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arXiv · 1403.5321

$L^2$-stability of solitary waves for the KdV equation via Pego and Weinstein's method

Tetsu Mizumachi·Nikolay Tzvetkov
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Abstract

In this article, we will prove $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV equation by using exponential stability property of the semigroup generated by the linearized operator. The proof follows the lines of recent stability argument of Mizumachi [Asymptotic stability of lattice solitons in the energy space, Comm. Math. Phys., (2009)] and Mizumachi, Pego and Quintero [Asymptotic stability of solitary waves in the Benney-Luke model of water waves, Differential Integral Equations, (2013)] which show stability in the energy class by using strong linear stability of solitary waves in exponentially weighted spaces. This gives an alternative proof of Merle and Vega [$L^2$ stability of solitons for KdV equation, Int. Math. Res. Not., (2003)] which shows $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV equation by using the Miura transformation. Our argument is a refinement of Pego and Weinstein [Asymptotic stability of solitary waves, Comm. Math. Phys., (1994)] that proves asymptotic stability of solitary waves in exponentially weighted spaces. We slightly improve the $H^1$-stability of the modified KdV equation as well.

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Tetsu Mizumachi, Nikolay Tzvetkov. 2014-03-20. $L^2$-stability of solitary waves for the KdV equation via Pego and Weinstein's method. https://arxiv.org/abs/1403.5321

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