Abstract

The asymptotic behavior of a piecewise-constant initial condition for the nonlinear Schrödinger (NLS) equation has been the subject of intense study recently by several investigators, both theoretically^1,2 and experimentally,^3,4 motivated in part: by the potential for technological applications inherent in such a system. This problem has been addressed theoretically through the use of a direct-scattering approach, which gives the set of eigenvalues λ as a consequence of requiring the continuity of a set of transversely counterpropagating waves v₁ and v₂ at the boundaries between constant regions of the initial condition. To date, progress in understanding the behavior of this system has been made by numerically solving the resulting set of transcendental equations, with a few special cases being solved analytically. In this paper we present a direct-scattering formalism that greatly simplifies the effort required to obtain the set of eigenvalues resulting from an arbitrary piecewise-constant initial condition,⁵ We apply this technique to the set of periodic initial conditions. In particular, as the piecewise-constant initial condition becomes periodic, we show that the eigenvalues of the asymptotic solution can arrange themselves in continua or bands, and we obtain expressions for the location of the bands and for the density-of-states functions associated with the bands. In addition, we show that even as the initial condition becomes periodic, under certain conditions a set of anomalous, discrete eigenvalues exist outside of the bands—a situation that cannot be qualitatively understood through the use of a Kronig-Penney-type of analogy.

© 1993 Optical Society of America