Stability of $N$ solitary waves for the generalized BBM equations

We consider the generalized BBM (Benjamin-Bona-Mahony) equations:\begin{equation}(1-\partial^2_x)u_t+{(u+u^p)}_x=0,\end{equation}for $p\geqslant 2$ integer, and the family of solitary wave solutions$\varphi_c(x-x_0-ct)$ of this equation.For any $p$, there exists a necessary and sufficient condition on thespeed $c>1$ so that a solitary wave solutionis nonlinearly stable (\cite{W-87}, \cite{S-S}). Following the approachof \cite{M-M-T.3} forthe generalized KdV equations, we prove that the sum of $N$ sufficientlydecoupled stable solitary wave solutionsis also stable in the energy space.The proof combines arguments of \cite{W-87} to prove the stability of asingle solitary wave,and monotonicity results of \cite{D-2}. We also obtain asymptotic stabilityresults following \cite{D-2}.Using the same tools, we then prove the existence and uniqueness of asolution behaving asymptotically in large time as the sum of $N$ givensolitary waves, following the method of \cite{Martel}.

Keywords

stability, asymptotic stability, solitary waves, regularized long-wave equation, BBM equation

2010 Mathematics Subject Classification

Primary 35B40, 35Q51. Secondary 35Q53.

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Published 1 January 2004