UC San Diego

We consider the initial value problem for various type of nonlinear Schrödinger equations with derivative nonlinearity which cannot be treated by normal perturbative arguments because of the loss in derivative from the nonlinearity.

The first part of the study involves finding the well-posedness in low regularity Sobolev spaces for different types of nonlinearities. The key idea is to capture a part of the solution that resembles the linear Schrödinger dynamic while keeping the remaining part spatial and frequency localized. With this, we can study the interactions between the truncations of the solution at different frequencies and obtain a meaningful perturbative analysis.

In the second part, we study the dynamic of the cubic nonlinear Schrödinger equation in the energy critical Sobolev space by projecting the solution onto different wave packets which are frequency and spatial localized at all time. As a result, we obtain the asymptotic behavior, modified scattering profile and asymptotic completeness of the solution without relying on the integrable structure of the equation.