Constraints of the KP hierarchy and the bilinear method

We consider generalizedk-constraints of the KP hierarchy where the Lax operatorL is forced to satisfy L _– ^k =q^–1r. We study the effect of those constraints on the bilinear equations.     

An explicit expression for the Korteweg-de Vries hierarchy

Several constructions and an explicit expression for the right-hand side of the KdV hierarchy are presented.     

On the geometry of soliton equations

The paper aims to suggest a geometric point of view in the theory of soliton equations. The belief is that a deeper understanding of the origin of these equations may provide a better understanding of their remarkable properties. According to the geometric point of view, soliton equations are the outcome of a specific reduction process of a bi-Hamiltonian manifold. The suggestion of the paper is to pay attention also to the unreduced form of soliton equations.This work has been supported by the Italian MURST and by the GNFM of the Italian CNR.     

A KdV equation in 2+1 dimensions: Painlevé analysis,solutions and similarity reductions

The nonlinear equationm _ty =(m _yxx +m _x m _y ) _x is throughly analyzed. The Painlevé test yields a positive result. The Bäckhand transformations are found and the Darboux-MoutardMatveev formalism arises in the context of this analysis. Some solutions and their interactions are also analyzed. The singular manifold equations are also used to determine symmetry reductions. This procedure can be related with the direct method of Clarkson and Kruskal.     

The KPI equation with unconstrained initial data

The solutionu(t, x, y) of the Kadomtsev-Petviashvili I (KPI) equation with given initial data u(0,x, y) belonging to the Schwartz space is considered. No additional special constraints, usually considered in the literature as dxu(0,x,y)=0 are required to be satisfied by the initial data. The spectral theory associated with KPI is studied in the space of the Fourier transform of the solutions. The variablesp={p ₁,p ₁} of the Fourier space are shown to be the most convenient spectral variables to use for spectral data. Spectral data are shown to decay rapidly at largep but to be discontinuous atp=0. Direct and inverse problems are solved with special attention to the behavior of all the quantities involved in the neighborhood oft=0 andp=0. It is shown in particular that the solutionu(t, x, y) has a time derivative discontinuous att = 0 and that at anyt 0 it does not belong to the Schwartz space no matter how small in norm and rapidly decaying at large distances the initial data are chosen.Work supported in part by Ministero delle Universitá e della Recerca Scientifica e Technologica, India.     

Graded differential equations and their deformations: A computational theory for recursion operators

An algebraic model for nonlinear partial differential equations (PDE) in the category ofn-graded modules is constructed. Based on the notion of the graded Frölicher-Nijenhuis bracket, cohomological invariants H ^* (A) are related to each object (A, ) of the theory. Within this framework, H ⁰ (A) generalizes the Lie algebra of symmetries for PDE's, while H ¹ (A) are identified with equivalence classes of infinitesimal deformations. It is shown that elements of a certain part of H ¹ (A) can be interpreted as recursion operators for the object (A, ), i.e. operators giving rise to infinite series of symmetries. Explicit formulas for computing recursion operators are deduced. The general theory is illustrated by a particular example of a graded differential equation, i.e. the Super KdV equation.Tverskoy-Yamskoy per. 14, Apt. 45, 125047 Moscow, Russia.     

KdV Equations and integrability detectors

We present a review of the various integrability detectors that have been developed based on the study of the singularities of the solutions of a given equation: the Painlevé method for continuous systems, and the singularity confinement approach for discrete ones. In each case the KdV equation was instrumental in the formulation of the conjectures relating the singularity structure to integrability.     

Colour calculus and colour quantizations

A colour calculus linked with an any discrete groupG is developed. Colour differential operators and colour jets are introduced. Algebras colour differential forms and de Rham complexes are constructed. For colour differential equations, Spencer complexes are constructed. Relations between colour commutative algebras and quantizations of usual algebras are considered.     

Integrable nonlinear evolution equations and dynamical systems in multidimensions

The investigation of nonlinear evolution equations and dynamical systems integrable in multidimensions constitutes at present our main research interest. Here we survey findings obtained recently as well as over time: solvable equations (both PDEs and ODEs) are reported, philosophical motivations and methodological approaches are outlined. For more detailed treatments, including the display and analysis of solutions, the interested reader is referred to the original papers.On leave while serving as Secretary General, Pugwash Conferences on Science and World Affairs, Geneva, London, Rome.     

Generalized self-dual Yang-Mills flows,explicit solutions and reductions

An evolution equation is added to the generalized self-dual Yang-Mills equations. The evolution equation contains terms of negative powers of the spectral parameter as well as terms of positive powers. The Darboux matrix method is used to obtain explicit solutions, especially single and multiple solitons. All integrable soliton equations in the framework of AKNS system (inR ⁿ⁺¹ orR ¹⁺¹) can be derived from the generalized Yang-Mills flows by reduction.Supported by Chinese research project Nonlinear Science and K.C. Wong Education Foundation, Hong Kong.     

The reductive perturbation method and the Korteweg-de Vries hierarchy

By using the reductive perturbation method of Taniuti with the introduction of an infinite sequence of slow time variables ₁, ₃, ₅, ..., we study the propagation of long surface-waves in a shallow inviscid fluid. The Korteweg-de Vries (KdV) equation appears as the lowest order amplitude equation in slow variables. In this context, we show that, if the lowest order wave amplitude ₀ satisfies the KdV equation in the time ₃, it must satisfy the (2n+1)th order equation of the KdV hierarchy in the time _n+1, withn=2,3,4, ... As a consequence of this fact, we show with an explicit example that the secularities of the evolution equations for the higher-order terms ( ₁, ₂, ...) of the amplitude can be eliminated when ₀ is a solitonic solution to the KdV equation. By reversing this argument, we can say that the requirement of a secular-free perturbation theory implies that the amplitude ₀ satisfies the (2n+ 1)th order equation of the KdV hierarchy in the time _2n+1. This essentially means that the equations of the KdV hierarchy do play a role in perturbation theory. Thereafter, by considering a solitary-wave solution, we show, again with an explicit, example that the elimination of secularities through the use of the higher order KdV hierarchy equations corresponds, in the laboratory coordinates, to a renormalization of the solitary-wave velocity. Then, we conclude that this procedure of eliminating secularities is closely related to the renormalization technique developed by Kodama and Taniuti.     

New features of soliton dynamics in 2+1 dimensions

Exponentially localized soliton solutions have been found recently for all the equations of the hierarchy related to the Zakharov-Shabat hyperbolic spectral problem in the plane. In particular theN ²-soliton solution of the Davey-Stewartson I equation is considered. It is shown that the boundaries fix the kinematics of solitons, while the dynamics of their mutual interaction is determined by the chosen initial condition. The interacting solitons can have, asymptotically, zero mass and can simulate quantum effects as inelastic scattering, fusion and fission, creation and annihilation.Work supported in part by M.U.R.S.T.     

Deformations of algebraic curves and integrable nonlinear evolution equations

A brief exposition of applications of the methods of algebraic geometry to systems integrable by the IST method with variable spectral parameters is presented. Usually, theta-functional solutions for these systems are generated by some deformations of algebraic curves. The deformations of algebraic curves are also related with theta-functional solutions of Yang-Mills self-duality equations which contain special systems with a variable spectral parameter as a special reduction. Another important situation in which the deformations of algebraic curves naturally occur is the KdV equation with string-like boundary conditions. Most important concrete examples of systems integrable by the IST method with variable spectral parameter having different properties within a framework of the behavior of moduli of underlying curves, analytic properties of the Baker-Akhiezer functions, and the qualitative behavior of the solutions, are vacuum axially symmetric Einstein equations, the Heisenberg cylindrical magnet equation, the deformed Maxwell-Bloch system, and the cylindrical KP equation.Dedicated to the memory of J.-L. Verdier     

Symmetries and conservation laws of Kadomtsev-Pogutse equations

Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials in action. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical interpretation are demonstrated.     

On thec-spectral sequence for systems of evolution equations

The groups E ₁ ^2,n–1 ( ) of Vinogradov'sC-spectral sequence for determined systems of evolution equations are considered. Presentation of these groups useful in practical computations is obtained. The group E ₁ ^2,1 ( ) is calculated for a system of Schrödinger type equations.     

The nth reduced BKP hierarchy, the string equation and BW 1+∞-constraints

We study the BKP hierarchy and its n-reduction, for the case that n is odd. This is related to the principal realization of the basic module of the twisted affine Lie algebra % MathType!MTEF!2!1!+-% feaafeart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn% hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr% 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqaqpepeea0dXdb9aqVe% 0larpepe0lb9cs0-LqLs-Jarpepeea0-qqVe0Firpepa0xar-xfr-x% fj-hmeGabaqaciGacaGaaeqabaWaaeaaeaaakeaadaWfGaqaaiGaco% hacaGGSbWaaSbaaSqaaiaac6gaaeqaaaqabKqaGhaacqWIh4ETaaGc% daahaaWcbeqaaiaacIcacaaIYaGaaiykaaaaaaa!3B2F!\[\mathop {{\mathop{\rm sl}\nolimits} _n }\limits^ ^{(2)} \]. We show that the following two statements for a BKP function are equivalent: (1) is is n-reduced and satisfies the string equation, i.e., L _-1=0, where L _-1 is an element of some natural Virasoro algebra. (2) satisfies the vacuum constraints of the BW ₁₊ algebra. Here BW ₁₊ is the natural analog of the W ₁₊ algebra, which plays a role in the KP case.The research of Johan van de Leur is financially supported by the Stichting Fundamenteel Onderzoek der Materie (FOM).     

Elliptic solutions of nonlinear integrable equations and related topics

The theory of elliptic solitons for the Kadomtsev-Petviashvili (KP) equation and the dynamics of the corresponding Calogero-Moser system is integrated. It is found that all the elliptic solutions for the KP equation manifest themselves in terms of Riemann theta functions which are associated with algebraic curves admitting a realization in the form of a covering of the initial elliptic curve with some special properties. These curves are given in the paper by explicit formulae. We further give applications of the elliptic Baker-Akhiezer function to generalized elliptic genera of manifolds and to algebraic 2-valued formal groups.Dedicated to the memory of J.-L. Verdier     

Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Monotonicity formula and small action regularity for Yang-Mills flows in higher dimensions

In this paper the monotonicity formula and the small action regularity theorem for the regular heat flow of Yang-Mills connections in higher dimensions are established.     

Connections on manifolds and new characteristic classes

We give an extension of Maslov-Arnold classes to a certain class of symplectic manifolds. It is proved that any such generalized class of minimal surfaces is equal to zero for a large class of stable minimal surfaces. We describe some applications to pseudo-Riemannian geometry and to the investigation of completely integrable Hamiltonian systems.