Weak Interaction between Solitary Waves of the Generalized KdV Equations

In this paper we study the large time behavior of two decoupled solitary waves of the generalized KdV equations ut+(uxx+f(u))x=0, where $f(u)=|u|^{p-1}u/p$ ($3\le p < 5$). We prove that if the speeds of the solitary waves are sufficiently close at the initial time, the leading wave becomes larger and the trailing wave becomes smaller, and the distance between two solitary waves becomes larger as $t\to\infty$.

[1]  W. Strauss.  Dispersion of low-energy waves for two conservative equations , 1974 .

[2]  C. Kenig,et al.  Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle , 1993 .

[3]  Nakao Hayashi,et al.  LARGE TIME ASYMPTOTICS OF SOLUTIONS TO THE GENERALIZED KORTEWEG-DE VRIES EQUATION , 1998 .

[4]  Jalal Shatah,et al.  Global existence of small solutions to nonlinear evolution equations , 1982 .

[5]  Frank Merle,et al.  Instability of solitons for the critical generalized Korteweg—de Vries equation , 2001 .

[6]  Michael I. Weinstein,et al.  Asymptotic stability of solitary waves , 1994 .

[7]  M. Weinstein,et al.  Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation , 1991 .

[8]  Tetsu Mizumachi,et al.  Large Time Asymptotics of Solutions Around Solitary Waves to the Generalized Korteweg-de Vries Equations , 2001, SIAM J. Math. Anal..

[9]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[10]  Shin-Ichiro Ei,et al.  The Motion of Weakly Interacting Pulses in Reaction-Diffusion Systems , 2002 .

[11]  A. Soffer,et al.  Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data , 1992 .

[12]  Shin-ichiro Ei,et al.  Renormalization-Group Method for Reduction of Evolution Equations; Invariant Manifolds and Envelopes , 2000 .

[13]  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 .

[14]  Michael I. Weinstein,et al.  Modulational Stability of Ground States of Nonlinear Schrödinger Equations , 1985 .

[15]  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.

[16]  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.

[17]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1992, math/9201261.

[18]  Terence Tao,et al.  Sharp global well-posedness for KdV and modified KdV on ℝ and , 2003 .

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

[20]  Gustavo Ponce,et al.  Global, small amplitude solutions to nonlinear evolution equations , 1983 .

[21]  P. Deift,et al.  A steepest descent method for oscillatory Riemann–Hilbert problems. Asymptotics for the MKdV equation , 1993 .

[22]  Yvan Martel,et al.  Asymptotic Stability of Solitons¶for Subcritical Generalized KdV Equations , 2001 .

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

[24]  Luc Molinet,et al.  On the Cauchy Problem for the Generalized Korteweg-de Vries Equation , 2003 .

[25]  J. Bona,et al.  On the stability of solitary-wave solutions of model equations for long waves , 1994 .

[26]  Frank Merle,et al.  A Liouville theorem for the critical generalized Korteweg–de Vries equation , 2000 .

[27]  G. Ponce,et al.  Nonlinear Small Data Scattering for the Generalized Korteweg-de Vries Equation , 1990 .

[28]  Ohta,et al.  Equation of motion for interacting pulses. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[29]  Frank Merle,et al.  Blow up in finite time and dynamics of blow up solutions for the L^2-critical generalized KdV equation , 2002 .

[30]  M. Weinstein,et al.  Multichannel nonlinear scattering theory for nonintegrable equations , 1989 .

[31]  Tai-Peng Tsai,et al.  Stability and Asymptotic Stability in the Energy Space of the Sum of N Solitons for Subcritical gKdV Equations , 2001 .

[32]  N. Hayashi,et al.  On the Modified Korteweg–De Vries Equation , 2001, International Seminar. Day on Diffraction. Proceedings (IEEE Cat. No.99EX367).

[33]  J. Ginibre,et al.  Existence and uniqueness of solutions for the generalized Korteweg de Vries equation , 1990 .

[34]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[35]  P. Schuur.  Asymptotic analysis of soliton problems , 1986 .

[36]  Sergiu Klainerman,et al.  Long-time behavior of solutions to nonlinear evolution equations , 1982 .

[37]  J. Maddocks,et al.  On the stability of KdV multi‐solitons , 1993 .