Differential and Integral Equations

Construction and characterization of solutions converging to solitons for supercritical gKdV equations

Vianney Combet

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Abstract

We consider the generalized Korteweg-de Vries equation $$ \partial_t u +\partial_x^3 u +\partial_x(u^p)=0, \quad (t,x)\in{\mathbb{R}}^2, $$ in the supercritical case $p>5$, and we are interested in solutions which converge to a soliton in large time in $H^1$. In the subcritical case ($p < 5$), such solutions are forced to be exactly solitons by variational characterization [3, 32], but no such result exists in the supercritical case. In this paper, we first construct a ``special solution" in this case by a compactness argument, i.e., a solution which converges to a soliton without being a soliton. Secondly, using a description of the spectrum of the linearized operator around a soliton [20], we construct a one-parameter family of special solutions which characterizes all such special solutions. In the case of the nonlinear Schrödinger equation, a similar result was proved in [7, 8].

Article information

Source
Differential Integral Equations, Volume 23, Number 5/6 (2010), 513-568.

Dates
First available in Project Euclid: 20 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356019309

Mathematical Reviews number (MathSciNet)
MR2654248

Zentralblatt MATH identifier
1240.35433

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35B40: Asymptotic behavior of solutions 35Q51: Soliton-like equations [See also 37K40] 37K40: Soliton theory, asymptotic behavior of solutions

Citation

Combet, Vianney. Construction and characterization of solutions converging to solitons for supercritical gKdV equations. Differential Integral Equations 23 (2010), no. 5/6, 513--568. https://projecteuclid.org/euclid.die/1356019309


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