The Linearization of the Initial-Boundary Value Problem of the Nonlinear Schrödinger Equation

Abstract

We consider the nonlinear Schrödinger (NLS) equation in the variable $q(x,t)$ with both x and t in $[ {0,\infty } )$. We assume that $q(x,0) = u(x)$ and $q(0,t) = v(t)$ are given, that $u(0) = v(0)$, and that $u(x)$ and $v(t)$ as well as their first two derivatives belong to $L_1 \cap L_2 (\mathbb{R}^ + )$. We show that the solution of this initial-boundary value problem can be reduced to solving a Riemann–Hilbert (RH) problem in the complex k-plane with jumps on $\operatorname{Im} (k^2 ) = 0$. This RH problem is equivalent to a linear integral equation which has a unique global solution. This linear integral equation is uniquely defined in terms of certain functions (scattering data) $b(k)$ and $c(k)$. The function $b(k)$ can be effectively computed in terms of $u(x)$. However, although the analytic properties of $c(k)$ are completely determined, the relationship between $c(k)$, $u(x)$ and $v(t)$ is highly nonlinear. In spite of this difficulty, we can give an effective description of the asymptotic behavior of $q(x,t)$ for large t. In particular, we show that as $t \to \infty $, solitons are generated moving away from the boundary. In addition, our formalism can be used to generate effectively pairs of functions $q(0,t)$ and $q_x (0,t)$ compatible with a given $q(x,0)$ as well as to determine the associated $q(x,t)$. It is important to emphasize that the analysis of this problem, in addition to techniques of exact integrability, requires the essential use of general partial differential equations (PDE) techniques.