Open AccessWell-posedness for a modified nonlinear Schrödinger equation modeling the formation of rogue waves
Author(s) -
Curtis Holliman,
Logan Hyslop
Publication year - 2021
Publication title -
open journal of mathematical analysis
Language(s) - English
Resource type - Journals
eISSN - 2616-8111
pISSN - 2616-8103
DOI - 10.30538/psrp-oma2021.0088
Subject(s) - sobolev space , mathematics , nonlinear system , mathematical analysis , contraction (grammar) , norm (philosophy) , cauchy problem , multiplier (economics) , initial value problem , modulation space , nonlinear schrödinger equation , exponent , mathematical physics , schrödinger equation , physics , quantum mechanics , medicine , political science , law , linguistics , philosophy , economics , macroeconomics
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.