## The nonlinear Schrödinger equation on the half-line

**A 2004 Preprint by
A. Fokas,
A. Its, and
L. Sung
**

- 2004:08
Assuming that the solution

*q(x, t)*of the nonlinear Schrödinger equation on the halfline exists, it has been shown 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 $q _ x(0, t)-pq(0, t) = 0$ where $p$ is a real constant.