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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.

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