A Riemann-Hilbert Problem for an Energy Dependent Schrödinger Operator

Details A Riemann-Hilbert Problem for an Energy Dependent Schrödinger Operator A Riemann-Hilbert Problem for an Energy Dependent Schrödinger Operator

1997-12-19

arxiv.org/abs/solv-int/9712013v1

Inverse Problems 12 (1996) 1003-1025

Nonlinear Sciences

Exactly Solvable and Integrable Systems

Scientific paper

10.1088/0266-5611/12/6/014

\We consider an inverse scattering problem for Schr\"odinger 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.

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