Linearizable Initial-Boundary Value Problems for the sine-Gordon Equation on the Half-Line

Abstract

A rigorous methodology for the analysis of initial boundary value problems on the half-line, $0<x<\infty$, $t>0$, for integrable nonlinear evolution PDEs has recently appeared in the literature. As an application of this methodology the solution $q(x,t)$ of the sine-Gordon equation can be obtained in terms of the solution of a $2\times 2$ matrix Riemann-Hilbert problem. This problem is formulated in the complex $k$-plane and is uniquely defined in terms of the so called spectral functions $a(k)$, $b(k)$, and $B(k)/A(k)$. The functions $a(k)$ and $b(k)$ can be constructed in terms of the given initial conditions $q(x,0)$ and $q_t(x,0)$ via the solution of a system of two {\it linear} ODE's, while for \emph{arbitrary} boundary conditions the functions $A(k)$ and $B(k)$ can be constructed in terms of the given boundary condition via the solution of a system of four {\it nonlinear} ODEs. In this paper we analyse two \emph{particular} boundary conditions: the case of constant Dirichlet data, $q(0,t) = \chi$, as well as the case that $q_x(0,t)$, $\sin (q(0,t)/2)$, and $\cos(q(0,t)/2)$ are linearly related by two constants $\chi_1$ and $\chi_2$. We show that for these particular cases, the system of the above nonlinear ODEs can be avoided, and $B(k)/A(k)$ can be computed explicitly in terms of $\{a(k),b(k),\chi\}$ and $\{a(k),b(k), \chi_1, \chi_2\}$ respectively. Thus these ``linearizable'' initial-boundary value problems can be solved with absolutely the same level of efficiency as the classical initial value problem of the line.