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We study the Ginzburg-Landau equations on Riemann surfaces of arbitrary genus. In particular: - we construct explicitly the (local moduli space of gauge-equivalent) solutions in a neighbourhood of the constant curvature ones; - classify holomorphic structures on line bundles arising as solutions to the equations in terms of the degree, the Abel-Jacobi map, and symmetric products of the surface; - determine the form of the energy and identify when it is below the energy of the constant curvature (normal) solutions., Comment: 37 pages

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.

In this paper, recent results on the behavior of roll patterns in a class of problems typified by high Prandtl number convection are presented. A key finding is that the Gaussian curvature of the “crumpled” phase surface, which consists of patches with an almost constant wave number, line defects on which most of the free energy is stored and point defects with nontrivial topologies; condenses onto line and point defects. This property allows considerable mathematical simplification in that the fourth order nonlinear diffusion equation governing stationary states can be effectively reduced to the linear Helmholtz equation. The observed patterns have much is common with the deformation of thin elastic sheets.

Analysis and computations are presented for singularities in the solution of the steady Boussinesq equations for two-dimensional, stratified flow. The results show that for codimension 1 singularities, there are two generic singularity types for general solutions, and only one generic singularity type if there is a certain symmetry present. The analysis depends on a special choice of coordinates, which greatly simplifies the equations, showing that the type is exactly that of one dimensional Legendrian singularities, generalized so that the velocity can be infinite at the singularity. The solution is viewed as a surface in an appropriate compactified jet space. Smoothness of the solution surface is proved using the Cauchy-Kowalewski Theorem, which also shows that these singularity types are realizable. Numerical results from a special, highly accurate numerical method demonstrate the validity of this geometric analysis. A new analysis of general Legendrian singularities with blowup, i.e., at which the derivative may be infinite, is also presented, using projective coordinates.

Certain Hamiltonian discretizations of the periodic focusing Nonlinear Schrodinger Equation (NLS) have been shown to be responsible for the generation of numerical instabilities and chaos. In this paper we undertake a dynamical systems type of approach to modeling the observed irregular behavior of a conservative discretization of the NLS. Using heuristic Mel'nikov methods, the existence of a pair of isolated homoclinic orbits is indicated for the perturbed system. The structure of the persistent homoclinic orbits that are predicted by the Mel'nikov theory possesses the same features as the wave form observed numerically in the perturbed system after the onset of chaotic behavior and appears to be the main structurally stable feature of this type of chaos. The Mel'nikov analysis implemented in the pde context appears to provide relevant qualitative information about the behavior of the pde in agreement with the numerical experiments. In a neighborhood of the persistent homoclinic orbits, the existence of a horseshoe is investigated and related with the onset of chaos in the numerical study.

We investigate the typical cycle lengths, the total number of cycles, and the number of finite cycles in random permutations whose probability involves cycle weights. Typical cycle lengths and total number of cycles depend strongly on the parameters, while the distributions of finite cycles are usually independent Poisson random variables., Comment: 22 pages, 2 figures

We present a heuristic proof that the nonlinear Schrodinger equation (NLS) - iϑΨ ϑt = 1 2 ΔΨ+ 1 2 (1-|Ψ| 2 )Ψ in 2 + 1 dimensions has a family of solutions which can be well approximated by a collection of point vortices for a planar incompressible fluid. The novelty of our approach is that we begin with a representation of the NLS as a compressible perturbation of Euler's equations for hydrodynamics.

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 study of the general projective curve X, defined by the vanishing of an irreducible polynomial p(x, y), originated in the discovery of addition formulas for integrals ∫ ∞ v r(x, y) dx of rational functions on such a curve

This paper shows that the Ablowitz-Ladik hierarchy of equations (a well-known integrable discretization of the Non-linear Schrodinger system) can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R (obtained by composing K and the inverse of J.) In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations., 33 pages

The Cross–Newell phase diffusion equation τ(k)Θ T =−∇· k → B(k), k → =∇Θ,| k → |=k , and its regularization describe patterns and defects far from onset in large aspect ratio systems with translational and rotational symmetry. In this paper we show how director field solutions of this equation can be used to describe features of global patterns. The ideas are illustrated in the context of a non-trivial case study of high Prandtl number convection in a large aspect ratio, shallow, elliptical container with heated sidewalls, for which we also have the results of simulation and experiment.

Labyrinthic patterns are observed both in systems where the uniform states are metastable, as a result of a front instability, and in systems displaying a cellular instability, when the band of excited Fourier modes is wide enough to support resonant interactions between modes lying on different shells. We show that the phase formalism is a suitable description for low-density labyrinthic patterns with a relatively long range correlation and is capable of describing both its smooth and singular structures. The point defects of roll patterns, the concave and convex disclinations, and the line singularities or phase grain boundaries across which the wavevector makes an order one transition, are found to be singular and weak solutions of the Cross-Newell phase diffusion equation, which take account of their energetics as well as their topologies.

One of the most elegant classical results about nonsingular algebraic curves is Riemann’s singularity theorem, which provides a beautiful interpretation of the singularities of Riemann’s theta function in terms of the geometry of special divisors. In recent times, soliton theory has suggested generalizations of the classical geometry. Integrable partial differential equations posed with periodic or scattering boundary conditions lead to Riemann surfaces of infinite, or even “continuously infinite”, genus, and to correspondingly huge Jacobians ([McKT], [EMcK]). More special boundary conditions have been studied via the theory of infinite-dimensional Lie groups and their homogeneous spaces. In this latter approach, one finds a large family of (possibly infinite-dimensional) algebraic varieties that are natural generalizations of the classical Jacobians. The Lie-theoretic aspects of these varieties are, of course, prominent in such work ([SW], [Sch]). In another direction, soliton problems have led naturally to singular limits of smooth Riemann surfaces. While, in the abstract, singular curves and their Jacobians are a subject of current research, soliton theory provides a concrete context in which to study such algebraic varieties.

The work in this paper pertains to the solutions of Nahm's equations, which arise in the Atiyah-Drinfield-Hitchin-Manin-Nahm construction of solutions to the Bogomol'nyi equations for static monopoles. This paper provides an explicit construction of the solution of Nahm's equations which satisfy regularity and reality conditions. The Lax form of Nahm's equations is reduced to a standard eigenvalue problem by a special gauge transformation. These equations may then be solved by the method of Baker-Krichever. This leads to a compact representation of the solutions of Nahm's equations. The regularity condition is shown to be related to the monodromy of the gauge reduced linear operator. Hitchin showed that the solutions of Nahm's equations can be characterized by an algebraic curve and some data on that curve. Here, this characterization reduces to a transcendental equation involving certain loop integrals of a meromorphic differential. Donaldson coordinatized the moduli space ofk-monopoles by a class of rational maps from the Riemann sphere to itself. The data of a Baker function is equivalent to this map. This method gives an “apriori” construction of the (known) two monopole solutions. We also give a generalization of the two monopole solution to a class of elliptic solutions of arbitrary charge. These solutions correspond to reducible curves with elliptic components and the associated Donaldson rational function has a simple partial fraction expansion.

Let there be given a finite-gap operator L = d 2 /d x 2 + q and its Baker function ψ( x, p ), which is analytic for p on a certain hyperelliptic curve C . It is shown that a sequence of Bäcklund transformations maps C to a projective space. This embedding can be interpreted as a matrix representation of the Hill equation by the Neumann system of constrained harmonic oscillators. The image curve, C' , lies on a rational ruled surface; the structure of this surface is explained by use of ideas due to Burchnall & Chaundy ( Proc. R. Soc. Lond . A 118, 557-583 (1928)). Baker functions and Bäcklund transformations are then used to define a (many-to-many) correspondence between effective divisors on the curve C and points lying on a quadric, or in the intersection of two or more quadrics. This relates the theory of the Hill equation to earlier work of Knörrer, Moser and Reid. It is then shown that the Kummer image of the Jacobian of C can be realized as a hypersurface in the space of momentum variables of the Neumann system. Further projects, such as extensions to non-hyperelliptic curves, are outlined.