FINITE GENUS SOLUTIONS TO THE ABLOWITZ-LADIK EQUATIONS

Generic wave train solutions to the complex Ablowitz-Ladik equations a re developed using methods of algebraic geometry. The inverse spectral transform is used to realize these solutions as potentials in a spati ally discrete linear operator. The manifold of wave trains is infinite -dimensional, but is stratified by finite-dimensional submanifolds ind exed by nonnegative integers g. Each of these strata is a foliation wh ose leaves are parametrized by the moduli space of (possibly singular) hyperelliptic Riemann surfaces of genus g. The generic leaf is a g-di mensional complex torus. Thus, each wave train is constructed from a f inite number of complex numbers comprising a set of spectral data, ind icating that the wave train has a finite number of degrees of freedom. Our construction uses a new Lax pair differing from that originally g iven by Ablowitz and Ladik. This new Lax pair allows a simplified cons truction that avoids some of the degeneracies encountered in previous analyses that make use of the original discretized AKNS Lax pair. Gene ric 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 a t a coinciding pair of branch points. The corresponding solutions in t he pinched case may also be derived from wave trains belonging to nons ingular surfaces using Backlund transformations. The problem of reduci ng the complex Ablowitz-Ladik equations to the focusing and defocusing versions of the discrete nonlinear Schrodinger equation is solved by specifying which spectral data correspond to focusing or defocusing po tentials. Within the class of finite genus complex potentials, spatial ly periodic potentials are isolated, resulting in a formula for the so lution to the spatially periodic initial-value problem. Formal modulat ion equations governing slow evolution of (g + 1)-phase wave trains ar e developed, and a gauge invariance is used to simplify the equations in the focusing and defocusing cases. In both of these cases, the modu lation equations can be either hyperbolic (suggesting modulational sta bility) or elliptic (suggesting modulational instability), depending u pon the local initial data. As has been shown to be the case with modu lation equations for other integrable systems, hyperbolic data will re main hyperbolic under the evolution, at least until infinite derivativ es develop. (C) 1996 John Wiley & Sons, Inc.