The Focusing NLS Equation on the Half-Line with Periodic Boundary Conditions

TL;DR

This paper develops a novel inverse scattering approach to solve the focusing NLS equation on the half-line with periodic boundary conditions, establishing the Dirichlet-to-Neumann map via a Riemann-Hilbert problem.

Contribution

It introduces an indirect method to characterize the Neumann boundary value for the focusing NLS on the half-line using a Riemann-Hilbert problem with partially unspecified jump data.

Findings

Constructed solutions with Schwartz initial data and decaying boundary perturbations.

Proved the solution satisfies the initial-boundary conditions and decay properties.

Provided explicit conditions on scattering data for the boundary derivatives.

Abstract

We consider the Dirichlet problem for the focusing NLS equation on the half-line, with given Schwartz initial data and boundary data $q(0,t)$ equal to an exponentially decaying perturbation $u(t)$ of the periodic boundary data $ a e^{2i\omega t + i \epsilon}$ at $x=0.$ It is known from PDE theory that this problem admits a unique solution (for fixed initial data and fixed $u$). On the other hand, the associated inverse scattering transform formalism involves the Neumann boundary value for $x=0$. Thus the implementation of this formalism requires the understanding of the "Dirichlet-to-Neumann" map which characterises the associated Neumann boundary value. We consider this map in an indirect way: we postulate a certain Riemann-Hilbert problem, on a specified contour but with partially unspecified jump data of some generality, and then prove that the solution of the initial-boundary value problem for the focusing NLS constructed through this Riemann-Hilbert problem satisfies all the required properties: the data $q(x,0)$ are Schwartz and $q(0,t)-a e^{2i\omega t + i \epsilon}$ is exponentially decaying. More specifically, we focus on the case $-3a^2 < \omega < a^2.$ By considering a large class of appropriate scattering data for the t-problem, we provide solutions of the above Dirichlet problem such that the data $q_x(0,t)$ is given by an exponentially decaying perturbation of the function $2iab e^{2i\omega t + i \epsilon},$ where $\omega = a^2-2b^2,~~b>0$.