Séminaire d'Analyse

On Hirota-Satsuma's equation
Didier Pilod
Thursday 14 January 2010 11:15 -  Tours -  Salle 2290 (Bât E2)

Résumé :
We are interested in the initial value problem associated to the Hirota-Satsuma equation u_t+u_x-2uu_t+2u_x\int_x^{\infty}u_tdx'-u_{txx}=0, \\ u(0)=u_0, \quad x\in \mathbb R, where $u$ is a real valued function. This equation models the unidirectional propagation of shallow water waves as the well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations. Here we show local well-posedness for initial data $u_0$ in the space $$\Omega_1=\{\phi \in H^1(\mathbb R) \ / \ -1 \notin \sigma(-\partial_x^2-2\phi)\}.$$ Equation \eqref{HS} also possesses solitary-wave solutions just as do the related KdV and BBM equations. Using an analysis of an associated Liapunov functional, nonlinear stability of these solitary waves is established.

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