Analysis & PDE

• Anal. PDE
• Volume 4, Number 4 (2011), 573-638.

Soliton dynamics for generalized $\mathrm{KdV}$ equations in a slowly varying medium

Claudio Muñoz

Abstract

We consider the problem of existence and global behavior of solitons for generalized Korteweg–de Vries equations (gKdV) with a slowly varying (in space) perturbation. We prove that such slowly varying media induce on the soliton dynamics large dispersive effects at large times. We also prove that, unlike the unperturbed case, there is no pure-soliton solution to the perturbed gKdV.

Article information

Source
Anal. PDE, Volume 4, Number 4 (2011), 573-638.

Dates
Revised: 3 June 2010
Accepted: 13 July 2010
First available in Project Euclid: 20 December 2017

https://projecteuclid.org/euclid.apde/1513731169

Digital Object Identifier
doi:10.2140/apde.2011.4.573

Mathematical Reviews number (MathSciNet)
MR2872119

Zentralblatt MATH identifier
1264.35188

Citation

Muñoz, Claudio. Soliton dynamics for generalized $\mathrm{KdV}$ equations in a slowly varying medium. Anal. PDE 4 (2011), no. 4, 573--638. doi:10.2140/apde.2011.4.573. https://projecteuclid.org/euclid.apde/1513731169

References

• N. Asano, “Wave propagation in non-uniform media”, Progr. Theoret. Phys. Suppl. 55 (1974), 52–79.
• T. B. Benjamin, “The stability of solitary waves”, Proc. Roy. Soc. $($London$)$ Ser. A 328 (1972), 153–183.
• H. Berestycki and P.-L. Lions, “Nonlinear scalar field equations, I: Existence of a ground state”, Arch. Rational Mech. Anal. 82:4 (1983), 313–345.
• J. L. Bona, W. G. Pritchard, and L. R. Scott, “Solitary-wave interaction”, Phys. Fluids 23 (1980), 438–441.
• J. L. Bona, P. E. Souganidis, and W. A. Strauss, “Stability and instability of solitary waves of Korteweg–de Vries type”, Proc. Roy. Soc. London Ser. A 411:1841 (1987), 395–412.
• T. Cazenave and P.-L. Lions, “Orbital stability of standing waves for some nonlinear Schrödinger equations”, Comm. Math. Phys. 85:4 (1982), 549–561.
• S. I. Dejak and B. L. G. Jonsson, “Long-time dynamics of variable coefficient modified Korteweg–de Vries solitary waves”, J. Math. Phys. 47:7 (2006), 072703, 16.
• S. I. Dejak and I. M. Sigal, “Long-time dynamics of KdV solitary waves over a variable bottom”, Comm. Pure Appl. Math. 59:6 (2006), 869–905.
• E. Fermi, J. Pasta, and S. Ulam, “Studies of nonlinear problems I”, technical report LA1940, Los Alamos National Labs, 1955. reprinted as pp. 143–156 in Nonlinear Wave Motion, edited by A. C. Newell, Amer. Math. Soc., Providence, RI, 1974.
• J.-C. Fernandez, C. Froeschlé, and G. Reinisch, “Adiabatic perturbations of solitons and shock waves”, Phys. Scripta 20:3-4 (1979), 545–551.
• J. Fr öhlich, S. Gustafson, B. L. G. Jonsson, and I. M. Sigal, “Solitary wave dynamics in an external potential”, Comm. Math. Phys. 250:3 (2004), 613–642.
• R. Grimshaw, “Slowly varying solitary waves. I. Korteweg - de Vries equation”, Proc. Roy. Soc. London Ser. A 368:1734 (1979), 359–375.
• R. Grimshaw, “Slowly varying solitary waves. II. Nonlinear Schrödinger equation”, Proc. Roy. Soc. London Ser. A 368:1734 (1979), 377–388.
• R. H. J. Grimshaw and S. R. Pudjaprasetya, “Generation of secondary solitary waves in the variable-coefficient Korteweg-de Vries equation”, Stud. Appl. Math. 112:3 (2004), 271–279.
• J. Holmer, “Dynamics of KdV solitons in the presence of a slowly varying potential”, preprint.
• J. Holmer and M. Zworski, “Soliton interaction with slowly varying potentials”, Int. Math. Res. Not. 2008:10 (2008), Art. ID rnn026.
• J. Holmer, J. Marzuola, and M. Zworski, “Fast soliton scattering by delta impurities”, Comm. Math. Phys. 274:1 (2007), 187–216.
• J. Holmer, J. Marzuola, and M. Zworski, “Soliton splitting by external delta potentials”, J. Nonlinear Sci. 17:4 (2007), 349–367.
• B. L. G. Jonsson, J. Fr öhlich, S. Gustafson, and I. M. Sigal, “Long time motion of NLS solitary waves in a confining potential”, Ann. Henri Poincaré 7:4 (2006), 621–660.
• H. Kalisch and J. L. Bona, “Models for internal waves in deep water”, Discrete Contin. Dynam. Systems 6:1 (2000), 1–20.
• V. I. Karpman and E. M. Maslov, “Perturbation theory for solitons”, Ž. Èksper. Teoret. Fiz. 73:2 (1977), 281–291. In Russian; translated in Soviet Phys. JETP 46:2 (1977), 537–559.
• D. J.. Kaup and A. C. Newell, 361 (1978), 413–446.
• C. E. Kenig, G. Ponce, and L. Vega, “Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle”, Comm. Pure Appl. Math. 46:4 (1993), 527–620.
• K. Ko and H. H. Kuehl, “Korteweg–de Vries soliton in a slowly varying medium”, Phys. Rev. Lett. 40:4 (1978), 233–236.
• D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular canal, and on a new type of stationary waves”, Philos. Mag. Ser. 5 39 (1895), 422–443.
• P. Lochak, “On the adiabatic stability of solitons and the matching of conservation laws”, J. Math. Phys. 25:8 (1984), 2472–2476.
• P. Lochak and C. Meunier, Multiphase averaging for classical systems, with applications to adiabatic theorems,, Applied Mathematical Sciences 72, Springer, New York, 1988.
• Y. Martel, “Asymptotic $N$-soliton-like solutions of the subcritical and critical generalized Korteweg-de Vries equations”, Amer. J. Math. 127:5 (2005), 1103–1140.
• Y. Martel and F. Merle, “Blow up in finite time and dynamics of blow up solutions for the $L\sp 2$-critical generalized KdV equation”, J. Amer. Math. Soc. 15:3 (2002), 617–664.
• Y. Martel and F. Merle, “Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation”, Ann. of Math. $(2)$ 155:1 (2002), 235–280.
• Y. Martel and F. Merle, “Asymptotic stability of solitons of the subcritical gKdV equations revisited”, Nonlinearity 18:1 (2005), 55–80.
• Y. Martel and F. Merle, “Description of two soliton collision for the quartic gKdV equation”, preprint, 2007. To appear in Ann. Math.
• Y. Martel and F. Merle, “Refined asymptotics around solitons for gKdV equations”, Discrete Contin. Dyn. Syst. 20:2 (2008), 177–218.
• Y. Martel and F. Merle, “Stability of two soliton collision for nonintegrable gKdV equations”, Comm. Math. Phys. 286:1 (2009), 39–79.
• Y. Martel and F. Merle, “Inelastic interaction of nearly equal solitons for the BBM equation”, Discrete Contin. Dyn. Syst. 27:2 (2010), 487–532.
• Y. Martel and F. Merle, “Inelastic interaction of nearly equal solitons for the quartic gKdV equation”, Invent. Math. 183:3 (2011), 563–648.
• Y. Martel, F. Merle, and T.-P. Tsai, “Stability and asymptotic stability in the energy space of the sum of $N$ solitons for subcritical gKdV equations”, Comm. Math. Phys. 231:2 (2002), 347–373.
• Y. Martel, F. Merle, and T. Mizumachi, “Description of the inelastic collision of two solitary waves for the BBM equation”, Arch. Ration. Mech. Anal. 196:2 (2010), 517–574.
• F. Merle, “Existence of blow-up solutions in the energy space for the critical generalized KdV equation”, J. Amer. Math. Soc. 14:3 (2001), 555–578.
• R. M. Miura, “The Korteweg-de Vries equation: a survey of results”, SIAM Rev. 18:3 (1976), 412–459.
• T. Mizumachi, “Weak interaction between solitary waves of the generalized KdV equations”, SIAM J. Math. Anal. 35:4 (2003), 1042–1080.
• C. Muñoz, “On the inelastic 2-soliton collision for gKdV equations with general nonlinearity”, Int. Math. Research Notices 2010 (2010), 1624–1719.
• C. Muñoz, “Inelastic character of solitons of slowly varying gKdV equations”, preprint, 2011. To appear in Comm. Math. Phys.
• C. Muñoz, “Dynamics of soliton-like solutions for slowly varying, generalized KdV equations: refraction versus reflection”, SIAM J. Math. Anal. 44, 1–60.
• C. Muñoz, “On the soliton dynamics under slowly varying medium for nonlinear Schrödinger equations”, Math. Annalen.
• A. C. Newell, Solitons in mathematics and physics, CBMS-NSF Regional Conference Series in Applied Mathematics 48, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1985.
• R. L. Pego and M. I. Weinstein, “Asymptotic stability of solitary waves”, Comm. Math. Phys. 164:2 (1994), 305–349.
• G. Perelman, “A remark on soliton-potential interactions for nonlinear Schrödinger equations”, Math. Res. Lett. 16:3 (2009), 477–486.
• L. Y. Shih, “Soliton-like interaction governed by the generalized Korteweg-de Vries equation”, Wave Motion 2:3 (1980), 197–206.
• F. Verhulst, Nonlinear differential equations and dynamical systems, Springer, Berlin, 2006.
• J. Wright, “Soliton production and solutions to perturbed Korteweg–de Vries equations”, Phys. Rev. A 21:1 (1980), 335–339.
• N. J. Zabusky and M. D. Kruskal, “Interaction of “solitons” in a collisionless plasma and recurrence of initial states”, Phys. Rev. Lett 15 (1965), 240–243.