The nonlinear Schrödinger equation on the half-line

Assuming that the solution q ( x , t ) of the nonlinear Schrödinger equation on the half-line exists, it has been shown in Fokas (2002 Commun. Math. Phys. 230 1–39) that q ( x , t ) can be represented in terms of the solution of a matrix Riemann–Hilbert (RH) problem formulated in the complex k -plane. The jump matrix of this RH problem has explicit x , t dependence and it is defined in terms of the scalar functions { a ( k ), b ( k ), A ( k ), B ( k )} referred to as spectral functions. The functions a ( k ) and b ( k ) are defined in terms of q 0 ( x ) = q ( x ,0), while the functions A ( k ) and B ( k ) are defined in terms of g 0 ( t ) = q (0, t ) and g 1 ( t ) = q x (0, t ). The spectral functions are not independent but they satisfy an algebraic global relation . Here we first prove that if there exist spectral functions satisfying this global relation, then the function q ( x , t ) defined in terms of the above RH problem exists globally and solves the nonlinear Schrödinger equation, and furthermore q ( x , 0) = q 0 ( x ), q (0, t ) = g 0 ( t ) and q x (0, t ) = g 1 ( t ). We then show that, given appropriate initial and boundary conditions, it is possible to construct such spectral functions through the solution of a nonlinear Volterra integral equation whose solution exists globally. We also show that for a particular class of boundary conditions it is possible to bypass this nonlinear equation and to compute the spectral functions using only the algebraic manipulation of the global relation; thus for this particular class of boundary conditions, which we call linearizable , the problem on the half-line can be solved as effectively as the problem on the line. An example of a linearizable boundary condition is q x (0, t ) − ρ q (0, t ) = 0 where ρ is a real constant.