Asymptotic behavior of the self-defocusing nonlinear Schrodinger equation for piecewise constant initial conditions

In this paper we use a transfer matrix method to calculate the asymptotic behavior of the nonlinear Schrödinger (NLS) equation in a self-defocusing medium for piecewise constant initial conditions. Treating initial conditions that consist of m repeated regions, we show that the eigenvalues associated with this problem appear in bands, and, as m tends to infinity, we obtain the eigenvalue density of states for these bands. Comparing results from the transfer matrix approach to the results for a Bloch function method, we show that the edges of a region with periodic initial conditions result in a finite number of additional eigenvalues that appear outside the bands.