Local regularity properties for 1D mixed nonlinear Schrödinger equations on half-line

Abstract

The main purpose of this paper is to consider the initial-boundary value problem for the 1D mixed nonlinear Schrodinger equation ut = iαuxx + βu2ūx + γ∣u∣2ux + i∣u∣2u on the half-line with inhomogeneous boundary condition. We combine Laplace transform method with restricted norm method to prove the local well-posedness and continuous dependence on initial and boundary data in low regularity Sobolev spaces. Moreover, we show that the nonlinear part of the solution on the half-line is smoother than the initial data.