Solitary waves and their linear stability in weakly coupled KdV equations

Abstract.We consider a system of weakly coupled KdV equations developed initially by Gear & Grimshaw to model interactions between long waves. We prove the existence of a variety of solitary wave solutions, some of which are not constrained minimizers. We show that such solutions are always linearly unstable. Moreover, the nature of the instability may be oscillatory and as such provides a rigorous justification for the numerically observed phenomenon of “leapfrogging.”

[1]  T. Kubota,et al.  Resonant Transfer of Energy between Nonlinear Waves in Neighboring Pycnoclines , 1980 .

[2]  J. Albert,et al.  Stability and symmetry of solitary-wave solutions to systems modeling interactions of long waves , 2000 .

[3]  Amitabha Bose,et al.  Symmetric and Antisymmetric Pulses in Parallel Coupled Nerve Fibres , 1995, SIAM J. Appl. Math..

[4]  B. Malomed.  “Leapfrogging” solitons in a system of coupled KdV equations , 1987 .

[5]  P. Lions.  The concentration-compactness principle in the Calculus of Variations , 1984 .

[6]  Frank Merle,et al.  Asymptotic stability of solitons of the subcritical gKdV equations revisited , 2005 .

[7]  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.

[8]  Guido Schneider,et al.  The long‐wave limit for the water wave problem I. The case of zero surface tension , 2000 .

[9]  J. Bona,et al.  Solitary waves in nonlinear dispersive systems , 2002 .

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

[11]  Roger H.J. Grimshaw,et al.  Weak and strong interactions between internal solitary waves , 1984 .

[12]  Orlando Lopes,et al.  Instability of travelling waves for weakly coupled KdV systems , 2008 .

[13]  J. Hale,et al.  Methods of Bifurcation Theory , 1996 .

[14]  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.

[15]  Michael M. Tom,et al.  A model system for strong interaction between internal solitary waves , 1992 .

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

[17]  Roy H. Goodman,et al.  Vector-soliton collision dynamics in nonlinear optical fibers. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.