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Dispersive and soliton perturbations of finite-genus solutions of the KdV equation: computational results. (English) Zbl 1331.35312

Summary: All solutions of the Korteweg-de Vries equation that are bounded on the real line are physically relevant, depending on the application area of interest. Usually, both analytical and numerical approaches consider solution profiles that are either spatially localized or (quasi-)periodic. In this paper, we discuss a class of solutions that is a nonlinear superposition of these two cases: their asymptotic state for large \(| x|\) is (quasi-)periodic, but they may contain solitons, with or without dispersive tails. Such scenarios might occur in the case of localized perturbations of previously present sea swell, for instance. Such solutions have been discussed from an analytical point of view only recently. We numerically demonstrate different features of these solutions.

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B20 Perturbations in context of PDEs

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References:

[1] Ablowitz, M.; Segur, H., Solitons and the Inverse Scattering Transform (1981), SIAM: SIAM Philadelphia, PA · Zbl 0472.35002
[2] Segur, H., The Korteweg-de Vries equation and water waves. Solutions of the equation. Part 1, J. Fluid Mech., 59, 04, 721-736 (1973) · Zbl 0265.76027
[3] Hammack, J. L.; Segur, H., The Korteweg-de Vries equation and water waves. II. Comparison with experiments, J. Fluid Mech., 65, 289-313 (1974) · Zbl 0373.76010
[4] Hammack, J. L.; Segur, H., The Korteweg-de Vries equation and water waves. III. Oscillatory waves, J. Fluid Mech., 84, 2, 337-358 (1978) · Zbl 0373.76011
[5] Trogdon, T.; Olver, S.; Deconinck, B., Numerical inverse scattering for the Korteweg-de Vries and modified Korteweg-de Vries equations, Physica D, 241, 1003-1025 (2012) · Zbl 1248.65108
[6] Deconinck, B.; Heil, M.; Bobenko, A.; van Hoeij, M.; Schmies, M., Computing Riemann theta functions, Math. Comput., 73, 1417-1442 (2004) · Zbl 1092.33018
[7] Frauendiener, J.; Klein, C., Hyperelliptic theta-functions and spectral methods: KdV and KP solutions, Lett. Math. Phys., 76, 249-267 (2006) · Zbl 1127.14032
[8] Lax, P. D., Periodic solutions of the KdV equation, Commun. Pure Appl. Math., 28, 141-188 (1975) · Zbl 0295.35004
[9] Trogdon, T.; Deconinck, B., A Riemann-Hilbert problem for the finite-genus solutions of the KdV equation and its numerical solution, Physica D, 251, 1-18 (2013) · Zbl 1278.37050
[10] Trogdon, T.; Deconinck, B., A numerical dressing method for the superposition of solutions of the KdV equation, Nonlinearity (2013), in press
[11] Egorova, I.; Grunert, K.; Teschl, G., On the Cauchy problem for the Korteweg-de Vries equation with steplike finite-gap initial data. I. Schwartz-type perturbations, Nonlinearity, 22, 6, 1431-1457 (2009) · Zbl 1171.35103
[12] Mikikits-Leitner, A.; Teschl, G., Long-time asymptotics of perturbed finite-gap Korteweg-de Vries solutions, J. Anal. Math., 116, 163-218 (2012) · Zbl 1308.35252
[13] McKean, H. P.; Trubowitz, E., Hillʼs surfaces and their theta functions, Bull. Am. Math. Soc., 84, 1042-1085 (1978) · Zbl 0428.34026
[14] Gardner, C. S.; Greene, J. M.; Kruskal, M. D.; Miura, R. M., Method for solving the Korteweg-de Vries equation, Phys. Rev. Lett., 19, 1095-1097 (1967) · Zbl 1103.35360
[15] Fokas, A. S., A Unified Approach to Boundary Value Problems (2008), SIAM: SIAM Philadelphia, PA · Zbl 1181.35002
[16] Doktorov, E. V.; Leble, S. B., A Dressing Method in Mathematical Physics, Math. Phys. Stud., vol. 28 (2007), Springer: Springer Dordrecht · Zbl 1142.35002
[17] Zakharov, V. E., On the dressing method, (Inverse Methods in Action (Montpellier, 1989), Inverse Probl. Theoret. Imaging (1990), Springer: Springer Berlin), 602-623
[18] Trogdon, T.; Deconinck, B., The solution of linear constant-coefficient evolution PDEs with periodic boundary conditions, Appl. Anal., 91, 529-544 (2012) · Zbl 1242.35011
[19] Olver, F. W.J.; Lozier, D. W.; Boisvert, R. F.; Clark, C. W., NIST Handbook of Mathematical Functions (2010), Cambridge University Press · Zbl 1198.00002
[20] Ablowitz, M. J.; Fokas, A. S., Complex Variables: Introduction and Applications (2003), Cambridge University Press · Zbl 1088.30001
[21] Trogdon, T.; Olver, S., Numerical inverse scattering for the focusing and defocusing nonlinear Schrödinger equations, Proc. R. Soc. A, 469 (2013) · Zbl 1372.65356
[22] Olver, S.; Trogdon, T., Nonlinear steepest descent and the numerical solution of Riemann-Hilbert problems, Commun. Pure Appl. Math. (2013), in press
[23] Olver, S., A general framework for solving Riemann-Hilbert problems numerically, Numer. Math., 122, 2, 305-340 (2012) · Zbl 1257.65014
[24] Olver, S., Numerical solution of Riemann-Hilbert problems: Painlevé II, Found. Comput. Math., 11, 153-179 (2010) · Zbl 1214.30026
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