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THE mKdV EQUATION ON THE HALF-LINE

Published online by Cambridge University Press:  30 April 2004

A. Boutet de Monvel
Affiliation:
Institut de Mathématiques de Jussieu, Case 7012, Université de Paris 7, 2 Place Jussieu, 75251 Paris, France ([email protected])
A. S. Fokas
Affiliation:
Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Cambridge CB3 0WA, UK ([email protected])
D. Shepelsky
Affiliation:
Mathematical Division, Institute for Low Temperature Physics, 47 Lenin Avenue, 61103 Kharkiv, Ukraine ([email protected])

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

Type
Research Article
Copyright
2004 Cambridge University Press

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