THE mKdV EQUATION ON THE HALF-LINE

Abstract

An initial boundary-value problem for the modified Korteweg–de Vries equation on the half-line, $0<x<\infty$, $t>0$, is analysed by expressing the solution $q(x,t)$ in terms of the solution of a matrix Riemann–Hilbert (RH) problem in the complex $k$-plane. This RH problem has explicit $(x,t)$ dependence and it involves certain functions of $k$ referred to as the spectral functions. Some of these functions are defined in terms of the initial condition $q(x,0)=q_0(x)$, while the remaining spectral functions are defined in terms of the boundary values $q(0,t)=g_0(t)$, $q_x(0,t)=g_1(t)$, and $q_{xx}(0,t)=g_2(t)$. The spectral functions satisfy an algebraic global relation which characterizes, say, $g_2(t)$ in terms of $\{q_0(x),g_0(t),g_1(t)\}$. It is shown that for a particular class of boundary conditions, the linearizable boundary conditions, all the spectral functions can be computed from the given initial data by using algebraic manipulations of the global relation; thus, in this case, the problem on the half-line can be solved as efficiently as the problem on the whole line.

AMS 2000 Mathematics subject classification: Primary 37K15; 35Q53. Secondary 35Q15; 34A55