property of near recurrence is closely related to the property of resolution into solitary waves. In a different, completely integrable context, Camassa and Lee [14] have numerical simulations indicative of this point. It will follow from this that in cases where the remainder r has significant energy (i.e. its L 2 -norm is a significant fraction of the L 2 -norm of the initial data), the recurrence is less noticeable. The paper is organized as follows. In Section 2, the numerical methods are introduced and convergence and accuracy ascertained. Section 3 is devoted to the recurrence for the generalized KdV equations. In Section 4 the recurrence for the generalized BBM-equation is presented. The overall conclusions are summarized in Section 5. 2 A Numerical Method It is our purpose to investigate numerically recurrence phenomenon for the nonlinear dispersive wave equations (4) and (5), with p a positive integer. The numerical schemes used are based on a q-stage Gauss-Legendre method for the temporal discretization and a finite-element method for the spatial discretization (see [7] and its precursor [6]). The following convergence theorem for such a method applied to (4) was proved by Bona et al. in [7]. Theorem 2.1. Suppose that, as the spatial discretization parameter h +, the temporal discretization k obeys the hypotheses 1. if p = 1, k = O(h 3/(2(q+2)) ) for q 2 or k = O(h 3/4 ) for q = 1, 2. if p = 2, kh 1/2 is sufficiently small for q 2 or k = O(h 3/4 ) for q = 1, 3

4 3. if p 3, kh 1 is sufficiently small for all q 1. Then for h sufficiently small, there exists a unique solution U n of the fully discrete numerical scheme, which is a q-stage Gauss-Legendre time stepping coupled with smooth splines of degree r 1 finite element approximation in space, such that max U n u n c(k 2 + h r ) for q = 1, (6) n J max U n u n c(k q+2 + h r ) for q 2, (7) n J where J is fixed, r 2, U n represents the numerical solution at time t = nk and u n represents the analytical solution at the same moment. A theorem of this nature is not only of academic interest. It tells us what to expect from a properly written and coded numerical scheme. If we do not see these rates of convergence, we can be pretty sure something is awry. In any event, as we are asking the reader to believe the results of our simulations, it behooves us to provide evidence. In the earlier work [7], such tests were performed on solitary-wave solutions. That was appropriate, as perturbations of solitary waves figured prominently. Here the simulations feature interaction of solitary-wave structures and so we deemed it more appropriate to test the code on multi-soliton solutions. Consider first the KdV equation rescaled in the form and pose it with the initial datum u t + 1uu x +.25u xxx = (8) u(x, ) = cosh(4 1(x.49)) + cosh(8 1(x.13)) (3 cosh(2 1(x +.23)) + cosh(6 1(x.25))) 2. Using Inverse Scattering theory [17, 18], the exact solution on R of (8) with the above initial datum is seen to be u(x, t) = cosh(4 1(x 4t.49)) + cosh(8 1(x 16t.13)) (3 cosh(2 1(x 28t +.23)) + cosh(6 1(x 12t.25))). 2 (9) As t, and along the line x 16t = constant, the asymptotic behavior of u(x, t) is 4.8 sech 2 (4 1(x 16t x 2 ). If instead, x 4t is fixed, the asymptotic behavior of u(x, t) will look like 1.2 sech 2 (2 1(x 4t x 1 )). Since the error term is exponentially small, it is straightforward to deduce that as t, u(x, t) 4.8 sech 2 (4 1(x 16t x 2 )) sech 2 (2 1(x 4t x 1 )) (1) The amplitudes of the two emerging solitons are thus 4.8 and 1.2, respectively. As is well-known, phase shifts appear during the interaction. The taller soliton has moved forward and the shorter one backward, compared to the positions 4

5 they would have reached if there had been no interaction. The phase shifts for the solution in (9) from the formula in [35] are forward shift of the larger soliton after interaction;.88685, backward shift of the smaller soliton after interaction; As a promising preliminary indication of accuracy, the phase shifts after the interaction in our numerical tests turn out to be forward shift of the larger soliton after interaction;.9, backward shift of the smaller soliton after interaction;.17. We compare the exact 2-soliton solution (9) of the KdV-equation on the real line with the numerical solution on the periodic domain [,1] for accuracy (see [4]). The outcome of this study is reported in Table 1 with h =.1 and k =.25. It shows good accuracy as expected from Theorem 2.1. Note that the taller soliton catches up and interacts with the shorter one around t =.31. Normalized L 2 error. 2.6E E E E E E E-6 Table 1: Error between the exact 2-soliton solution and the numerical solution 3 Recurrence for the generalized KdV equations Equation (4) with p = 2 is usually called the modified-kdv-equation, and like the KdV-equation (3), it is in a particular sense completely integrable. For p 3, as far as one can tell, there is no analog of the complete integrability that holds when p = 1 or 2 (see e.g.[9]). Results of McLeod and Olver [28] indicate that (5) with p = 1 is not integrable, and numerical simulations show convincingly that the solitary-wave solutions of (5) for various values of p are not solitons, which is to say they do not interact exactly. Indeed, they are not all stable for p > 4 (see [8], [11], and [25]). On the other hand, the KAM-theory of Kolmogoroff, Arnold, Moser and Siegel (see [2]) states that a weakly perturbed, completely integrable system retains the property of possessing quasi-periodic solutions. Hence at least in certain parameter regimes, the recurrence of initial states might be expected to persist for suitable Hamiltonian perturbations of (4) or (5) with p = 2, say. A Hamiltonian for the generalized KdV-equation (4) is 5

6 H(u, u x ) = 1 [ ] ɛu2 x (x, t) p + 2 up+2 (x, t) dx. (11) It is far from clear that the case p = 3, say, corresponds to a small perturbation of an integrable case p = 1 or 2! We begin with some numerical simulations involving the exact multi-soliton solutions obtained from Inverse Scattering theory. The relevant formulas may be found, for example, in Whitham s text [35]. The exact solutions are ideal for testing the implementation of the numerical scheme in a context that is quite relevant to recurrence phenomena. The following natural quantity will be used throughout in our discussion of recurrence. It is the L 2 (, 1)-norm of the difference between the solution at time t and the initial value, modulo translation and periodicity. Definition 3.1. Let u(x, t) be a solution to one of the evolution equations studied here, emanating from nonzero initial data u(x, ). The shape difference is f(t) u(x,) 2, where f(t) = min θ(t) 1 1 ( u(x + θ(t), ) u(x, t)) 2dx. (12) We take θ() = so that f() =. A recurrence time T r > is a time at which the shape difference is again small. Of course, T r will be at least a local minimum of f. While it is not obvious how to make this notion more precise, it will become clear that in practice these recurrence times do exist. The recurrence time t r > closest to is the first recurrence time. It will transpire that later recurrence times are often more or less integer multiples of t r. 3.1 The KdV equation (p = 1) In the following experiments, we report recurrence results for these test cases: 1. two-solition, 2. four-soliton with dispersive tail. Test 1. (two-soliton) By rescaling, (1) is written in the form u t + 1uu x u xxx = (13) and posed with initial datum u(x, ) = 3.6sech 2 (1 1(x.5)). 6

7 As in (1), the exact solution of (13) on R is u(x, t) = cosh(2 1(x 4t.5)) + cosh(4 1(x 16t.5)) (3 cosh(1 1(x 28t.5)) + cosh(3 1(x 12t.5))) 2. As in (11), as t, (14) u(x, t) 4.8sech 2 (2 1(x 16t x 2 )) + 1.2sech 2 (1 1(x 4t x 1 )). (15) The recurrence results are shown in Table 2 when the problem is posed as a periodic initial-value problem with the initial datum effectively having compact support in the period domain. T r shape difference Table 2: Two-soliton initial data. T r is the computed value of the recurrence time with h =.2 and k = The positions of 2 solitons Figure 1: Two-soliton initial data. The dark line in the figure at the right represents the peak of the smaller soliton. The lighter line tracks the peak of the taller soliton. Remark 3.1. From Table 2 and the graph of the shape difference (see Figure 1), one sees a definite value.79 of the recurrence time and a small linear growth in the shape difference at multiples of the recurrence time. The number of solitons is now increased to ascertain how well the program functions in a more complex environment. Test 2. (four-soliton plus dispersive wave) We solve u u + 1uu x u xxx = (16) 7

8 with u(x, t) = 9.sech 2 (1 1(x.5)). Using the Inverse Scattering theory, the solution of (16) is seen to have the form u(x, t) sech 2 ( (x 36t x 1 )) sech 2 ( (x 16t x 2 )) sech 2 ( (x 4t x 3 )) sech 2 ( (x 4t x 4 )) + dispersive tail (17) as t. The recurrence result for (16) is in Table 3 and Figure 2. T r shape difference Table 3: Test 2. T r is the computed value of the recurrence time with h =.1 and k = Figure 2: Test 2. Recurrence results for initial data with four-soliton plus a dispersive tail in the KdV-equation 3.2 The modified KdV equation (p = 2) In this subsection, recurrence is studied in the context of the modified KdV equation. Directing attention back to the generalized KdV-equation, the case p = 2 is known to be completely integrable. In each numerical experiment, we report numerically generated approximations just as in the previous section. We approximated solutions of the mkdv equation u t + u 2 u x u xxx = (18) 8

9 with the initial condition u(x, ) = a cos(2π(x.5)) 2, x 1, (19) taking the six cases: a=.6,.75,.875, 1., 1.1, and The recurrence times are given in Tables 4-6. Snapshots of the solutions at recurrence times are available in [36]. a=.6 a= Table 4: a=.6 and.75. T r is the computed value of the recurrence time with h=.2 and k= shape difference Figure 3: The left figure represents the recurrence result for u(x, ) =.6 cos 2 (2π(x.5)). The right figure represents the recurrence result for u(x, ) =.75 cos 2 (2π(x.5)). a=.875 a= Table 5: a=.875 and 1.. T r is the computed value of the recurrence time with h=.2 and k=.1. 9

10 Figure 4: The left figure represents the recurrence result for u(x, ) =.875 cos 2 (2π(x.5)). The right figure represents the recurrence result for u(x, ) = 1. cos 2 (2π(x.5)). a=1.1 a= Table 6: a=1.1 and T r is the computed value of the recurrence time with h=.2 and k= Figure 5: The left figure represents the recurrence result for u(x, ) = 1.1 cos 2 (2π(x.5)). The right figure represents the recurrence result for u(x, ) = 1.25 cos 2 (2π(x.5)). 1

11 amplitude of initial data t r Table 7: The amplitudes of the initial data a cos 2 (2π(x.5)) and their corresponding first recurrence times. A search is made for a relationship between the first recurrence time and the amplitude a. Such a relation is obtained for the fixed value.121 of ɛ and variations of a as indicated. The data is summarized in Table 7. Using least squares fitting of t r = A a B + C, the following equation emerges: t r = (amplitude) (2) It appears that B = 4 3 and C = 22. This result is shown in Figure y = *x**( ) y = *x**( 4/3) The fisrt recurrence time The fisrt recurrence time Amplitude Amplitude Figure 6: The relationship between the first recurrence time of the mkdv equation and various amplitudes of cos 2 (2π(x.5)), according to the data in Table 7 with the solid line being the least squares fit. The left plot shows the full fit to t r = A a B + C; the right plot fits only t r = A a

12 3.3 The generalized KdV equation with p = 3 In this subsection, recurrence is studied in the context of the generalized KdV equation with p = 3. The case p = 2 is known to be integrable, as mentioned before. However, the evidence suggests that for p 3, the evolution equation (4) is not integrable. We use the two-stage diagonally implicit Runge-Kutta (DIRK) method in time and the finite-element method with 1-periodic smooth splines of order 4 (degree 3) in space. We approximated solutions of the equation u t + u 3 u x u xxx = (21) numerically via the scheme just mentioned with the initial conditions u(x, ) = a cos 2 (2π(x.5)), x 1. (22) The amplitude a was given the values: a=.8,.825,.85,.875,.9,.925, and.95. The results are shown in Tables 8-11 and Figures 7-1. a=.8 a= Table 8: a=.8 and.825. T r is the computed value of the recurrence time with h =.2 and k = Figure 7: The left figure is the recurrence result for a=.8. The right figure is the recurrence result for a=

13 a=.85 a= Table 9: a=.85 and.875. T r is the computed value of the recurrence time with h =.2 and k = Figure 8: The left figure is the recurrence result for a=.85. The right figure is the recurrence result for a=.875. a=.9 a= Table 1: a=.9 and.925. T r is the computed value of the recurrence time with h =.2 and k = Figure 9: The left figure is the recurrence result for a=.9. The right figure is the recurrence result for a=

14 T r shape difference Table 11: a=.95. T r is the computed value of the recurrence time with h =.1 and k = amplitude t=2.331 Figure 1: The left figure is the recurrence result for a=.925. The right figure is the solution at the first recurrence time. amplitude of initial data t r Table 12: The amplitudes of the cos 2 initial data and the corresponding first recurrence time of the solution. 14

15 There is clear evidence of recurrence for (22)-(23). Using least squares fitting of t r = A a B +C on the data summarized in Table 12, there appears the equation for which B 11 1 t r = (amplitude) , (23) and C 3 (see Fig. 11). 3.5 y = *x**( ) y = 5.5*x**( 11/1) 3 The fisrt recurrence time The fisrt recurrence time Amplitude Amplitude Figure 11: The relationship between the first recurrence of GKdV and various amplitudes of cos 2 (2π(x.5)) according to the data in Table 12 with least squares fitting of t r = A a B + C on the left and t r = A a on the right. Remark 3.2. From the information acquired in Sections 3.2 and 3.3 (Figs. 6 and 11), one is tempted to hazard the guess t r c p a p 2 +2p 4 p 2 +2p 5 + dp, where c p and d p are constants related to p. 4 Recurrence for the generalized BBM-equation In this section, recurrence is studied in the context of the BBM-equation. In 1972, Benjamin et al. [3] found that, under the same approximations and assumptions that originally led Boussinesq and Korteweg and de Vries to their equation, the partial differential equation u t + u x + uu x ɛu xxt = (24) could equally well be justified as a model. The above equation is called the BBMequation or regularized-long-wave (RLW) equation. While the KdV equation 15

16 is an integrable equation and has infinitely many conservation laws [26], Olver [28] proved that the BBM-equation has only three conservation laws. Bona, Pritchard, and Scott [1] presented a quantitative comparison between the solutions to the initial-value problem for KdV and BBM. In [9], the same authors studied the interaction between two solitary waves of the BBM-equation. They showed that there were dispersive tails generated after the interaction. Thus it is provisionally inferred that the BBM-equation is not integrable. BBM-type equations are not temporally stiff as are explicit discretizations of the KdV-type equations. J. Pasciak [29], B.-Y. Guo and V. Manoranjan [19], and D. Sloan [31] have developed spectral methods for BBM-equation. Pasciak [29] used a fourth order Runge-Kutta method for the temporal discretization. Here we use a finite element method in space and a 2-stage Gauss-Legendre method in time. The three conservation laws for the GBBM-equation u t + u x + u p u x ɛu xxt = (25) are (see [32]) 1 u(x, t)dx = c 1, (26) 1 1 u 2 (x, t)dx + ɛ u 2 (x, t) + 1 u 2 x (x, t)dx = c 2, (27) 2 (p + 1)(p + 2) up+2 (x, t)dx = c 3. (28) As the second conservation law (27) is equivalent to the H 1 -norm, it seems natural to use that functional to measure recurrence in the equations (25). Definition 4.1. Let u(x, t) be a solution emanating from initial data u(x, ). g(t) The shape difference for the BBM-type equation is u(x,), where H 1 and g(t) = +ɛ { 1 min θ(t) 1 1 u(x, ) H 1 = ( u(x + θ(t), ) u(x, t) ) 2dx ( ux (x + θ(t), ) u x (x, t) ) 2 dx } The BBM-equation (p=1) First, solutions of the BBM-equation (29) (u 2 (x, ) + ɛu 2 x (x, ) ) dx. (3) u t + u x + uu x 1 4 u xxt = (31) 16

17 were approximated numerically with the initial condition u(x, ) = a cos 2 (2π(x.5)), x 1. (32) Experiments were run with a =.75,.875, 1, 1.25, 1.5, and 2. Results appear in Tables and Figures a=.75 a= Table 13: a =.75 and.875. T r is the computed value of the recurrence time with h =.4 and k = Figure 12: The left figure is the recurrence result for a=.75. The right figure is the recurrence result for a=.875. a=1. a= Table 14: a=1. and T r is the computed value of the recurrence time with h =.2 and k =.1. 17

18 Figure 13: The left figure is the recurrence result for a=1.. The right figure is the recurrence result for a=1.25. a=1.5 a= Table 15: a=1.5 and 2.. T r is the computed value of the recurrence time with h =.2 and k = Figure 14: The left figure is the recurrence result for a=1.5. The right figure is the recurrence result for a=2.. 18

19 amplitude of the initial data t r Table 16: The amplitudes of the initial data and their corresponding first recurrence times. A relationship may be deduced between the first recurrence time and the amplitude of the cos 2 initial data for the BBM-equation with fixed ɛ =.1. From the data, summarized in Table 16, least squares fitting gives the equation (see Figure 15) of t r = A a B + C, t r = (amplitude) (33) 2 y = *x**( ) The fisrt recurrence time Amplitude Figure 15: The relationship between the first recurrence of BBM and the amplitude of cos 2 (2π(x.5)). 4.2 The modified BBM-equation (p=2) In this subsection, recurrence is studied in the context of the modified BBMequation. Solutions of the mbbm-equation u t + u x + u 2 u x ɛu xxt =, (34) were approximated numerically, where ɛ = 1 4, with the initial condition u(x, ) = a cos 2 (2π(x.5)), x 1. (35) 19

20 Six simulations were run with a =.75,.85,.9, 1, 1.15, and Results are shown in Tables and Figures a=.75 a= Table 17: a=.75 and.85. T r is the computed value of the recurrence time with h =.4 and k = Figure 16: The left figure is the recurrence result for a=.75. The right figure is the recurrence result for a=.85. a=.9 a= Table 18: a=.9. T r is the computed value of the recurrence time with h =.2 and k =.1. As before, a relationship is inferred between the first recurrence time and the amplitude of the cos 2 initial data for the modified BBM-equation with fixed ɛ =.1. From the data in Table 2, least squares fitting yields which is plotted in Figure 19. t r = (amplitude) , (36) 2

21 Figure 17: The left figure is the recurrence result for a=.9. The right figure is the recurrence result for a=1.. a=1.15 a= Table 19: a=1.15 and T r is the computed value of the recurrence time with h =.2 and k = Figure 18: The left figure is the recurrence result for a=1.15. The right figure is the recurrence result for a=1.25. amplitude of the initial data t r Table 2: The amplitudes of the initial data and their corresponding first recurrence times. 21

22 8.5 y = *x**( ) The fisrt recurrence time Amplitude Figure 19: The relationship between the first recurrence of mbbm and the amplitude of cos 2 (2π(x.5)). The solid line is the least-squares approximation of the data. 4.3 The generalized BBM-equation with p = 3 In this subsection, the generalized BBM-equation u t + u x + u 3 u x ɛu xxt = (37) is considered with u(x, ) = cos 2 (2π(x.5)), different values of ɛ and periodic boundary conditions on [,1]. In the cubic case where the nonlinearity is u 3 u x, this term is much smaller that the other terms when u < 1, and so the equation is essentially linear. Recurrence on relatively short time scales requires nonlinearity, however. In this section we have elected to vary the parameter ɛ instead of the amplitude of the initial data. Simple rescaling shows that varying ɛ is equivalent to varying the period of the initial data. The 9 cases ɛ =.5,.6,.75,.825,.95,.1,.175,.35, and.5 are integrated numerically. The outcomes are displayed in Tables and Figures

23 ɛ=.5 ɛ= Table 21: ɛ=.5 and ɛ=.6. T r is the computed value of the recurrence time with h=.2 and k= Figure 2: The left figure is the recurrence result for ɛ=.5. The right figure is the recurrence result for ɛ=.6. ɛ=.75 ɛ= Table 22: ɛ=.75 and ɛ=.825. T r is the computed value of the recurrence time with h =.2 and k = Figure 21: The left figure is the recurrence result for ɛ=.75. The right figure is the recurrence result for ɛ=

24 ɛ=.95 ɛ= Table 23: ɛ=.95 and ɛ=.1. T r is the computed value of the recurrence time with h=.2 and k= Figure 22: The left figure is the recurrence result for ɛ=.95. The right figure is the recurrence result for ɛ=.1. ɛ=.175 ɛ= Table 24: ɛ=.175 and ɛ=.35. T r is the computed value of the recurrence time with h=.2 and k=.1. The data is summarized in Table 26. Using least squares fitting, there obtains the equation: t r = 4.84(1ɛ) (38) This is plotted with the data from Table 26 in Figure

25 Figure 23: The left figure is the recurrence result for ɛ=.175. The right figure is the recurrence result for ɛ=.35. T r shape difference Table 25: ɛ=.5. T r is the computed value of the recurrence time with h =.2 and k = amplitude t=.96 Figure 24: The left figure is the recurrence result for ɛ=.5. The right figure is the solution at the first recurrence time. 25

26 1ɛ t r Table 26: The relationship between the first recurrence time of GBBM and various values of ɛ. 8 y = 4.84*x**(.841) The fisrt recurrence time ε Figure 25: The various ɛ and their corresponding first recurrence times. 26

27 5 Conclusions For the KdV- and mkdv-equations, recurrence is connected to the property of resolution into solitary waves together with the exact interaction of solitary waves. Studies not reported here show that we do have resolution into solitary waves for the GKdV-equation with p = 3 and for the GBBM-equations. However, it is also known that the solitary-wave solutions of these equations do not interact exactly. Rather, a very small amount of energy is lost to dispersion each time they interact. However, the numerical results presented herein support the notion that these solitary-wave solutions obtain near recurrence of initial states. Because of dissipative energy loss, we expect the recurrence to eventually weaken as less energy is concentrated in solitary waves. Because of the very small relative loss of energy, this decay of recurrence is expected to occur on much longer time scales than those appearing here. Indeed, as far as modelling real wave phenomena is concerned, the time scale for decay of recurrence appears considerably longer than the time scale of formal validity of the model (see e.g. [5]). References [1] M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Transform, SIAM Studies in Appl. Math.: Philadelphia, [2] V. I. Arnold, Mathematical methods of classical mechanics, Springer- Verlag: Berlin, [3] T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. London. Ser. A 272 (1972) [4] J. L. Bona, Convergence of periodic wavetrains in the limit of large wavelength, Appl. Sci. Res. 37 (1981) [5] J. L. Bona, T. Colin and D. Lannes, Long wave approximations for water waves, preprint (24). [6] J. L. Bona, V. A. Dougalis and O. A. Karakashian, Fully discrete Galerkin methods for the Korteweg-de Vries equation, J. Comp. Math. Applic. 12A (1986) [7] J. L. Bona, V. A. Dougalis, O. A. Karakashian and W. R. McKinney, Conservative, high-order numerical schemes for the generalized Kortewegde Vries equation, Philos. Trans. Royal Soc. London. Ser. A 351 (1995) [8] J. L. Bona, W. R. McKinney and J. M. Restrepo, Stable and unstable solitary-wave solutions of the generalized regularized long-wave equation, J. Nonlinear Science 1 (2)

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