The nonlinear Schrödinger equation on the half-line

Abstract

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 q0(x) = q(x,0), while the functions A(k) and B(k) are defined in terms of g0(t) = q(0,t) and g1(t) = qx(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) = q0(x), q(0, t) = g0(t) and qx(0, t) = g1(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 qx(0, t) − ρq(0, t) = 0 where ρ is a real constant.