Differential and Integral Equations

Soliton dynamics for a non-Hamiltonian perturbation of mKdV

Quanhui Lin

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Abstract

We study the dynamics of soliton solutions to the perturbed mKdV equation $\partial_t u = \partial_x(-\partial_x^2 u -2u^3) + \epsilon V u$, where $V\in \mathcal{C}^1_b(\mathbb{R})$ and $0 < \epsilon\ll 1$. This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(\epsilon {\langle} t\rangle^{1/2})$ close to a soliton on an $O(\epsilon^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O(\epsilon^{-1/2})$. Over this time scale, the perturbation can impart an $O(1)$ influence on the soliton position.

Article information

Source
Differential Integral Equations, Volume 26, Number 1/2 (2013), 81-104.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1355867507

Mathematical Reviews number (MathSciNet)
MR3058698

Zentralblatt MATH identifier
1289.35291

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 37K05: Hamiltonian structures, symmetries, variational principles, conservation laws

Citation

Lin, Quanhui. Soliton dynamics for a non-Hamiltonian perturbation of mKdV. Differential Integral Equations 26 (2013), no. 1/2, 81--104. https://projecteuclid.org/euclid.die/1355867507


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