 
Summary: CONVERGENCE OF NUMERICAL SCHEMES FOR SHORT
WAVE LONG WAVE INTERACTION EQUATIONS
PAULO AMORIM1, M´ARIO FIGUEIRA1
Abstract. We consider the numerical approximation of a system of partial
differential equations involving a nonlinear Schr¨odinger equation coupled with
a hyperbolic conservation law. This system arises in models for the interac
tion of short and long waves. Using the compensated compactness method,
we prove convergence of approximate solutions generated by semidiscrete fi
nite volume type methods towards the unique entropy solution of the Cauchy
problem. Some numerical examples are presented.
1. Introduction
1.1. Interaction equations of short and long waves. The nonlinear interac
tion between short waves and long waves has been studied in a variety of physical
situations. In [4], D.J. Benney presents a general theory, deriving nonlinear differ
ential systems involving both short and long waves. The short waves u(x, t) are
described by a nonlinear Schr¨odinger equation and the long waves v(x, t) satisfy a
quasilinear wave equation, eventually with a dispersive term. In its most general
form, the interaction is described by the nonlinear system
(1.1)
itu + ic1xu + xxu = u v + u2
