Spatial plane waves for the nonlinear Schr\"odinger equation: local existence and stability results

TL;DR

This paper introduces new functional spaces based on spatial plane waves for the nonlinear Schrödinger equation on , establishing well-posedness and stability results, including global solutions with large norms, and develops a novel plane wave transform.

Contribution

The paper constructs new functional spaces using spatial plane waves for the NLS, proving well-posedness and stability, and introduces the plane wave transform with broad PDE applications.

Findings

Established global well-posedness in new spaces including large norms.

Developed the plane wave transform and proved its properties.

Solved classical linear PDEs using the plane wave transform.

Abstract

We consider the Cauchy problem for the nonlinear Schr\"odinger equation on $\mathbb{R}^2$, $iu_t + u_{xx} + u_{yy} + \lambda|u|^\sigma u =0$, $\lambda\in \mathbb{R}$, $\sigma>0$. We introduce new functional spaces over which the initial value problem is well-posed. Their construction is based on \textit{spatial plane waves} (cf. arXiv:1510.08745). These spaces contain $H^1(\mathbb{R}^2)$ and do not lie within $L^2(\mathbb{R}^2)$. We prove several global well-posedness and stability results over these new spaces, including a new global well-posedness result of $H^1$ solutions with indefinitely large $H^1$ and $L^2$ norms. Some of these results are proved using a new functional transform, the \textit{plane wave transform}. We develop a suitable theory for this transform, prove several properties and solve classical linear PDE's with it, highlighting its wide range of application.