A Riemann-Hilbert approach to the initial-boundary problem for derivative nonlinear Schrödinger equation

We use the Fokas method to analyze the derivative nonlinear Schrödinger (DNLS) equation iqt(x,t)=−qxx(x,t)+(rq2)x on the interval [0, L]. 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 ξ. This problem has explicit (x,t) dependence, and it has jumps across {ξ ∈ℂ| Imξ4=0}. The relevant jump matrices are explicitely given in terms of the spectral functions {a(ξ),b(ξ)},{A(ξ),B(ξ)}, and {A(ξ),B(ξ)}, which in turn are defined in terms of the initial data q0(x)=q(x,0), the boundary data g0(t)=q(0,t),g1(t)=qx(0,t), and another boundary values f0(t)=q(L,t),f1(t)=qx(L,t). The spectral functions are not independent, but related by a compatibility condition, the so-called global relation.