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Complex Billiard Hamiltonian Systems and Nonlinear Waves

The relationships between phase shifts, monodromy effects, and billiard solutions are studied in the context of Riemannian manifolds for both integrable ordinary and partial differential equations. The ideas are illustrated with the three wave interaction, the nonlinear Schrödinger equation, a coupled Dym system and the coupled nonlinear Schrödinger equations.
Mark S. Alber, Gregory G. Luther, Jerrold E. Marsden

Automorphic Pseudodifferential Operators

The theme of this paper is the correspondence between classical modular forms and pseudodifferential operators (ΨDO’s) which have some kind of automorphic behaviour. In the simplest case, this correspondence is as follows. Let Γ be a discrete subgroup of PSL 2(ℝ) acting on the complex upper half-plane H in the usual way, and f(z) a modular form of even weight k on Γ. Then there is a unique lifting from f to a Γ-invariant ΨDO with leading term f(z)-k/2, where ∂ is the differential operator \( \frac{d}{{dz}} \). This lifting and the fact that the product of two invariant ΨDO’s is again an invariant ΨDO imply a non-commutative multiplicative structure on the space of all modular forms whose components are scalar multiples of the so-called Rankin-Cohen brackets (canonical bilinear maps on the space of modular forms on Γ defined by certain bilinear combinations of derivatives; the definition will be recalled later). This was already discussed briefly in the earlier paper [Z], where it was given as one of several “raisons d’être” for the Rankin-Cohen brackets.
Paula Beazley Cohen, Yuri Manin, Don Zagier

On τ-Functions of Zakharov—Shabat and Other Matrix Hierarchies of Integrable Equations

Matrix hierarchies are: multi-component KP, general Zakharov—Shabat (ZS) and its special cases, e.g., AKNS. The ZS comprises all integrable systems having a form of zero-curvature equations with rational dependence of matrices on a spectral parameter. The notion of a τ-function is introduced here in the most general case along with formulas linking τ-functions with wave Baker functions. The method originally invented by Sato et al. for the KP hierarchy is used. This method goes immediately from definitions and does not require any assumption about the character of a solution, being the most general. Applied to the matrix hierarchies, this involves considerable sophistication. The paper is self-contained and does not expect any special prerequisite from a reader.
L. A. Dickey

On the Hamiltonian Representation of the Associativity Equations

We demonstrate that for an arbitrary number n of primary fields the equations of the associativity can be rewritten in the form of (n — 2) pairwise commuting systems of hydrodynamic type, which appear to be nondiagonalizable but integrable. We propose a natural Hamiltonian representation of the systems under study in the cases n = 3 and n = 4.
E. V. Ferapontov, O. I. Mokhov

A Plethora of Integrable Bi-Hamiltonian Equations

This paper discusses several algorithmic ways of constructing integrable evolution equations based on the use of multi-Hamiltonian structures. The recognition that integrable soliton equations, such as the Korteweg-deVries (KdV) and nonlinear Schrodinger (NLS) equations, can be constructed using a biHamiltonian method dates back to the late 1970’s. An extension of the method was proposed by the first author and Fuchssteiner in the early 1980’s and was used to derive integrable generalizations of the KdV and of the modified KdV. However it was not until these models reappeared in physical problems, and their novel solutions such as compactons and peakons were discovered, that the method achieved recognition. In this paper, we describe the basic approach to constructing a wide variety of integrable bi-Hamiltonian equations. In addition to usual soliton equations, these new hierarchies include equations with nonlinear dispersion which support novel types of solitonic solutions.
A. S. Fokas, P. J. Olver, P. Rosenau

Hamiltonian Structures in Stationary Manifold Coordinates

We consider the restriction of isospectral flows to stationary manifolds. Specifically, we present a systematic construction of Hamiltonian structures written in stationary manifold coordinates, which demonstrates the close relationship between the Hamiltonian formulations of nonlinear evolution equation (PDE) and its stationary reduction. We illustrate these ideas in the context of the KdV and 5 th order KdV equations.
We then apply these ideas to the Boussinesq hierarchy, associated with the (trace free) 3 rd order Lax operator, together with the Sawada-Kotera and Kaup-Kupershmidt reductions.
We use our results to study the integrable cases of the Hénon Heiles equation.
Allan P. Fordy, Simon D. Harris

Compatibility in Abstract Algebraic Structures

Compatible Hamiltonian pairs play a crucial role in the structure theory of integrable systems. In this paper we consider the question of how much of the structure given by compatibility is bound to the situation of hamiltonian dynamic systems and how much of that can be transferred to a complete abstract situation where the algebraic structures under consideration are given by bilinear maps on some module over a commutative ring. Under suitable modification of the corresponding definitions, it turns out that notions like, compatible, hereditary, invariance and Virasoro algebra may be transferred to the general abstract setup. Thus the same methods being so successful in the area of integrable systems, may be applied to generate suitable abelian algebras and hierarchies in very general algebraic structures.
Benno Fuchssteiner

A Theorem of Bochner, Revisited

Many hierarchies of the theory of solitons possess symmetries which do not belong to the hierarchy itself. These symmetries are known under the various names of additional, master or conformal symmetries. They were discovered by Fokas, Fuchssteiner and Oevel [9], [10], [25], Chen, Lee and Lin [4] and Orlov and Schulman [26]. They are intimately related to the bihamiltonian nature of the equations of the theory of solitons which was pioneered in the work of Magri [23] and Gel’fand and Dorfman [11].
F. Alberto Grünbaum, Luc Haine

Obstacles to Asymptotic Integrability

We study nonintegrable effects appearing in the higher order corrections of an asymptotic perturbation expansion for a given nonlinear wave equation, and show that the analysis of the higher order terms provides a sufficient condition for asymptotic integrability of the original equation. The nonintegrable effects, which we call “obstacles” to the integrability, are shown to result in an inelasticity in soliton interaction. The main technique used in this paper is an extension of the normal form theory developed by Kodama and the approximate symmetry approach proposed by Mikhailov. We also discuss the case of the KP equation with the higher order corrections, a quasi-two dimensional extension of weakly dispersive nonlinear waves.
Y. Kodama, A. V. Mikhailov

Infinitely-Precise Space-Time Discretizations of the Equation ut + uux = 0

The classical Volterra system u n,t = constu n (u n+i u n−i ) is time-discretized in four different ways such that each one of the infinity of conservation laws of the Volterra system is preserved exactly. Since in the space-continuous limit the Volterra system turns into the basic nonlinear infinite-dimensional dynamical system u t + uu x = 0, the Volterra conservation laws are discretizations of the conservation laws (u m /m) t + [(u m+1/(m+1)] x = 0, mN.
B. A. Kupershmidt

Trace Formulas and the Canonical 1-Form

This paper studies the canonical 1-form of symplectic geometry in the context of the (defocussing) cubic Schrodinger system. The phase space is populated by pairs QP of smooth functions of period 1, equipped with the classical 1-form QdP = ∫ 0 1 [Q(x)dP(x)]dx. The introduction of canon- ically paired coordinates Q n P n : n ∈ ℤ, as in Sections 2 and 6 below, suggests the identity QdP = Σ Q n dP n , up to an additive exact form, and this may be verified, as in Sections 5 and 6, with the help of new trace formulas, derived in what I believe to be a new way; see, especially Section 4, nos. 4 and 5. The discussion could be carried over to sine/sh-Gordon, etc.; compare Section 7 where this is done for KdV.
Henry P. McKean

On some “Schwarzian” Equations and their Discrete Analogues

Some integrable discrete analogues of the Schwarzian KdV (Krichever-Novikov) equation and of other Möbius-invariant equations, are discussed together with their Miura chains relating them to associated equations like the lattice KdV and lattice modified KdV equation. Furthermore, the similarity solutions of such lattice equations are considered, as well as reductions to discrete Painlevé equations.
Frank Nijhoff

Poisson Brackets for Integrable Lattice Systems

Poisson brackets associated with Lax operators of lattice systems are considered. Linear brackets originate from various r-matrices on the algebra of (pseudo-) shift symbols. Quadratic brackets are investigated which provide Hamiltonian formulations for various reductions of the (modified) Lattice KP hierarchy.
W. Oevel

On the r-Matrix Structure of the Neumann System and its Discretizations

A novel impetus to the construction of integrable discretisations of given integrable continuous-time hamiltonian systems has been given in recent years by a number of relevant findings: we mention the successful application to differential-difference integrable hierarchies of the stationary flow or restricted flow approach [1], the results obtained toward the identification of integrable mappings of the standard type [2], the discovery of Backlund transformations for Calogero-Moser and Rujsenaars systems [3], including relativistic Toda [4], and finally the construction of non-autonomous mappings which are endowed with a proper discrete analog of the Painleve’ property [5]. Of course, time-discretisation is a highly non-unique procedure, even if we restrict considerations to integrability-preserving difference schemes. One may just ask to get an integrable Poisson map such that the discrete dynamics it describes goes into the continuous one in a suitable asymptotic limit, together with integrals of motion and Poisson structure, or require that Poisson structure and integrals of motion be exactly preserved by the discretisation. Stationary or restricted flow technique typically lead to discretisation of the former type, while Backlund transformations provide an example of the latter one. In the present paper, we compare two integrable discretisations of the Neumann system, both belonging to the first family; indeed, one of them is obtained by applying the restricted flow technique to the Toda lattice hierarchy.
Orlando Ragnisco, Yuri B. Suris

Multiscale Expansions, Symmetries and the Nonlinear Schrödinger Hierarchy

We study the propagation of quasi-monocromatic, nondissipative and weakly nonlinear waves, modelled by partial differential equations in 1 + 1 dimensions, using a multitime expansion. In the case of pure radiation, we show that the asymptotic character of this expansion is guaranted by requiring that the modulation of the leading amplitude of the waves satisfy the nonlinear Schrodinger hierarchy of evolution equations with respect to the slow space-time variables characteristic of the problem. The theory of the symmetries of integrable systems plays a crucial role in the derivation of this result.
Paolo Maria Santini

On a Laplace Sequence of Nonlinear Integrable Ernst-Type Equations

Invariance under Laplace-Darboux-type transformations is established for the 2+1-dimensional Loewner-Konopelchenko-Rogers integrable system. This is exploited to derive a chain of novel, integrable Ernst-type equations which contain an arbitrary parameter.
W. K. Schief, C. Rogers

Classical and Quantum Nonultralocal Systems on the Lattice

We classify nonultralocal Poisson brackets for 1-dimensional lattice systems and describe the corresponding regularizations of the Poisson bracket relations for the monodromy matrix. A nonultralocal quantum algebras on the lattices for these systems are constructed. For some class of such algebras an ultralocalization procedure is proposed. The technique of the modified Bethe-Anzatz for these algebras is developed and is applied to the nonlinear sigma model problem.
Michael Semenov-Tian-Shansky, Alexey Sevostyanov
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