May  2010, 27(2): 487-532. doi: 10.3934/dcds.2010.27.487

Inelastic interaction of nearly equal solitons for the BBM equation

1. 

Université de Versailles Saint-Quentin-en-Yvelines and IUF, Laboratoire de mathématiques de Versailles, UMR CNRS 8100, 45, av. des Etats-Unis, 78035 Versailles cedex, France

2. 

Université de Cergy-Pontoise and IHES, Laboratoire de mathématiques, UMR CNRS 8088, 2, av. Adolphe Chauvin, 95302 Cergy-Pontoise cedex

Received  October 2009 Revised  February 2010 Published  February 2010

This paper is concerned with the interaction of two solitons of nearly equal speeds for the (BBM) equation. This work is an extension of [31] addressing the same question for the quartic (gKdV) equation. We consider the (BBM) equation, for $\lambda \in [0,1)$,

$ (1-\lambda \partial_x^2) \partial_t u + \partial_x (\partial_x^2 u - u + u^2) =0. $     (BBM)

Solitons are solutions of the form $ R_{\mu,x_0}(t,x)=Q_{\mu}(x-\mu t -x_0), $ for $\mu > -1$, $x_0\in \RR$.
   For $\mu_0>0$ small, let $U(t,x)$ be the unique solution of (BBM) such that

$ \lim_{t\to -\infty} \||U(t) - Q_{-\mu_0}(.+\mu_0 t) - Q_{\mu_0}(.-\mu_0 t )\||_{H^1} = 0. $

First, we prove that $U(t)$ remains close to the sum of two solitons, for all time $t\in \RR$,

$ U(t,x) = Q_{\mu_1(t)}(x-y_1(t)) + Q_{\mu_2(t)}(x-y_2(t)) + $ε$\ (t) \quad where \quad |\|\varepsilon(t)\| | \leq \mu_0^{2^-}, $

with $ y_1(t)-y_2(t)> 2 |\ln \mu_0| + O(1), $ which means that at the main order the situation is similar to the integrable KdV case. However, we show that the collision is perfectly elastic if and only if $\lambda=0$ (i.e. only in the integrable case).

Citation: Yvan Martel, Frank Merle. Inelastic interaction of nearly equal solitons for the BBM equation. Discrete & Continuous Dynamical Systems - A, 2010, 27 (2) : 487-532. doi: 10.3934/dcds.2010.27.487
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