The Initial-Boundary-Value Problems for the Hirota Equation on the Half-Line

Abstract

An initial boundary-value problem for the Hirota equation on the half-line, 0 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) = q0(x), while the remaining spectral functions are defined in terms of the boundary values q(0, t) = g0(t), qx(0, t) = g1(t) and qxx(0, t) = g2(t). The spectral functions satisfy an algebraic global relation which characterizes, say, g2(t) in terms of {q0(x), g0(t), g1(t)}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.