Well-posedness for a modified nonlinear Schrodinger equation modeling the formation of rogue waves

Abstract

The Cauchy problem for a higher order modification of the nonlinear Shcrodinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent $\ge 0$. This result is achieved by demonstrating that the associated integral operator is a contraction on a Bourgain space that has been adapted to the particular linear symbol present in the equation. the ctraction is proved by using microlocal analysis and a new trilinear estimate.