Showing papers on "Integrable system published in 1983"

Nonlinear Schrödinger equations and simple Lie algebras

Abstract: We associate a system of integrable, generalised nonlinear Schrodinger (NLS) equations with each Hermitian symmetric space. These NLS equations are considered as reductions of more general systems, this time associated with a reductive homogeneous space. The nonlinear terms are related to the curvature and torsion tensors of the appropriate geometrical space. The Hamiltonian structure is investigated using “r-matrix” techniques and shown to be “canonical” for all these equations. Throughout the reduction procedure this Hamiltonian structure does not degenerate. Each of the above systems of equations is gauge equivalent to a generalised ferromagnet. Reductions of the latter are discussed in terms of the corresponding NLS type equations.

Local Hamiltonians for integrable quantum models on a lattice. II

Abstract: This paper discusses a method of constructing local Hamiltonians for integrable lattice models proposed by Tarasov, Takhtadzhyan, and Faddeev. The method is generalized to the case of inhomogeneous models. Another model, inhomogeneous, is considered which describes the interaction of spin impurities with a model of the type of a nonlinear lattice Schroedinger model.

Necessary Condition for the Existence of Algebraic First Integrals. II: Condition for Algebraic Integrability

Abstract: Necessary condition for the existence of a sufficient number of algebraic first integrals is given for a class of dynamical systems. It is proved that in order that a given system is algebraically integrable, all possible Kowalevski's exponents, which characterize a singularity of the solution, must be rational number. For example, the classical 3-body problem and the Henon-Heiles system are shown to be not algebraically integrable.

Two-dimensional Exactly and Completely Integrable Dynamical Systems (Monopoles, Instantons, Dual Models, Relativistic Strings, Lund Regge Model, Generalized Toda Lattice, Etc.)

Abstract: An investigation of two-dimensional exactly and completely integrable dynamical systems associated with the local part of an arbitrary Lie algebra\(\mathfrak{g}\) whose grading is consistent with an arbitrary integral embedding of 3d-subalgebra in\(\mathfrak{g}\) has been carried out. We have constructed in an explicit form the corresponding systems of nonlinear partial differential equations of the second order and obtained their general solutions in the sense of a Goursat problem. A method for the construction of a wide class of infinite-dimensional Lie algebras of finite growth has been proposed.

A new class of integrable systems

Abstract: We present a family of dynamical systems associated with the motion of a particle in two space dimensions. These systems possess a second integral of motion quadratic in velocities (apart from the Hamiltonian) and are thus completely integrable. They were found through the derivation and subsequent resolution of the integrability condition in the form of a partial differential equation (PDE) for the potential. A most important point is that the same PDE was derived through considerations on the analytic structure of the singularities of the solutions (‘‘weak‐Painleve property’’).

All Self-Dual Multimonopoles for Arbitrary Gauge Groups

Abstract: The ADHM formalism is adapted to self-dual multimonopoles for arbitrary charge and arbitrary gauge group. Each configuration is characterized by a solution of a certain ordinary non-linear differential equation, which has chances to be completely integrable. For axially symmetric configurations it reduces to the integrable Toda lattice equations. The construction of the potential requires the solution of a further ordinary linear differential equation.

The complete Whittaker theorem for two-dimensional integrable systems and its application

Abstract: The authors prove, for a particle moving in a plane under the influence of a conservative force, that when the motion is constrained by a 'second' invariant quadratic in the velocities, then the potential allows separability of the Hamilton-Jacobi equation in rectangular, polar, elliptical cylinder or parabolic cylinder coordinates. This link shows the intimate connection between quadratic invariants and the two-dimensional Hamilton-Jacobi equation. They give examples of the utility of parabolic cylinder coordinates in cases of recent study.

Perturbation Theory for Classical Hamiltonian Systems

Abstract: the relation between the Kolmogorov-Arnold-Moser theory of the non resonant motions in nearly integrable Hamiltonian systems and the renormalization group methods is pointed out. It is followed by a very detailed proof of a version of the KAM theorem based on dimensional estimates (in which no attention is paid to obtaining best constants).

Nonlinear lattice and soliton theory

Abstract: Since the notion of stable pulses, known as solitons, plays a central role in the phenomena of wave propagation in nonlinear systems, an exposition of this topic is developed in some detail. It is known that the equations of motion of the one-dimensional lattice of particles with exponential interaction are integrable, namely, they admit exact solutions, and this system is equivalent to an LC circuit with certain nonlinear capacitance. In addition, a closely related partial differential equation called the Korteweg-de Vries (KdV) equation is also integrable. Special emphasis is placed on these integrable systems.

Nonlinear lattice and soliton theory

Abstract: Since the notion of stable pulses, known as solitons, plays a central role in the phenomena of wave propagation in nonlinear systems, an exposition of this topic is developed in some detail. It is known that the equations of motion of the one-dimensional lattice of particles with exponential interaction are integrable, namely, they admit exact solutions, and this system is equivalent to an LC circuit with certain nonlinear capacitance. In addition, a closely related partial differential equation called the Korteweg-de Vries (KdV) equation is also integrable. Special emphasis is placed on these integrable systems.

Rational solutions of the Zakharov-Shabat equations and completely integrable systems of N particles on a line

Abstract: One constructs all the decreasing rational solutions of the Kadomtsev-Petviashvili equations. The presented method allows us to identify the motion of the poles of the obtained functions with the motion of a system of N particles on a line with a Hamiltonian of the Calogero-Moser type. Thus, this Hamiltonian system is imbedded in the theory of the algebraic-geometric solutions of the Zakharov-Shabat equations.

Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces

Abstract: The Hamilton–Jacobi and Laplace–Beltrami equations on the Hermitian hyperbolic space HH(2) are shown to allow the separation of variables in precisely 12 classes of coordinate systems The isometry group of this two‐complex‐dimensional Riemannian space, SU(2,1), has four mutually nonconjugate maximal abelian subgroups These subgroups are used to construct the separable coordinates explicitly All of these subgroups are two‐dimensional, and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones The symmetry reduction of the free HH(2) Hamiltonian by a maximal abelian subgroup of SU(2,1) reduces this Hamiltonian to one defined on an O(2,1) hyperboloid and involving a nontrivial singular potential Separation of variables on HH(2) and more generally on HH(n) thus provides a new method of generating nontrivial completely integrable relativistic Hamiltonian systems

Restricted multiple three‐wave interactions: Integrable cases of this system and other related systems

Abstract: Restricted multiple three‐wave interactions, in which a set of wave triads interact through one shared wave, are discussed. It is shown that this system is integrable when all triads have equal coupling coefficients regardless of the frequency mismatches. This system is then used as a starting point from which to determine integrable cases of a more general class of three‐wave interactions.

Nonlinear Transformations and Integrable Evolution Equations

Abstract: The transformation properties of the partial differential equations integrable by the inverse scattering transform method are considered. It is shown that the nonlinear transformations characteristic to the integrable equations (symmetry groups, Backlund-transformations) and integrable equations themselves are contained in certain universal nonlinear transformations group which is defined by linear spectral problem.

Integrable Reductions of Manycomponent Magnetic Systems in (1,1) Dimensions

Abstract: A generalized many component Heisenberg spin chain with phonon interaction is proposed Some reductions of the proposed model leading to different real magnetic systems such as many chained magnetic crystals with nontrivial interchain couplings, a mixture of many chained ferro and antiferromagnets, a "colour" generalized Pierels-Hubbard model, etc, are studied It has been shown that the dynamics of all the above real models are close to some integrable systems and coincide with them in certain limits Such integrable systems are the coupled generalised system of Yajima and Oikawa and U(p, q) nonlinear Schrodinger equation, already well studied

Regularized Integrable Version of the One-Dimensional Quantum Sine-Gordon Model

Abstract: We derive a regularized exactly solvable version of the one-dimensional quantum Sine-Gordon model proceeding from the exact solution of the U(1)-symmetric Thirring model. The ground state and the excitation spectrum are obtained in the region β2 < 8 π.

Laws of composition of Bäcklund transformations and the universal form of completely integrable systems in dimensions two and three.

Abstract: Backlund transformations are defined as operations on solutions of a Riemann boundary value problem (vector bundles over P1) that add apparent singularities. For solutions of difference and differential linear spectral problems, Backlund transformations are presented in explicit form through the Christoffel formula and its generalizations. Identities satisfied by iterations of elementary Backlund transformations are represented in the form of the law of addition or as the three-dimensional difference equation of Hirota's type. Matrix two-dimensional isospectral deformation equations are imbedded into three-dimensional scalar systems of Kadomtzev-Petviashvili (law of addition) form. Two-dimensional matrix systems correspond to reductions of Kadomtzev-Petviashvili equations with pseudodifferential operators satisfying algebraic equations.