The derivative nonlinear Schrödinger equation on the half-line

Abstract

We analyze the derivative nonlinear Schrödinger equation i q t + q x x = i ( | q | 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 ζ . The jump matrix has explicit x , t dependence and is given in terms of the spectral functions a ( ζ ) , b ( ζ ) (obtained from the initial data q 0 ( x ) = q ( x , 0 ) ) as well as A ( ζ ) , B ( ζ ) (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 Schrödinger equation with the prescribed initial and boundary values.