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arXiv1994

Integrable Quantum Mappings

We discuss the canonical structure of a class of integrable quantum mappings, i.e. iterative canonical transformations that can be interpreted as a discrete dynamical system. As particular examples we consider quantum mappings associated with the lattice analogues of the KdV and MKdV equations. These mappings possess a non-ultralocal quantum Yang-Baxter structure leading to the existence of commuting families of exact quantum invariants. We derive the associated quantum Miura transformations between these mappings and the corresponding quantum bi-Hamiltonian structure.

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arXiv1994

Analytic Solutions of the Vector Burgers' Equation

The well-known analytical solution of Burgers' equation is extended to curvilinear coordinate systems in three-dimensions by a method which is much simpler and more suitable to practical applications than that previously used. The results obtained are applied to incompressible flow with cylindrical symmetry, and also to the decay of an initially linearly increasing wind.

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arXiv1994

The Integrable Dynamics of Discrete and Continuous Curves

We show that the following geometric properties of the motion of discrete and continuous curves select integrable dynamics: i) the motion of the curve takes place in the N dimensional sphere of radius R, ii) the curve does not stretch during the motion, iii) the equations of the dynamics do not depend explicitly on the radius of the sphere. Well known examples of integrable evolution equations, like the nonlinear Schroedinger and the sine-Gordon equations, as well as their discrete analogues, are derived in this general framework.

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arXiv1995

Dual Field Approach to Correlation Functions in the Heisenberg Xxz Spin Chain

We study zero temperature correlation functions of the spin-$1\over 2$ Heisenberg XXZ model in the critical regime $-1< Δ\leq 1$ in a magnetic field by means of the {\tenit Dual Field Approach}. We show for one particular example how to derive determinant representations for correlation functions and how to use these to embed the correlation functions in integrable systems of integro-difference equations (IDE). These IDE are associated with a Riemann-Hilbert problem.

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arXiv1995

Conservation Laws in Field Dynamics or Why Boundary Motion is Exactly Integrable?

An infinite number of conserved quantities in the field dynamics $ϕ_t = L U(ϕ) + ρ$ for a linear Hermitian (or anti-Hermitian) operator $L$, an arbitrary function $U$ and a given source $ρ$ are presented. These integrals of motion are the multipole moments of the potential created by $ϕ$ in the far-field. In the singular limit of a bistable scalar field $ϕ= ϕ_{\pm}$ (i.e. Ising limit) this theory describes a dissipative boundary motion (such as Stefan or Saffman-Taylor problem that is the continuous limit of the DLA-fractal growth) and can be exactly integrable. These conserved quantities are the polynomial conservation laws attributed to the integrability. The criterion for integrability is the uniqueness of the inverse potential problem's solution.

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arXiv1995

Approximate Lie Group Analysis of a Model Advection Equation on an Unstructured Grid

A technique of ``approximate group analysis'' recently developed by Baikov, Gazizov and Ibragimov is applied to a differential approximation (otherwise referred to as an equivalent differential equation) corresponding to the finite difference approximation of a nonlinear advection equation on unstructured grid. We determine which groups from the infinite variety of groups admitted by a nonlinear advection equation ``survive'' the discretization. The situations arising for different choices of an arbitrary function (local speed of propagation) are also studied.

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arXiv1995

Binary nonlinearization for the Dirac systems

A Bargmann symmetry constraint is proposed for the Lax pairs and the adjoint Lax pairs of the Dirac systems. It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville integrable Hamiltonian system and that under the control of the spatial part, the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are interpreted as a hierarchy of commutative, finite dimensional Liouville integrable Hamiltonian systems whose Hamiltonian functions consist of a series of integrals of motion for the spatial part. Moreover an involutive representation of solutions of the Dirac systems exhibits their integrability by quadratures. This kind of symmetry constraint procedure involving the spectral problem and the adjoint spectral problem is referred to as a binary nonlinearization technique like a binary Darboux transformation.

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arXiv1996

A stability analysis for the Korteweg-de Vries equation

In this paper the stability of the Korteweg-de Vries (KdV) equation is investigated. It is shown analytically and numerically that small perturbations of solutions of the KdV-equation introduce effects of dispersion, hence the perturbation propagates with a different velocity then the unperturbed solution. This effect is investigated analytically by formulating a differential equation for perturbations of solutions of the KdV-equation. This differential equation is solved generally using an Inverse Scattering Technique (IST) using the continuous part of the spectrum of the Schrödinger equation. It is shown explicitly that the perturbation consist of two parts. The first part represents the time-evolution of the perturbation only. The second part represents the interaction between the perturbation and the unperturbed solution. It is shown explicitly that singular non-dispersive solutions of the KdV-equation are unstable.

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arXiv1996

A Convexity Theorem in the Scattering Theory for the Dirac Operator

The Dirac operator enters into zero curvature representation for the cubic nonlinear Schrödinger equation. We introduce and study a conformal map from the upper half-plane of the spectral parameter of the Dirac operator into itself. The action variables turn out to be limiting boundary values of the imaginary part of this map. We describe the image of the momentum map (convexity theorem) in the simplest case of a potential from the Schwartz class. We apply this description to the invariant manifolds for the nonlinear Schrödinger equation.

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arXiv1997

Quantum Coupled Nonlinear Schrödinger System with Different Masses

In this letter, I have considered one-dimensional quantum system with different masses $m$ and $M$, which does not appear integrable in general. However I have found an exact two-body wave function and due to the extension of the integrable system to more general system, it was concluded that the rapidity or quasi-momentum in the integrable system should be regarded as a modification of velocity rather than that of momentum. I have also considered the three-body wave function and argued its integrable condition.

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arXiv1997

Bispectral Operators, Dual Isomonodromic Deformations and the Riemann-Hilbert Dressing Method

A comparison is made between bispectral systems and dual isomonodromic deformation equations. A number of examples are given, showing how bispectral systems may be embedded into isomonodromic ones. Sufficiency conditions are given for the construction of rational solutions of isomonodromic deformation equations through the Riemann-Hilbert problem dressing method, and these are shown, in certain cases, to reduce to bispectral systems.

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arXiv1997

The Stackel systems and algebraic curves

We show how the Abel-Jacobi map provides all the principal properties of an ample family of integrable mechanical systems associated to hyperelliptic curves. We prove that derivative of the Abel-Jacobi map is just the Stäckel matrix, which determines $n$-orthogonal curvilinear coordinate systems in a flat space. The Lax pairs, $r$-matrix algebras and explicit form of the flat coordinates are constructed. An application of the Weierstrass reduction theory allows to construct several flat coordinate systems on a common hyperelliptic curve and to connect among themselves different integrable systems on a single phase space.

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arXiv1998

Hierarchy of Higher Dimensional Integrable System

Integrable equations in ($1 + 1$) dimensions have their own higher order integrable equations, like the KdV, mKdV and NLS hierarchies etc. In this paper we consider whether integrable equations in ($2 + 1$) dimensions have also the analogous hierarchies to those in ($1 + 1$) dimensions. Explicitly is discussed the Bogoyavlenskii-Schiff(BS) equation. For the BS hierarchy, there appears an ambiguity in the Painlevé test. Nevertheless, it may be concluded that the BS hierarchy is integrable.

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arXiv1998

Schlesinger Transformations for Linearisable Equations

We introduce the Schlesinger transformations of the Gambier equation. The latter can be written, in both the continuous and discrete cases, as a system of two coupled Riccati equations in cascade involving an integer parameter n. In the continuous case the parameter appears explicitly in the equation while in the discrete case it corresponds to the number of steps for singularity confinement. Two Schlesinger transformations are obtained relating the solutions for some value $n$ to that corresponding to either n+1 or n+2.

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arXiv1998

Linear r-Matrix Algebra for a Hierarchy of One-Dimensional Particle Systems Separable in Parabolic Coordinates

We consider a hierarchy of many-particle systems on the line with polynomial potentials separable in parabolic coordinates. The first non-trivial member of this hierarchy is a generalization of an integrable case of the Hénon-Heiles system. We give a Lax representation in terms of $2\times 2$ matrices for the whole hierarchy and construct the associated linear r-matrix algebra with the r-matrix dependent on the dynamical variables. A Yang-Baxter equation of dynamical type is proposed. Classical integration in a particular case is carried out and quantization of the system is discussed with the help of separation variables. This paper was published in the rary issues: Sfb 288 Preprint No. 110, Berlin and Nonlinear Mathematical Physics, {\bf 1(3)}, 275-294 (1994)

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arXiv2003

On the bilinear equations for Fredholm determinants appearing in random matrices

It is shown how the bilinear differential equations satisfied by Fredholm determinants of integral operators appearing as spectral distribution functions for random matrices may be deduced from the associated systems of nonautonomous Hamiltonian equations satisfied by auxiliary canonical phase space variables introduced by Tracy and Widom. The essential step is to recast the latter as isomonodromic deformation equations for families of rational covariant derivative operators on the Riemann sphere and interpret the Fredholm determinants as isomonodromic $τ$-functions.

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arXiv1999

Self-similar solutions of NLS-type dynamical systems

We study self-similar solutions of NLS-type dynamical systems. Lagrangian approach is used to show that they can be reduced to three canonical forms, which are related by Miura transformations. The fourth Painleve equation (PIV) is central in our consideration - it connects Heisenberg model, Volterra model and Toda model to each other. The connection between the rational solutions of PIV and Coulomb gas in a parabolic potential is established. We discuss also the possibility to obtain an exact solution for optical soliton i.e. of the NLS equation with time-dependent dispersion.

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