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Finite Genus Solutions to the Ablowitz-Ladik Equations

Peter D. Miller and Nicholas M. Ercolani
Program in Applied Mathematics
The University of Arizona
Tucson, AZ 85721 USA

Igor M. Krichever
The Landau Institute for Theoretical Physics
Moscow, Russia

C. David Levermore
Program in Applied Mathematics
The University of Arizona
Tucson, AZ 85721 USA


Generic wave train solutions to the complex Ablowitz-Ladik equations are developed using methods of algebraic geometry. The inverse spectral transform is used to realize these solutions as potentials in a spatially discrete linear operator. The manifold of wave trains is infinite-dimensional, but is stratified by finite-dimensional submanifolds indexed by nonnegative integers g. Each of these strata is a foliation whose leaves are parametrized by the moduli space of (possibly singular) hyperelliptic Riemann surfaces of genus g. The generic leaf is a g-dimensional complex torus. Thus, each wave train is constructed from a finite number of complex numbers comprising a set of spectral data, indicating that the wave train has a finite number of degrees of freedom. Our construction uses a new Lax pair differing from that originally given by Ablowitz and Ladik. This new Lax pair allows a simplified construction that avoids some of the degeneracies encountered in previous analyses that make use of the original discretized AKNS Lax pair. Generic wave trains are built from Baker-Akhiezer functions on nonsingular Riemann surfaces having distinct branch points, and the construction is extended to handle singular Riemann surfaces that are pinched off at a coinciding pair of branch points. The corresponding solutions in the pinched case may also be derived from wave trains belonging to nonsingular surfaces using Baecklund transformations. The problem of reducing the complex Ablowitz-Ladik equations to the focusing and defocusing versions of the discrete nonlinear Schrödinger equation is solved by specifying which spectral data correspond to focusing or defocusing potentials. Within the class of finite genus complex potentials, spatially periodic potentials are isolated, resulting in a formula for the solution to the spatially periodic initial-value problem. Formal modulation equations governing slow evolution of (g+1)-phase wave trains are developed, and a gauge invariance is used to simplify the equations in the focusing and defocusing cases. In both of these cases, the modulation equations can be either hyperbolic (suggesting modulational stability) or elliptic (suggesting modulational instability), depending on the local initial data. As has been shown to be the case with modulation equations for other integrable systems, hyperbolic data will remain hyperbolic under the evolution, at least until infinite derivatives develop.