A Riemann-Hilbert Problem for an Energy Dependent Schrödinger Operator

Abstract

\We consider an inverse scattering problem for Schrödinger operators with energy dependent potentials. The inverse problem is formulated as a Riemann-Hilbert problem on a Riemann surface. A vanishing lemma is proved for two distinct symmetry classes. As an application we prove global existence theorems for the two distinct systems of partial differential equations $u_t+(u^2/2+w{)}_x=0,w_t\pm u_{xxx}+(uw{)}_x=0$ for suitably restricted, complementary classes of initial data.

References in corpus (1)

• An integrable shallow water equation with peaked solitons

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