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We study extensions of the classical Toda lattices at several different space-time scales. These extensions are from the classical tridiagonal phase spaces to the phase space of full Hessenberg matrices, referred to as the Full Kostant-Toda Lattice. Our formulation makes it natural to make further Lie-theoretic generalizations to dual spaces of Borel Lie algebras. Our study brings into play factorizations of Loewner-Whitney type in terms of canonical coordinatizations due to Lusztig. Using these coordinates we formulate precise conditions for the well-posedness of the dynamics at the different space-time scales. Along the way we derive a novel, minimal box-ball system for Full Toda that doesn't involve any capacities or colorings, as well as an extension of O'Connell's ODEs to Full Toda.

Maps are polygonal cellular networks on Riemann surfaces. This paper analyzes the construction of closed form general representations for the enumerative generating functions associated to maps of fixed but arbitrary genus. The method of construction developed here involves a novel asymptotic symbol calculus for difference operators based on the relation between spectral asymptotics for Hermitian random matrices and asymptotics of orthogonal polynomials with exponential weights. These closed form expressions have a universal character in the sense that they are independent of the explicit valence distribution of the cellular networks within a broad class. Nevertheless the valence distributions may be recovered from the closed form generating functions by a remarkable unwinding identity in terms of Appell polynomials generated by Bessel functions. Our treatment reveals the generating functions to be solutions of nonlinear conservation laws and their prolongations. This characterization enables one to gain insights that go beyond more traditional methods that are purely combinatorial. Universality results are connected to stability results for characteristic singularities of conservation laws that were studied by Caflisch, Ercolani, Hou and Landis, Multi-valued solutions and branch point singularities for nonlinear hyperbolic or elliptic systems, Commun. Pure Appl. Math. 46 (1993) 453–499, as well as directly related to universality results for random matrix spectra.

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

We propose a novel complex-analytic method for sums of i.i.d. random variables that are heavy-tailed and integer-valued. The method combines singularity analysis, Lindel\"of integrals, and bivariate saddle points. As an application, we prove three theorems on precise large and moderate deviations which provide a local variant of a result by S. V. Nagaev (1973). The theorems generalize five theorems by A. V. Nagaev (1968) on stretched exponential laws $p(k) = c\exp( -k^\alpha)$ and apply to logarithmic hazard functions $c\exp( - (\log k)^\beta)$, $\beta>2$; they cover the big jump domain as well as the small steps domain. The analytic proof is complemented by clear probabilistic heuristics. Critical sequences are determined with a non-convex variational problem., Comment: 32 pages, 3 figures

This article discusses numerical and analytical results on grain boundaries, which are line defects that separate roll patterns oriented in different directions. The work is set in the context of a canonical pattern-forming system, the Swift–Hohenberg (SH) equation, and of its phase diffusion equation, the regularized Cross–Newell equation. It is well known that, as the angle made by the rolls on each side of a grain boundary is decreased, dislocations appear at the core of the defect. Our goal is to shed some light on this transition, which provides an example of defect formation in a system that is variational. Numerical results of the SH equation that aim to analyse the phase structure of far-from-threshold grain boundaries are presented. These observations are then connected to properties of the associated phase diffusion equation. Outcomes of this work regarding the role played by phase derivatives in the creation of defects in pattern-forming systems, about the role of harmonic analysis in understanding the phase structure in such systems, and future research directions are also discussed. This article is part of the theme issue ‘Stability of nonlinear waves and patterns and related topics’.

This paper develops a deeper understanding of the structure and combinatorial significance of the partition function for Hermitian random matrices. The coefficients of the large N expansion of the logarithm of this partition function, also known as the genus expansion (and its derivatives), are generating functions for a variety of graphical enumeration problems. The main results are to prove that these generating functions are, in fact, specific rational functions of a distinguished irrational (algebraic) function, z0(t). This distinguished function is itself the generating function for the Catalan numbers (or generalized Catalan numbers, depending on the choice of weight of the parameter t). It is also a solution of the inviscid Burgers equation for certain initial data. The shock formation, or caustic, of the Burgers characteristic solution is directly related to the poles of the rational forms of the generating functions.As an intriguing application, one gains new insights into the relation between certain derivatives of the genus expansion, in a double-scaling limit, and the asymptotic expansion of the first Painleve transcendent. This provides a precise expression of the Painleve asymptotic coefficients directly in terms of the coefficients of the partial fractions expansion of the rational form of the generating functions established in this paper. Moreover, these insights point towards a more general program relating the first Painleve hierarchy to the higher order structure of the double-scaling limit through the specific rational structure of generating functions in the genus expansion. The paper closes with a discussion of the relation of this work to recent developments in understanding the asymptotics of graphical enumeration.As a by-product, these results also yield new information about the asymptotics of recurrence coefficients for orthogonal polynomials with respect to exponential weights, the calculation of correlation functions for certain tied random walks on a 1D lattice, and the large time asymptotics of random matrix partition functions.

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 we derive a hierarchy of differential equations which uniquely determine the coefficients in the asymptotic expansion, for large $N$, of the logarithm of the partition function of $N \times N$ Hermitian random matrices. These coefficients are generating functions for graphical enumeration on Riemann surfaces. The case that we particularly consider is for an underlying measure that differs from the Gaussian weight by a single monomial term of degree $2\nu$. The coupling parameter for this term plays the role of the independent dynamical variable in the differential equations. From these equations one may deduce functional analytic characterizations of the coefficients in the asymptotic expansion. Moreover, this ode system can be solved recursively to explicitly construct these coefficients as functions of the coupling parameter. This analysis of the fine structure of the asymptotic coefficients can be extended to multiple coupling parameters and we present a limited illustration of this for the case of two parameters., Comment: 41 pages, 1 table

We introduce a set of reduced Maxwell–Bloch equations that incorporates the effects of a permanent dipole and inhomogeneous broadening. We demonstrate the integrability of this reduction by providing a Lax pair, which is found using the pseudo-potential technique. An appropriate Backlund transformation is employed to produce a family of solitonic solutions.

We study the limiting behavior as e ↘ 0 of solutions u e to the Dirichlet problem e 2 Δ u e − u e = 0 on Ω , u e | ∂ Ω = e − θ / e , where Ω ¯ is a bounded domain and θ a given smooth function on its boundary ∂ Ω . We provide a natural criterion on θ in order to obtain an estimate e ∂ ν u e ( x ) u e ( x ) ≤ C ∞ , x ∈ ∂ Ω , independent of e as e ↘ 0 , where ∂ ν u e denotes the normal derivative of u e . The results of this paper serve to significantly strengthen the analysis of asymptotic minimizers for a Ginzburg–Landau variational problem for irrotational vector fields (gradient vector fields) known as the regularized Cross–Newell variational problem in the pattern formation literature. In particular, this yields estimates on the asymptotic energy of these minimizers for general Dirichlet viscosity boundary conditions. The class of boundary conditions for this variational problem to which our methods apply is quite general (even including domains which are general Riemannian manifolds with boundary). This, for instance, provides a first step for extending the Ginzburg–Landau type model we consider to the larger class of vector fields that are locally gradient (often called director fields).

We give a rigorous construction of complete families of biorthonormal polynomials associated to a planar measure of the form e −n(V(x)+W(y)−2τxy) d x d y for polynomial V and W. We are further able to show that the zeroes of these polynomials are all real and distinct. A complex analytical construction of the biorthonormal polynomials is given in terms of a non-local Riemann–Hilbert problem which, given our prior result, provides an avenue for developing uniform asymptotics for the statistical distributions of these zeroes as n becomes large. The biorthonormal polynomials considered here play a fundamental role in the analysis of certain random multi-matrix models. We show that the evolutions of the recursion matrices for the polynomials induced by linear deformations of V and W coincide with a semi-infinite generalization of the completely integrable full Kostant–Toda lattice. This connection could be relevant for understanding aspects of scaling limits for the multi-matrix model.

We obtain the Lax pair, hierarchy of commuting flows and Backlund transformations for a reduced Maxwell–Bloch (RMB) system. This system is of particular interest for the description of unipolar, nonoscillating electromagnetic solitons (also called “electromagnetic bubbles” ).

and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order ``self-dual'' equation.

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.

The moment formulas that globally characterize the zero-dispersion limit of the Korteweg-deVries (KdV) equation are known to be expressed in terms of the solution of a maximization problem. Here we establish a direct relation between this maximizer and the zero-dispersion limit of the logarithm of the Jost functions associated with the inverse spectral transform. All the KdV conserved densities are encoded in the spatial derivative of these functions, known as Weyl functions. We show the Weyl functions are densities of measures that converge in the weak sense to a limiting measure. This limiting measure encodes all of the weak limits of the KdV conserved densities. Moreover, we establish the weak limit of spectral measures associated with the Dirichlet problem.

A meromorphic solution to the Burgers equation with complex viscosity is analysed. The equation is linearized via the Cole - Hopf transform which allows for a careful study of the behaviour of the singularities of the solution. The asymptotic behaviour of the solution as the dispersion coefficient tends to zero is derived. For small dispersion, the time evolution of the poles is found by numerically solving a truncated infinite-dimensional Calogero-type dynamical system. The initial data are provided by high-order asymptotic approximations of the poles at the critical time for the dispersionless solution via the method of steepest descents. The solution is reconstructed using the pole expansion and the location of the poles. The oscillations observed via the singularities are compared to those obtained by a classical stationary phase analysis of the solution as the dispersion parameter . A uniform asymptotic expansion as of the dispersive solution is derived in terms of the Pearcey integral in a neighbourhood of the caustic. A continuum limit of the pole expansion and the Calogero system is obtained, yielding a new integral representation of the solution to the inviscid Burgers equation.

The phenomenon of spatio-temporal phase modulation made possible by resonance is investigated in detail through the analysis of an example problem. A simple family of exact solutions to the Ablowitz-Ladik equations is found to be modulationally stable in some regimes. This family of solutions is determined by fixing antiperiod 2 boundary conditions, which determines two wavenumbers. Within the family of solutions, the frequencies do not depend on amplitude; this feature ensures that the antiperiod 2 boundary conditions will be enforced under modulation. The family of solutions is described by four parameters, two being actions that foliate the phase space, and two being macroscopically observable functions of the phase constants. The modulation of the actions is described by a closed hyperbolic system of first order equations, which is consistent with the full set of four genus 1 modulation equations. The modulation of the phase information, easily observed due to the presence of two resonances, is described by two more equations that are driven by the actions. The results are confirmed by numerical experiments.

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.

1991 Mathematics Subject Classification: Primary: 30F10. Secondary: 35Q53. Lecture given by H. Knorrer at the Banach Center Colloquium on 6th May 1993. Partially supported by the Alexander von Humboldt Stiftung and the Sonderforschungsbereich 288; on leave of absence from St. Petersburg Branch of Steklov Mathematical Institute, St. Petersburg, Russia. Supported by the National Science Foundation (DMS-9001897) and the Arizona Center for Mathematical Sciences, sponsored by AFOSR Contract FQ 86711-900589. The paper is in final form and no version of it will be published elsewhere.

The objective of this manuscript is to provide an evidence-based analysis of the current status and future perspectives of robotic laparoendoscopic single-site surgery (R-LESS). A PubMed search has been performed for all relevant urological literature regarding natural orifice transluminal endoscopic surgery (NOTES) and laparoendoscopic single-site surgery (LESS). All clinical and investigative reports for robotic LESS and NOTES procedures in the urological literature have been considered. A significant number of clinical urological procedures have been successfully completed utilizing R-LESS procedures. The available experience is limited to referral centers, where the case volume is sufficient to help overcome the challenges and learning curve of LESS surgery. The robotic interface remains the best fit for LESS procedures but its mode of use continues to evolve in attempts to improve surgical technique. We stand today at the dawn of R-LESS surgery, but this approach may well become the standard of care in the near future. Further technological development is needed to allow widespread adoption of the technique.

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

In this paper we rigorously show the existence and smoothness ine of traveling wave solutions to a periodic Korteweg-deVries equation with a Kuramoto-Sivashinsky-type perturbation for sufficiently small values of the perturbation parametere. The shape and the spectral transforms of these traveling waves are calculated perturbatively to first order. A linear stability theory using squared eigenfunction bases related to the spectral theory of the KdV equation is proposed and carried out numerically. Finally, the inverse spectral transform is used to study the transient and asymptotic stages of the dynamics of the solutions.

The purpose of this paper is the derivation of reduced, finite-dimensional dynamical systems that govern the near-integrable modulations ofN-phase, spatially periodic, integrable wavetrains. The small parameter in this perturbation theory is the size of the nonintegrable perturbation in the equation, rather than the amplitude of the solution, which is arbitrary. Therefore, these reduced equations locally approximate strongly nonlinear behavior of the nearly integrable PDE. The derivation we present relies heavily on the integrability of the underlying PDE and applies, in general, to anyN-phase periodic wavetrain. For specific applications, however, a numerical pretest is applied to fix the truncation orderN. We present one example of the reduction philosophy with the damped, driven sine-Gordon system and summarize our present progress toward application of the modulation equations to this numerical study.

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.

This note describes the explicit construction of the 4( k −1)-dimensional moduli space of k -monopoles in center-of-mass coordinates. This space is known to have a natural hyper-Kahler structure. We show how the construction of this hyper-Kahler metric can be reduced to solving a gauge orthogonality condition.

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

In this paper the homoclinic geometric structure of the integrable sine-Gordon equation under periodic boundary conditions is developed. Specifically, focus is given to orbits homoclinic to N-tori. Simple examples of such homoclinic orbits are constructed and a physical interpretation of these states is given. A labeling is provided which identifies and catalogues all such orbits. These orbits are related in a one-to-one manner to linearized instabilities. Explicit formulas for all homoclinic orbits are given in terms of Backlund transformations.

The evaluation of the ventricular function has a key role in determining therapy and prognosis in patients with heart disease. The standard analysis is limited to quantification of tele-diastolic and tele-systolic frames. An analysis of wall kinetic in each frame of cardiac cycle might allow further information about contractility and diastolic function. The developed algorithms are aimed to achieve an automatic detection of left ventricular wall kinetic during a cardiac cycle in routine clinical activity.

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.

An understanding of the functional readjustments that the brain undergoes during the early days after a stroke would give us a major insight into how and how much neurons are capable to react to an insult. Thirty-two patients affected by an acute monohemispheric ischemic stroke were enrolled in the study. Magnetoencephalography was used to record the somatosensory-evoked fields (SEF) generated in response to median nerve stimulation. Latency, strength, and position of the related early cortical components (M20 and M30) were studied both separately within each hemisphere, and in terms of interhemispheric differences. Interhemispheric cross-correlations among SEF waveshapes in the two hemispheres were also investigated. Overall, except for some source displacement possibly induced by the perilesional edema, results did not demonstrate any unusual neural recruitment. The severity of the clinical picture was found related to the sources' strengths (both as absolute values and as interhemispheric differences), to excessive interhemispheric differences in SEF waveshapes and in the M30 latencies. Signs of an enhanced excitability were present in the affected hemisphere (AH) following a cortical lesion, usually in combination with preserved hand functionality. An enhanced excitability of the unaffected hemisphere (UH) was paired with larger lesions with cortical involvement; signs compatible with an abnormal transcallosal transmission and intracortical function of inhibitory GABAergic interneurons in the AH were found subtending UH enhancement. Spared responsiveness from Brodmann's area (BA) 2 and posterior parietal areas despite an altered response from BA 3b was found in six patients, combined to high hand functionality. Present results in acute phase increase the knowledge of the mechanisms governing brain adaptation/reaction capabilities, for future efforts to establish therapeutic and rehabilitative procedures.

This is a report on recent work that examines the behaviour of a class of nonlinear partial differential equations which axe considered to provide a good qualitative model of significant aspects of pattern formation and defects in a diverse range of physical systems. This work was done in collaboration with C. Bowman, R. Indik, A. C. Newell at the University of Arizona and with T. Passot at the Observatoire de Nice. The details of the formal and numerical results mentioned in this introduction will appear in [15] and details of the analytical results mentioned in the last section will appear in [9].

This study reports psychological symptoms assessed in 327 patients with fibromyalgia (FS) in a multicenter investigation. Two self-report scales, in their validated Italian translations, were used for screening: the CES-D (the Center of Epidemiologic Studies-Depression) developed at the NIMH for measuring depression and the Illness Behavior Questionnaire (IBQ) developed by Pilowsky and Spence. The cutoff point of 23 in the CES-D scores revealed about 49% of the fibromyalgic patients as depressed. In analyzing patterns of illness behavior patients with FS showed a high score on IBQ scales of disease conviction, psychological versus somatic focusing and denial. CES-D scores showed significant correlations with illness behavior scales. These results and their implications for the treatment of fibromyalgic patients are discussed.

The equations for the (finite, nonperiodic) Toda lattice can, as is well known, be written in Lax form, $$\dot X(t) = [X(t),\Pi X(t)]$$ Here X is a symmetric tridiagonal matrix and Π is the projection onto the skew-symmetric summand in the decomposition of X into skew-symmetric plus upper triangular. The eigenvalues of X are constants of motion of (1), and the Toda lattice turns out to be a completely integrable Hamiltonian system.

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.

We model Fomenko’s topological classification of 2 degree of freedom integrable stratifications in an infinite dimensional soliton system. Specifically, the analyticity of the Floquet discriminant Δ(q, ») in both of its arguments provides a transparent realization of a Bott function and of the remaining building blocks of the stratification; in this manner, Fomenko’s structure theorems are expressed through the inverse spectral transform. Thus, soliton equations are shown to provide natural representatives of the classification in the context of PDE’s.

The characterization ofreal, N phase, quasiperiodic solutions of the sine-Gordon equation has been an open problem. In this paper we achieve this result, employing techniques of classical algebraic geometry which have not previously been exploited in the soliton literature. A significant by-product of this approach is a naturalalgebraic representation of the full complex isospectral manifolds, and an understanding of how the real isospectral manifolds are embedded. By placing the problem in this general context, these methods apply directly to all soliton equations whose multiphase solutions are related to hyperelliptic functions.