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

TL;DR

This work addresses the Dirichlet-to-Neumann map for the focusing NLS equation on the half-line with Dirichlet data $q(0,t)=a e^{2i\omega t+i\epsilon}+u(t)$, where $u(t)$ decays exponentially, by embedding the problem in a Riemann-Hilbert framework. It introduces a two-sheeted spectral curve with $\Omega(k)=2(k-b)X(k)$ and $X(k)=\sqrt{(k+b)^2+a^2}$ and analyzes a zero-zone spectral data setting through a RH problem on a contour $\Sigma$ defined by $\mathrm{Im}\,\Omega(k)=0$, building on the Lax pair formulation. The main result shows that the RH solution reproduces a physically admissible solution with $q(x,0)$ in the Schwartz class and $q(0,t)-a e^{2i\omega t+i\epsilon}$ decaying exponentially, and that for a broad class of $t$-problem scattering data, $q_x(0,t)$ is an exponentially decaying perturbation of $2 i a b e^{2i\omega t+i\epsilon}$ with $\omega=a^2-2b^2$, $b>0$, in the parameter range $-3a^2<\omega<a^2$. This highlights both a constructive route to the Dirichlet-to-Neumann map and the role of exponentially decaying boundary perturbations in stabilizing the inverse scattering formulation for this problem.

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^{2iωt + i ε}$ 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 $q(0,t)-a e^{2iωt + i ε}$ is exponentially decaying. More specifically, we focus on the case $-3a^2 < ω< a^2.$ 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 $2iab e^{2iωt + i ε},$ where $ω= a^2-2b^2,~~b>0$.

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