SOME MODELS FOR THE INTERACTION OF LONG AND SHORT WAVES IN DISPERSIVE MEDIA. PART II: WELL-POSEDNESS

The (in)validity of a system coupling the cubic, nonlinear Schrodinger equation (NLS) and the Korteweg-de Vries equation (KdV) commonly known as the NLS-KdV system for studying the interaction of long and short waves in dispersive media was discussed in part I of this work [N.V. Nguyen and C. Liu, Water Waves, 2:327-359, 2020]. It was shown that the NLS-KdV system can never be obtained from the full Euler equations formulated in the study of water waves, nor even the linear Schrodinger-Korteweg de Vries system where the two equations in the system appear at the same scale in the asymptotic expansion for the temporal and spatial variables. A few alternative models were then proposed for describing the interaction of long and short waves. In this second installment, the Cauchy problems associated with the alternative models introduced in part I are analyzed. It is shown that all of these models are locally well-posed in some Sobolev spaces. Moreover, they are also globally well-posed in those spaces for a range of suitable parameters.