An asymptotic approach to solitary wave instability and critical collapse in long-wave KdV-type evolution equations

Abstract Instability development and critical collapse of solitary waves are considered in the framework of generalized Korteweg-de Vries (KdV) equations in one and two dimensions. An analytical theory of the solitary wave dynamics and generation of radiation is constructed for the critical case when the solitary waves are weakly unstable. Characteristic types of the global, essentially nonlinear evolution of the unstable solitary waves are analyzed for some typical generalized KdV equations. The scalling laws of the self-similar wave field transformation are found analytically for the power-like KdV equation in the critical case p = 4. The asymptotic approach is also developed for the modified Zakharov-Kuznetsov equation in two dimensions and the rate of the singularity formation is found to be smaller than in one dimension due to diffractive wave effects.

[1]  Hiroaki Ono,et al.  Weak Non-Linear Hydromagnetic Waves in a Cold Collision-Free Plasma , 1969 .

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

[3]  Mark J. Ablowitz,et al.  Solitons and the Inverse Scattering Transform , 1981 .

[4]  Mark J. Ablowitz,et al.  Wave collapse and instability of solitary waves of a generalized Kadomtsev-Petviashvili equation , 1994 .

[5]  K. Spatschek,et al.  On the stability of collapse in the critical case , 1992 .

[6]  Yuji Kodama,et al.  Perturbations of solitons and solitary waves , 1981 .

[7]  G. Whitham,et al.  Linear and Nonlinear Waves , 1976 .

[8]  D. Pelinovsky,et al.  Asymptotic theory of plane soliton self-focusing in two-dimensional wave media , 1995 .

[9]  J. Juul Rasmussen,et al.  Blow-up in Nonlinear Schroedinger Equations-I A General Review , 1986 .

[10]  J. Juul Rasmussen,et al.  Blow-up in Nonlinear Schroedinger Equations-II Similarity structure of the blow-up singularity , 1986 .

[11]  D. Pelinovsky,et al.  On a wave field transformation described by the two-dimensional Kadomtsev-Petviashvili equation , 1995 .

[12]  T. Rassias,et al.  Differential Geometry, Calculus of Variations, and Their Applications , 1985 .

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

[14]  Collapsing states of generalized Korteweg-de Vries equations , 1989 .

[15]  K. Spatschek,et al.  Stability theorem for KdV-type equations , 1984, Journal of Plasma Physics.

[16]  Pelinovsky,et al.  Nonlinear theory of oscillating, decaying, and collapsing solitons in the generalized nonlinear Schrödinger equation. , 1996, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[17]  H. Mitsudera,et al.  Slowly Varying Solitary Wave Solutions of the Perturbed Korteweg‐de Vries Equation Revisited , 1993 .

[18]  E. Kuznetsov.  Soliton stability in equations of the KdV type , 1984 .