Stability and Instability of the KDV Solitary Wave Under the KP-I Flow

We consider the KP-I and gKP-I equations in $${{\mathbb{R}}\,\times\,({\mathbb{R}}/2\pi{\mathbb{Z}})}$$. We prove that the KdV soliton with subcritical speed 0 < c < c* is orbitally stable under the global KP-I flow constructed by Ionescu and Kenig (Ann Math Stud 163:181–211, 2007). For supercritical speeds c > c*, in the spirit of the work by Duyckaerts and Merle (GAFA 18:1787–1840, 2009), we sharpen our previous instability result and construct a global solution which is different from the solitary wave and its translates and which converges to the solitary wave as time goes to infinity. This last result also holds for the gKP-I equation.

[1]  Vladimir E. Zakharov,et al.  Degenerative dispersion laws, motion invariants and kinetic equations , 1980 .

[2]  Michael I. Weinstein,et al.  Eigenvalues, and instabilities of solitary waves , 1992, Philosophical Transactions of the Royal Society of London. Series A: Physical and Engineering Sciences.

[3]  O. V. Besov,et al.  Integral representations of functions and imbedding theorems , 1978 .

[4]  E. M. Epshtein.  ''Anomalous'' magnetoresistance of ionic semiconductors at low temperatures , 1975 .

[5]  Emmanuel Grenier,et al.  On the nonlinear instability of Euler and Prandtl equations , 2000 .

[6]  Vladimir E. Zakharov.  Instability and nonlinear oscillations of solitons , 1975 .

[7]  N. Tzvetkov,et al.  Transverse nonlinear instability for two-dimensional dispersive models , 2006, math/0612494.

[8]  Frederic Rousset,et al.  Transverse nonlinear instability of solitary waves for some Hamiltonian PDE's , 2008 .

[9]  B. Kadomtsev,et al.  On the Stability of Solitary Waves in Weakly Dispersing Media , 1970 .

[10]  Frank Merle,et al.  Dynamic of Threshold Solutions for Energy-Critical NlS , 2007, 0710.5915.

[11]  N. Tzvetkov,et al.  A simple criterion of transverse linear instability for solitary waves , 2009, 0910.3789.

[12]  Robert L. Pego,et al.  On the transverse instability of solitary waves in the Kadomtsev-Petviashvili equation , 1997 .

[13]  Jean-Claude Saut,et al.  Global Well-Posedness for the KP-I Equation on the Background of a Non-Localized Solution , 2007 .

[14]  David Chiron,et al.  The KdV/KP-I Limit of the Nonlinear Schrödinger Equation , 2010, SIAM J. Math. Anal..

[15]  J. Shatah,et al.  Stability theory of solitary waves in the presence of symmetry, II☆ , 1990 .

[16]  Vianney Combet.  Construction and characterization of solutions converging to solitons for supercritical gKdV equations , 2009 .

[17]  M. Weinstein.  Lyapunov stability of ground states of nonlinear dispersive evolution equations , 1986 .

[18]  David Lannes,et al.  Large time existence for 3D water-waves and asymptotics , 2007, math/0702015.

[19]  C. Kenig,et al.  Chapter 9. Local and Global Wellposedness of Periodic KP-I Equations , 2009 .

[20]  P. Souganidis,et al.  Stability and instability of solitary waves of Korteweg-de Vries type , 1987, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.

[21]  Tetsu Mizumachi,et al.  Stability of the line soliton of the KP-II equation under periodic transverse perturbations , 2010, 1008.0812.

[22]  Frederic Rousset,et al.  Transverse instability of the line solitary water-waves , 2009, 0906.2487.