A continuum limit of a finite discrete nonlinear Schroedinger system

Abstract

A continuum limit of a discrete nonlinear Schrodinger system of ordinary differential equations is analyzed. The central question is the relation between the solution of the formally derived limiting system of partial differential equations and the limiting behavior of the solutions to the discrete systems. By setting appropriate boundary conditions on the initial data, a finite subchain decouples, and this system is known to be integrable and solvable by an inverse spectral method. In this thesis, it is found that subunitary data give rise to eigenvalues which are unitary and weighting constants which are positive. This enables one to apply the methods of Lax and Levermore, that is, to formulate a variational maximization problem whose solution characterizes the limiting behavior of the inverse spectral solutions of the discrete systems. We convert the maximization problem into a Riemann-Hilbert problem on the unit circle. The "one-gap" ansatz is studied for times before breaking of the formal PDEs. It is shown that, with certain assumptions on the asymptotic behavior of various objects involved and certain restrictions on the initial data, the solutions of the discrete systems do indeed converge to the solution of the formally limiting system for a certain length of time. WKB analysis is performed, and formulas for the spectral density and asymptotic weighting exponent are proposed for a certain class of data. The spectral density is confirmed by some rigorous results and numerical results. The asymptotics of the weighting constant are found to be quite complicated, and the formula which at least qualitatively matches numerical computations is not a result of asymptotic arguments. Rather, it is obtained through a formal analogy with asymptotic results from other problems in Lax-Levermore theory in which the spectral density and the asymptotic norming exponent are known to be symbolically related. These formulas do give the expected result for the solution to the maximization problem.