The derivative nonlinear Schrödinger equation on the half-line

We analyze the derivative nonlinear Schrödinger equation iqt+qxx=i(|q|2q)x on the half-line using the Fokas method. Assuming that the solution q(x,t)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 ζζ. The jump matrix has explicit x,tx,t dependence and is given in terms of the spectral functions a(ζ)a(ζ), b(ζ)b(ζ) (obtained from the initial data q0(x)=q(x,0)q0(x)=q(x,0)) as well as A(ζ)A(ζ), B(ζ)B(ζ) (obtained from the boundary values g0(t)=q(0,t)g0(t)=q(0,t) and g1(t)=qx(0,t)g1(t)=qx(0,t)). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation. Given initial and boundary values {q0(x),g0(t),g1(t)}{q0(x),g0(t),g1(t)} such that there exist spectral functions satisfying the global relation, we show that the function q(x,t)q(x,t) defined by the above Riemann–Hilbert problem exists globally and solves the derivative nonlinear Schrödinger equation with the prescribed initial and boundary values.