The derivative nonlinear Schr\"odinger equation on the half-line

Abstract

We analyze the derivative nonlinear Schr\"odinger equation $iq_t + q_{xx} = i(|q|^2q)_x$ on the half-line using the Fokas method. Assuming that the solution $q(x,t)$ exists, we show that it can be represented in terms of the solution of a matrix Riemann-Hilbert problem formulated in the plane of the complex spectral parameter $\zeta$. The jump matrix has explicit $x,t$ dependence and is given in terms of the spectral functions $a(\zeta)$, $b(\zeta)$ (obtained from the initial data $q_0(x) = q(x,0)$) as well as $A(\zeta)$, $B(\zeta)$ (obtained from the boundary values $g_0(t) = q(0,t)$ and $g_1(t) = q_x(0,t)$). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values $\{q_0(x), g_0(t), g_1(t)\}$ such that there exist spectral function satisfying the global relation, we show that the function $q(x,t)$ defined by the above Riemann-Hilbert problem exists globally and solves the derivative nonlinear Schr\"odinger equation with the prescribed initial and boundary values.