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

The Cauchy problem for a higher order modification of the nonlinear Schrödinger equation (MNLS) on the line is shown to be well-posed in Sobolev spaces with exponent \(s > \frac{1}{4}\). 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 contraction is proved by using microlocal analysis and a trilinear estimate that is shown via the \([k; Z]\)-multiplier norm method developed by Terence Tao.