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

An initial boundary-value problem for the Hirota 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 $\{q_0(x), g_0(t), g_1(t)\}$. The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.