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 < x < ∞, 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)}. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.

Keywords

Hirota equation / Riemann-Hilbert problem / Initial-boundary value problem / Global relation

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