Vanishing viscosity with short wave–long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave–long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.