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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 Backlund transformations. The problem of reducing 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 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 upon 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.

Multi-valued solutions are constructed for 2 × 2 first-order systems using a generalization of the hodograph transformation. The solution is found as a complex analytic function on a complex Riemann surface for which the branch points move as part of the solution. The branch point singularities are envelopes for the characteristics and thus move at the characteristic speeds. We perform an analysis of stability of these singularities with respect to perturbations of the initial data. The generic singularity types are folds, cusps, and nondegenerate umbilic points with non-zero 3-jet. An isolated singularity is generically a square root branch point corresponding to a fold. Two types of collisions between singularities are generic: At a “tangential” collision between two singularities moving at the same characteristic speed, a cube root branch point is formed, corresponding to a cusp. A “non-tangential” collision, between two square root branch points moving at different characteristic speeds, remains a square root branch point at the collision and corresponds to a nondegenerate umbilic point. These results are also valid for a diagonalizable n-th order system for which there are exactly two speeds. © 1993 John Wiley & Sons, Inc.

The modulational stability of real, two-phase sine-Gordon wavetrains are studied. There are three classes of such waves; we find the kink-kink trains are stable, while the breather trains and kink-radiation trains are unstable to modulations. These results continue the investigations of Flaschka, Forest, and McLaughlin for the kdV equation and of Forest and McLaughlin for the sing-Gordon and sine-Gordon equations. In a previous paper the sine-Gordon two-phase modulation theory could only be carried to an intermediate stage. Here we use recent results of Ercolani and Forest to complete this project.