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

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 $\omega =a^2-2b^2,b>0$.

This old article needs to be significantly revised. Hopefully it will be, with new results and set of authors, soon