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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 |
$ (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).
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