On the Stability of Periodic Multi-Solitons of the KdV Equation

In this paper we obtain the following stability result for periodic multi-solitons of the KdV equation: We prove that under any given semilinear Hamiltonian perturbation of small size \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon > 0$$\end{document}ε>0, a large class of periodic multi-solitons of the KdV equation, including ones of large amplitude, are orbitally stable for a time interval of length at least \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O(\varepsilon ^{-2})$$\end{document}O(ε-2). To the best of our knowledge, this is the first stability result of such type for periodic multi-solitons of large size of an integrable PDE.

[1]  M. Berti,et al.  Quasi-Periodic Standing Wave Solutions of Gravity-Capillary Water Waves , 2016, Memoirs of the American Mathematical Society.

[2]  M. Berti,et al.  Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle , 2018 .

[3]  Yunfeng Shi,et al.  The Stability of Full Dimensional KAM tori for Nonlinear Schr\"odinger equation , 2017, 1705.01658.

[4]  L. Biasco,et al.  Periodic orbits accumulating onto elliptic tori for the (N + 1)-body problem , 2008 .

[5]  Maciej Dunajski,et al.  Integrable Systems , 2012 .

[6]  P. Lax.  INTEGRALS OF NONLINEAR EQUATIONS OF EVOLUTION AND SOLITARY WAVES. , 1968 .

[7]  N. Zabusky,et al.  Interaction of "Solitons" in a Collisionless Plasma and the Recurrence of Initial States , 1965 .

[8]  C. Eugene Wayne,et al.  Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory , 1990 .

[9]  Dario Bambusi,et al.  Birkhoff normal form for partial differential equations with tame modulus , 2006 .

[10]  C. S. Gardner,et al.  Korteweg‐de Vries Equation and Generalizations. II. Existence of Conservation Laws and Constants of Motion , 1968 .

[11]  M. Procesi,et al.  Long time stability of small finite gap solutions of the cubic nonlinear Schrödinger equation on T2 , 2017, Journal of Differential Equations.

[12]  Xiaoping Yuan,et al.  A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations , 2011 .

[13]  J. Bernier,et al.  Long Time Dynamics for Generalized Korteweg–de Vries and Benjamin–Ono Equations , 2020, Archive for Rational Mechanics and Analysis.

[14]  R. Bikbaev,et al.  On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. krichever , 1993 .

[15]  T. Kappeler,et al.  Large KAM Tori for Quasi-linear Perturbations of KdV , 2019, Archive for Rational Mechanics and Analysis.

[16]  Luigi Chierchia,et al.  KAM Tori for 1D Nonlinear Wave Equations¶with Periodic Boundary Conditions , 1999, chao-dyn/9904036.

[17]  D. Korteweg,et al.  XLI. On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves , 1895 .

[18]  T. Kappeler,et al.  On the extension of the frequency maps of the KdV and the KdV2 equations , 2016, 1605.06690.

[19]  T. Tao,et al.  Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation , 2010 .

[20]  C. Procesi,et al.  A KAM algorithm for the resonant non-linear Schrödinger equation , 2012, 1211.4242.

[21]  Jean Bourgain,et al.  On diffusion in high-dimensional Hamiltonian systems and PDE , 2000 .

[22]  Dario Bambusi,et al.  Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations , 1999 .

[23]  Dario Bambusi,et al.  Birkhoff Normal Form for Some Nonlinear PDEs , 2003 .

[24]  Jean Bourgain,et al.  QUASI-PERIODIC SOLUTIONS OF HAMILTONIAN PERTURBATIONS OF 2D LINEAR SCHRODINGER EQUATIONS , 1998 .

[25]  M. Procesi,et al.  A note on growth of Sobolev norms near quasiperiodic finite-gap tori for the 2D cubic NLS equation , 2019, Rendiconti Lincei - Matematica e Applicazioni.

[26]  Leon A. Takhtajan,et al.  Hamiltonian methods in the theory of solitons , 1987 .

[27]  Sergei Kuksin,et al.  Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum , 1987 .

[28]  C. S. Gardner,et al.  Korteweg-devries equation and generalizations. VI. methods for exact solution , 1974 .

[29]  T. Kappeler,et al.  Normal Form Coordinates for the KdV Equation Having Expansions in Terms of Pseudodifferential Operators , 2018, Communications in Mathematical Physics.

[30]  E. Faou,et al.  Rational Normal Forms and Stability of Small Solutions to Nonlinear Schrödinger Equations , 2018, Annals of PDE.