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D.E. Pelinovsky and R.H.J. Grimshaw

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

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Physica D 98, 139-155 (1996)

**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
scaling 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.

**Keywords:**

NONLINEAR SCHRODINGER EQUATION, KADOMTSEV-PETVIASHVILI EQUATION,
BLOW-UP, STABILITY, CIRCULATION, SOLITONS,
MOMENTUM, STATES, ENERGY, MASS