Zero-dispersion limit of the short-wave–long-wave interaction equations

The purpose of this paper is to study the zero-dispersion limit of the water wave interaction equations which arise in modelling surface waves in the present of both gravity and capillary modes. This topic is also of interest in plasma physics. For the smooth solution, the limiting equation is given by the compressible Euler equation with a nonlocal pressure caused by the long wave. For weak solution, when the coupling coefficient λ is small order of ε, λ=o(ε), the wave map equation is derived and the scattering sound wave is shown to satisfy a linear wave equation.