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A discrete time relativistic Toda lattice

Four integrable symplectic maps approximating two Hamiltonian flows from the relativistic Toda hierarchy are introduced. They are demostrated to belong to the same hierarchy and to examplify the general scheme for symplectic maps on groups equiped with quadratic Poisson brackets. The initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flows are found for all the maps.

On Integrable Models and their Interrelations

We present an elementary discussion of the Calogero-Moser model. This gives us an opportunity to illustrate basic concepts of the dynamical integrable models. Some ideas are also presented regarding interconnections between integrable models based on the relation established between the Calogero-Moser model and the truncated KP hierarchy of Burgers-Hopf type.

The Gambier Mapping

We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained through a system of Riccati equations. The same holds true in the discrete case also. We use the singularity confinement criterion in order to study the integrability of this new mapping.

Discrete soliton equations and convergence acceleration algorithms

Some of the well-known convergence acceleration algorithms, when viewed as two-variable difference equations, are equivalent to discrete soliton equations. It is shown that the $\eta-$algorithm is nothing but the discrete KdV equation. In addition, one generalized version of the $\rho-$algorithm is considered to be integrable discretization of the cylindrical KdV equation.

Constructive building of the Lax pair in the non-linear sigma model

A derivation of the Lax pair for the (1+1)-dimensional non-linear sigma-model is described. Its main benefit is to have a clearer physical origin and to allow the study of a generalization to higher dimensions.

Explicit and Exact Solutions to a Kolmogorov-Petrovskii-Piskunov Equation

Some explicit traveling wave solutions to a Kolmogorov-Petrovskii-Piskunov equation are presented through two ans\"atze. By a Cole-Hopf transformation, this Kolmogorov-Petrovskii-Piskunov equation is also written as a bilinear equation and further two solutions to describe nonlinear interaction of traveling waves are generated. B\"acklund transformations of the linear form and some special cases are considered.

Algebra of Non-Local Charges in Supersymmetric Non-Linear Sigma Models

We propose a graphic method to derive the classical algebra (Dirac brackets) of non-local conserved charges in the two dimensional supersymmetric non-linear $O(N)$ sigma model. As in the purely bosonic theory we find a cubic Yangian algebra. We also consider the extension of graphic methods to other integrable theories.

Remarks on the Whitham equations

We survey some topics involving the Whitham equations, concentrating on the role of the product of the wave function and its adjoint in averaging and in producing Cauchy kernels and differentials on Riemann surfaces. There are also some new results.

A discrete time peakons lattice

A discretization of the peakons lattice is introduced, belonging to the same hierarchy as the continuous--time system. The construction examplifies the general scheme for integrable discretization of systems on Lie algebras with $r$--matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow is found. A variational (Lagrangian) formulation is also given.

On evolution of multiphase nonlinear modulated waves

We present a fundamental solution to an initial value problem for the KdV-Whitham system in an explicit integral form. Monotonically decreasing initial data with finite number of breaking points are considered. Generating function for the commuting flows of the averaged KdV hieararchy producing the analytical solutions to the KdV-Whitham system is constructed.

Integrable Models of the CFT on Hyper-Elliptic Surfaces

In this letter, we continue the work we started at a previous paper and we propose new series of integrable models in quantum field theory. These models are obtained as perturbed models of the minimal conformal field theories on the hyper-elliptic surfaces by particular relevant operators of the untwisted sector. The quantum group symmetry of the models is also discussed.

Discrete time Bogoyavlensky lattices

Discretizations of the Bogoyavlensky lattices are introduced, belonging to the same hierarchies as the continuous--time systems. The construction exemplifies the general scheme for integrable discretization of systems on Lie algebras with $r$--matrix Poisson brackets. An initial value problem for the difference equations is solved in terms of a factorization problem in a group. Interpolating Hamiltonian flow is found.

The Bi-Hamiltonian Structure of the Perturbation Equations of KdV Hierarchy

The bi-Hamiltonian structure is established for the perturbation equations of KdV hierarchy and thus the perturbation equations themselves provide also examples among typical soliton equations. Besides, a more general bi-Hamiltonian integrable hierarchy is proposed and a remark is given for a generalization of the resulting perturbation equations to $1+2$ dimensions.

Bilinear structure and Schlesinger transforms of the $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$ equations

We show that the recently derived ($q$-) discrete form of the Painlev\'e VI equation can be related to the discrete P$_{\rm III}$, in particular if one uses the full freedom in the implementation of the singularity confinement criterion. This observation is used here in order to derive the bilinear forms and the Schlesinger transformations of both $q$-P$_{\rm III}$ and $q$-P$_{\rm VI}$.

Localized Induction Hierarchy and Weingarten Systems

We describe a method of constructing Weingarten systems of triply orthogonal coordinates, related to the localized induction equation hierarchy of integrable geometric evolution equations

Some Classes of Solutions to the Toda Lattice Hierarchy

We apply an analogue of the Zakharov-Shabat dressing method to obtain infinite matrix solutions to the Toda lattice hierarchy. Using an operator transformation we convert some of these into solutions in terms of integral operators and Fredholm determinants. Others are converted into a class of operator solutions to the $l$-periodic Toda hierarchy.

From Discrete to Continuous Painlev\'e Equations: A Bilinear Approach

We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new and interesting result is the bilinearization of the P$_{\rm VI}$ equation, something that was missing till now.

Cellular Automata and Ultra-Discrete Painlev\'e Equations

Starting from integrable cellular automata we present a novel form of Painlev\'e equations. These equations are discrete in both the independent variable and the dependent one. We show that they capture the essence of the behavior of the Painlev\'e equations organizing themselves into a coalescence cascade and possessing special solutions. A necessary condition for the integrability of cellular automata is also presented.

Lax representation for two--particle dynamics splitting on two tori

Lax representation in terms of $2\times 2$ matrices is constructed for a separable multiply--periodic system splitting on two tori. Hyperelliptic Kleinian functions and their reduction to elliptic functions are used.

Hyperelliptic Kleinian functions and applications

We develop the theory of hyperelliptic Kleinian functions. As applications we consider construction of the explicit matrix realization of the hyperelliptic Kummer varieties, differential operators to have the hyperelliptic curve as spectral variety, solution of the KdV equations by Kleinian functions.

Dynamical r-matrix for the elliptic Ruijsenaars-Schneider system

The classical r-matrix structure for the generic elliptic Ruijsenaars-Schneider model is presented. It makes the integrability of this model as well as of its discrete-time version that was constructed in a recent paper manifest.

The Hubbard model on a complete graph: Exact Analytical results

We derive the analytical expression of the ground state of the Hubbard model with unconstrained hopping at half filling and for arbitrary lattice sites.

Ferromagnetic ground states of the Hubbard model on a complete graph

We use group theory to derive the exact analytical expression of the ferromagnetic ground states of the Hubbard model on a complete graph for arbitrary lattice sites f and for arbitrary fillings $N$. We find that for $t>0$ and for $N=f+1$ the ground state is maximally ferromagnetic with total spin $S=(f-1)/2$. For $N > f+1$ the ground state is still ferromagnetic but becomes degenerate with respect to $S$.

SO(4) invariant basis functions for strongly correlated Fermi systems

We show how to construct SO(4) invariant functions for strongly correlated Fermi systems on lattices of finite sizes. We illustrate the method on the case of the 1D Hubbard chain with four and with six sites.

Approximate Controlability by Control Constraints for Infinite Dimensional Systems

For linear infinite systems the approximate controllability problem by control constraints is considered. Controllability conditions represented via system parameters are obtained. Partial differential control systems and control systems with delays are considered as an example.

Elliptic Ruijsenaars-Schneider and Calogero-Moser hierarchies are governed by the same r-matrix

We demonstrate that in a certain gauge the elliptic Ruijsenaars--Schneider models admit Lax representation governed by the same dynamical $r$--matrix as their non--relativistic counterparts (Calogero--Moser models). This phenomenon was previously observed for the rational and hyperbolic models.

Algebraic properties of the 1+1 dimensional Heisenberg spin field model

The Estabrook-Wahlquist prolongation method is applied to the (compact and noncompact) continuous isotropic Heisenberg model in 1 + 1 dimensions. Using a special realization (an algebra of the Kac-Moody type) of the arising incomplete prolongation Lie algebra, a whole family of nonlinear field equations containing the original Heisenberg system is generated.

Integrable discretizations of the spin Ruijsenaars-Schneider models

Integrable discretizations are introduced for the rational and hyperbolic spin Ruijsenaars--Schneider models. These discrete dynamical systems are demonstrated to belong to the same integrable hierarchies as their continuous--time counterparts. Explicit solutions are obtained for arbitrary flows of the hierarchies, including the discrete time ones.

The Painlev\'e Test of Higher Dimensional KdV Equation

We argue the integrability of the generalized KdV(GKdV) equation using the Painlev\'e test. For $d( \le 2)$ dimensional space, GKdV equation passes the Painlev\'e test but does not for $d \geq 3$ dimensional space. We also apply the Ablowitz-Ramani-Segur's conjecture to the GKdV equation in order to complement the Painlev\'e test.

New integrable systems related to the relativistic Toda lattice

New integrable lattice systems are introduced, their different integrable discretization are obtained. B\"acklund transformations between these new systems and the relativistic Toda lattice (in the both continuous and discrete time formulations) are established.

A new integrable system related to the Toda lattice

A new integrable lattice system is introduced, and its integrable discretizations are obtained. A B\"acklund transformation between this new system and the Toda lattice, as well as between their discretizations, is established.

On completeness of the Moutard transformations

In this paper we solve positively the problem of (local) density of the "potentials" M(x,y) of the Moutard equation, u_{xy} = M(x,y) u, u=u(x,y), (used in many papers for construction of exact solutions of (2+1)-dimensional integrable systems) obtainable from a given initial potential with consecutive Moutard transformations.

Symmetries of KdV and Loop Groups

A simple version of the Segal-Wilson map from the SL(2,C) loop group to a class of solutions of the KdV hierarchy is given, clarifying certain aspects of this map. It is explained how the known symmetries, including Backlund transformations, of KdV arise from simple, field independent, actions on the loop group. A variety of issues in understanding the algebraic structure of Backlund transformations are thus resolved.

Some survey remarks on Whitham theory and EM duality

The nature of the BA function and its adjoint for KP-Toda is traced through the averaging method in generating the Whitham equations, differentials, and symplectic forms, with connections to EM duality.

Determinant Structure of the Rational Solutions for the Painlev\'e II Equation

Two types of determinant representations of the rational solutions for the Painlev\'e II equation are discussed by using the bilinear formalism. One of them is a representation by the Devisme polynomials, and another one is a Hankel determinant representation. They are derived from the determinant solutions of the KP hierarchy and Toda lattice, respectively.

Some further curiosities from the world of integrable lattice systems and their discretizations

Unexpected relations are found between the Toda lattice, the relativistic Toda lattice and the Bruschi--Ragnisco lattice, as well as between their integrable discretizations.

Gibbs' States for Moser-Calogero Potentials

We present two independent approaches for computing the thermodynamics for classical particles interacting via the Moser--Calogero potential. Combining the results we propose the form of equation of state or, what is equivalent, the asymptotics of the Jacobian between volume elements corresponding two symplectic structures on the phase space.

On the dynamics of rational solutions for 1-D generalized Volterra system

The Hirota bilinear formalism and soliton solutions for a generalized Volterra system is presented. Also, starting from the soliton solutions, we obtain a class of nonsingular rational solutions using the "long wave" limit procedure of Ablowitz and Satsuma, and appropriate "gauge" transformations. Their properties are also discussed and it is shown that these solutions interact elastically with no phase shift.

Fusion Hierarchies with Open Boundaries and Exactly Solvable Models

The formulation of integrable models with open boundary conditions and the functional relations of fused transfer matrices are discussed. It is shown that finite-size corrections to the transfer matrices and unitarity relations of free energies can be obtained from the functional relations. Unitarity relations of surface free energies presented in previous papers are also reviewed.

On Simplest Hamiltonian Systems

Simple Hamiltonian systems, such as mathematical pendulum or Euler equations for rigid body, are solved without computation. It is nothing but a joke but maybe you will find it nice.

A geometrical method towards first integrals for dynamical systems

We develop a method, based on Darboux' and Liouville's works, to find first integrals and/or invariant manifolds for a physically relevant class of dynamical systems, without making any assumption on these elements' form. We apply it to three dynamical systems: Lotka--Volterra, Lorenz and Rikitake.

Discrete version of the Chazy class III equation

We study the discretisation of the Chazy class III equation by two means: a discrete Painlev\'e test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.

Common Algebraic Structure for the Calogero-Sutherland Models

We investigate common algebraic structure for the rational and trigonometric Calogero-Sutherland models by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis for the rational case.

Quasi-point separation of variables for the Henon-Heiles system and a system with quartic potential

We examine the problem of integrability of two-dimensional Hamiltonian systems by means of separation of variables. The systematic approach to construction of the special non-pure coordinate separation of variables for certain natural two-dimensional Hamiltonians is presented. The relations with SUSY quantum mechanics are discussed.

Virasoro Symmetry Algebra of Dirac Soliton Hierarchy

A hierarchy of first-degree time-dependent symmetries is proposed for Dirac soliton hierarchy and their commutator relations with time-dependent symmetries are exhibited. Meantime, a hereditary structure of Dirac soliton hierarchy is elucidated and a Lax operator algebra associated with Virasoro symmetry algebra is given.

The Kowalewski top: a new Lax representation

The 2x2 monodromy matrices for the Kowalewski top on the Lie algebras e(3), so(4) and so(3,1) are presented. The corresponding quadratic R-matrix structure is the dynamical deformation of the standard R-matrix algebras. Some tops and Toda lattices related to the Kowalewski top are discussed.

N=2 KP and KdV hierarchies in extended superspace

We give the formulation in extended superspace of an $N=2$ supersymmetric KP hierarchy using chirality preserving pseudo-differential operators. We obtain two quadratic hamiltonian structures, which lead to different reductions of the KP hierarchy. In particular we find two different hierarchies with the $N=2$ classical super-${\cal W}_n$ algebra as a hamiltonian structure. The relation with the formulation in $N=1$ superspace is carried out.

Automorphisms of sl(2) and dynamical r-matrices

Two outer automorphisms of infinite-dimensional representations of $sl(2)$ algebra are considered. The similar constructions for the loop algebras and yangians are presented. The corresponding linear and quadratic $R$-brackets include the dynamical $r$-matrices.

Contractions of Integrable Equations

The contraction is applied to obtaining of integrable systems associated with nonsemisimple algebras. The effect of contraction is splitting off some components from initial system without loss of integrability.

Bilinearization of Discrete Soliton Equations and Singularity Confinement

Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.

A new method to test discrete Painlev\'e equations

Necessary discretization rules to preserve the Painlev\'e property are stated. A new method is added to the discrete Painlev\'e test, which perturbs the continuous limit and generates infinitely many no-log conditions.

Lax Representations and Zero Curvature Representations by Kronecker Product

It is showed that Kronecker product can be applied to construct not only new Lax representations but also new zero curvature representations of integrable models. Meantime a different characteristic between continuous and discrete zero curvature equations is pointed out.

An orthogonal basis for the $B_N$-type Calogero model

We investigate algebraic structure for the $B_N$-type Calogero model by using the exchange-operator formalism. We show that the set of the Jack polynomials whose arguments are Dunkl-type operators provides an orthogonal basis.

Complex Blow-Up in Burgers' Equation: an Iterative Approach

We show that for a given holomorphic noncharacteristic surface S in two-dimensional complex space, and a given holomorphic function on S, there exists a unique meromorphic solution of Burgers' equation which blows up on S. This proves the convergence of the formal Laurent series expansion found by the Painlev\'e test. The method used is an adaptation of Nirenberg's iterative proof of the abstract Cauchy-Kowalevski theorem.

Some explicit solutions of the Lam\'e and Bourlet type equations

Some special solutions to the multidimensional Lam\'e and Bourlet type equations are constructed in an explicit form.

Spinless Calogero-Sutherland model with twisted boundary condition

In this work, the spinless Calogero-Sutherland model with twisted boundary condition is studied. The ground state wavefunctions, the ground state energies, the full energy spectrum are provided in details.

The orthogonal eigenbasis and norms of eigenvectors in the Spin Calogero-Sutherland Model

Using a technique based on the Yangian Gelfand-Zetlin algebra and the associated Yangian Gelfand-Zetlin bases we construct an orthogonal basis of eigenvectors in the Calogero-Sutherland Model with spin, and derive product-type formulas for norms of these eigenvectors.

The L-Matrix for the Massive Thirring Model

As the new results for the massive Thirring model the L-matrix and the algebraic relations for its action angle variables are given. So it is shown most directly that this model which describes self-interacting relativistic Fermions in one-dimensional space is a quantum integrable system.

Correlators of the phase model

We introduce the phase model on a lattice and solve it using the algebraic Bethe ansatz. Time-dependent temperature correlation functions of phase operators and the "darkness formation probability" are calculated in the thermodynamical limit. These results can be used to construct integrable equations for the correlation functions and to calculate there asymptotics.

Solutions of (2+1)-dimensional spin systems

We use the methods of group theory to reduce the equations of motion of two spin systems in (2+1) dimensions to sets of coupled ordinary differential equations. We present solutions of some classes of these sets and discuss their physical significance.

Geometric Discretisation of the Toda System

The Laplace sequence of the discrete conjugate nets is constructed. The invariants of the nets satisfy, in full analogy to the continuous case, the system of difference equations equivalent to the discrete version of the generalized Toda equation.

Multidimensional Quadrilateral Lattices are Integrable

The notion of multidimensional quadrilateral lattice is introduced. It is shown that such a lattice is characterized by a system of integrable discrete nonlinear equations. Different useful formulations of the system are given. The geometric construction of the lattice is also discussed and, in particular, it is clarified the number of initial--boundary data which define the lattice uniquely.

The Whitham equations revisited

We survey some topics involving the Whitham equations, concentrating on the role of the Baker Akhiezer function in averaging. Some connections to symplectic geometry and Seiberg-Witten theory are indicated.

Some kernels on a Riemann surface

We discuss certain kernels on a Riemann surface, constructed mainly via Baker Akhiezer functions, and indicate relations to dispersionless theory.

Darboux Transformation for the Manin-Radul Supersymmetric KdV equation

In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to the Korteweg-de Vries equation. Soliton type solutions are constructed by dressing the vacuum and we present some relevant plots.

Separation of variables for the Dn type periodic Toda lattice

We prove separation of variables for the most general (Dn type) periodic Toda lattice with 2x2 Lax matrix. It is achieved by finding proper normalisation for the corresponding Baker-Akhiezer function. Separation of variables for all other periodic Toda lattices associated with infinite series of root systems follows by taking appropriate limits.

Twisted Quantum Lax Equations

We give the construction of twisted quantum Lax equations associated with quantum groups. We solve these equations using factorization properties of the corresponding quantum groups. Our construction generalizes in many respects the Adler-Kostant-Symes construction for Lie groups and the construction of M. A. Semenov Tian-Shansky for the Lie-Poisson case.

The structures underlying soliton solutions in integrable hierarchies

We point out that a common feature of integrable hierarchies presenting soliton solutions is the existence of some special ``vacuum solutions'' such that the Lax operators evaluated on them, lie in some abelian subalgebra of the associated Kac-Moody algebra. The soliton solutions are constructed out of those ``vacuum solitons'' by the dressing transformation procedure.

The Fourier method for the linearized Davey-Stewartson I equation

The linearized Davey-Stewartson equation with varing coefficients is solved by Fourier method. The approach uses the inverse scattering transform for the Davey-Stewartson equation.

The Yangian Symmetry in the Spin Calogero Model and its Applications

By using the non-symmetric Hermite polynomials and a technique based on the Yangian Gelfand-Zetlin bases, we decompose the space of states of the Calogero model with spin into irreducible Yangian modules, construct an orthogonal basis of eigenvectors and derive product-type formulas for norms of these eigenvectors.

Some eigenstates for a model associated with solutions of tetrahedron equation

Here we present some eigenstates for a 2+1-dimensional model associated with a solution of the tetrahedron equation. The eigenstates include those "particle-like" (namely one-particle and two-particle ones), constructed in analogy with the usual 1+1-dimensional Bethe ansatz, and some simple "string-like" ones.

The Solution of the N=2 Supersymmetric f-Toda Chain with Fixed Ends

The integrability of the recently introduced N=2 supersymmetric f-Toda chain, under appropriate boundary conditions, is proven. The recurrent formulae for its general solutions are derived. As an example, the solution for the simplest case of boundary conditions is presented in explicit form.

Is the classical Bukhvostov-Lipatov model integrable? A Painlev\'e analysis

In this work we apply the Weiss, Tabor and Carnevale integrability criterion (Painlev\'e analysis) to the classical version of the two dimensional Bukhvostov-Lipatov model. We are led to the conclusion that the model is not integrable classically, except at a trivial point where the theory can be described in terms of two uncoupled sine-Gordon models.

Coupled Nonlinear Schr\"{o}dinger equation and Toda equation (the Root of Integrability)

We consider the relation between the discrete coupled nonlinear Schr\"{o}dinger equation and Toda equation. Introducing complex times we can show the intergability of the discrete coupled nonlinear Schr\"{o}dinger equation. In the same way we can show the integrability in coupled case of dark and bright equations. Using this method we obtain several integrable equations.

Rational Solutions for the Discrete Painlev\'e II Equation

The rational solutions for the discrete Painlev\'e II equation are constructed based on the bilinear formalism. It is shown that they are expressed by the determinant whose entries are given by the Laguerre polynomials. Continuous limit to the Devisme polynomial representation of the rational solutions for the Painlev\'e II equation is also discussed.

Fully Supersymmetric Hierarchies From A Energy Dependent Super Hill Operator

A super Hill operator with energy dependent potentials is proposed and the associated integrable hierarchy is constructed explicitly. It is shown that in the general case, the resulted hierarchy is multi-Hamiltonian system. The Miura type transformations and modified hierarchies are also presented.

Asymptotics for Solution to the Cauchy Problem for Volterra Lattice with Step-Like Initial Values

The connection between modulated Riemann surface of genus one and solution to Volterra lattice that tends to constants at infinity is studied. The main term of asymptotics for large time of solution to the mentioned Cauchy problem is written out.

Quasi-BiHamiltonian Systems and Separability

Two quasi--biHamiltonian systems with three and four degrees of freedom are presented. These systems are shown to be separable in terms of Nijenhuis coordinates. Moreover the most general Pfaffian quasi-biHamiltonian system with an arbitrary number of degrees of freedom is constructed (in terms of Nijenhuis coordinates) and its separability is proved.

An Integral Operator Solution to the Matrix Toda Equations

In previous work the author found solutions to the Toda equations that were expressed in terms of determinants of integral operators. Here it is observed that a simple variant yields solutions to the matrix Toda equations. As an application another derivation is given of a differential equation of Sato, Miwa and Jimbo for a particular Fredholm determinant.

On the point transformations for the second order differential equations. I

Point transformations for the ordinary differential equations of the form $y''=P(x,y)+3 Q(x,y) y'+3 R(x,y) (y')^2+S(x,y) (y')^3$ are considered. Some classical results are resumed. Solution for the equivalence problem for the equations of general position is described.

A collection of integrable systems of the Toda type in continuous and discrete time, with 2x2 Lax representations

A fairly complete list of Toda-like integrable lattice systems, both in the continuous and discrete time, is given. For each system the Newtonian, Lagrangian and Hamiltonian formulations are presented, as well as the 2x2 Lax representation and r-matrix structure. The material is given in the "no comment" style, in particular, all proofs are omitted.

Some eigenstates for a model associated with solutions of tetrahedron equation. III. Tetrahedral Zamolodchikov algebras and perturbed strings

This paper continues the series begun with works solv-int/9701016 and solv-int/9702004. Here we show how to construct eigenstates for a model based on tetrahedron equation using the tetrahedral Zamolodchikov algebras. This yields, in particular, new eigenstates for the model on infinite lattice -- `perturbed', or `broken', strings.

Non-standard Construction of Hamiltonian Structures

Examples of the construction of Hamiltonian structures for dynamical systems in field theory (including one reputedly non-Hamiltonian problem) without using Lagrangians, are presented. The recently developed method used requires the knowledge of one constant of the motion of the system under consideration and one solution of the symmetry equation.

WDVV and DZM

We show how the WDVV equations and the DZM system can be characterized via a background family of functions.

Convergent Normal Forms of Symmetric Dynamical Systems

It is shown that the presence of Lie-point-symmetries of (non-Hamiltonian) dynamical systems can ensure the convergence of the coordinate transformations which take the dynamical sytem (or vector field) into Poincar\'e-Dulac normal form.

Complex Analysis of a Piece of Toda Lattice

We study a small piece of two dimensional Toda lattice as a complex dynamical system. In particular the Julia set, which appears when the piece is deformed, is shown analytically how it disappears as the system approaches to the integrable limit.

The Correspondence between Discrete Surface and Difference Geometry of the KP-hierarchy

The correspondence between two geometrical descriptions of the KP-hierarchy, one by discrete surface and another by difference analogue of differential geometry, is given.

Dual Resonance Model Solves the Yang-Baxter Equation

The duality of dual resonance models is shown to imply that the four point string correlation function solves the Yang-Baxter equation. A reduction of transfer matrices to $A_l$ symmetry is described by a restriction of the KP $\tau$ function to Toda molecules.

Singularity analysis towards nonintegrability of nonhomogeneous nonlinear lattices

We show non-integrability of the nonlinear lattice of Fermi-Pasta-Ulam type via the singularity analysis(Picard-Vessiot theory) of normal variational equations of Lam\'e type.

Non-commutative and commutative integrability of generic Toda flows in simple Lie algebras

In this paper we prove the complete integrability of Toda flows on generic coadjoint orbits in simple Lie algebras.

Huygens' Principle in Minkowski Spaces and Soliton Solutions of the Korteweg-de Vries Equation

A new class of linear second order hyperbolic partial differential operators satisfying Huygens' principle in Minkowski spaces is presented. The construction reveals a direct connection between Huygens' principle and the theory of solitary wave solutions of the Korteweg-de Vries equation.

Some eigenstates for a model associated with solutions of tetrahedron equation. IV. String-particle marriage

This paper continues the series begun with works solv-int/9701016, solv-int/9702004 and solv-int/9703010. Here we construct more sophisticated strings, combining ideas from those papers and some considerations involving solutions of tetrahedron equation due to Sergeev, Mangazeev and Stroganov.

Trilinear representation and the Moutard transformation for the Tzitzeica equation

In the paper we present a trilinear form and a Darboux-type transformation to an equation considered by Tzitzeica in 1910. This equation equivalent to the Bullough-Dodd-Jiber-Shabat equation. Soliton solutions are constructed by dressing the trivial solution.

Computation of conserved densities for systems of nonlinear differential-difference equations

A new method for the computation of conserved densities of nonlinear differential-difference equations is applied to Toda lattices and discretizations of the Korteweg-de Vries and nonlinear Schrodinger equations. The algorithm, which can be implemented in computer algebra languages such as Mathematica, can be used as an indicator of integrability.

Rational solutions to d-PIV

We study the rational solutions of the discrete version of Painleve's fourth equation d-PIV. The solutions are generated by applying Schlesinger transformations on the seed solutions -2z and -1/z. After studying the structure of these solutions we are able to write them in a determinantal form that includes an interesting parameter shift that vanishes in the continuous limit.

On superintegrable systems closed to geodesic motion

In this work we consider superintegrable systems in the classical $r$-matrix method. By using other authomorphisms of the loop algebras we construct new superintegrable systems with rational potentials from geodesic motion on $R^{2n}$.

On Adomian's Decomposition Method for Solving Differential Equations

We show that with a few modifications the Adomian's method for solving second order differential equations can be used to obtain the known results of the special functions of mathematical physics. The modifications are necessary in order to take correctly into account the behaviour of the solutions in the neighborhood of the singular points.

Integrability of Riccati equations and the stationary KdV equations

Using the S.Lie's infinitesimal approach we establish the connection between integrability of a one-parameter family of the Riccati equations and the stationary KdV hierarchy.

Stationary mKdV hierarchy and integrability of the Dirac equations by quadratures

Using the Lie's infinitesimal method we establish that the Dirac equation in one variable is integrable by quadratures if the potential V(x) is a solution of one of the equations of the stationary mKdV hierarchy.

R-Matrices and Generalized Inverses

Four results are given that address the existence, ambiguities and construction of a classical R-matrix given a Lax pair. They enable the uniform construction of R-matrices in terms of any generalized inverse of $ad L$. For generic $L$ a generalized inverse (and indeed the Moore-Penrose inverse) is explicitly constructed. The R-matrices are in general momentum dependent and dynamical. The construction applies equally to Lax matrices with spectral parameter.