THE mKdV EQUATION ON THE HALF-LINE

An initial boundary-value problem for the modified Korteweg-de Vries equation on the half-line, 0 < x < infinity, 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.