Small data well-posedness for derivative nonlinear Schrödinger equations

Donlapark Ponnoprat

Abstract: We study the generalized derivative nonlinear Schrödinger equation i∂tu+Δu=P(u,u¯¯¯,∂xu,∂xu¯¯¯), where P is a polynomial, in Sobolev spaces. It turns out that when deg P≥3, the equation is locally well-posed in H12 when each term in P contains only one derivative, otherwise we have a local well-posedness in H32. If deg P≥5, the solution can be extended globally. By restricting to equations of the form i∂tu+Δu=∂xP(u,u¯¯¯) with deg P≥5, we were able to obtain the global well-posedness in the critical Sobolev space.