The derivative nonlinear Schrodinger equation on the half-line

We analyze the derivative nonlinear Schrodinger equation iq(t) + q(xx) = i (vertical bar q vertical bar(2)q)(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. 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 functions 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 Schrodinger equation with the prescribed initial and boundary Values. (C) 2008 Elsevier B.V. All rights reserved.