Showing papers on "Integrable system published in 1987"

Hamiltonian methods in the theory of solitons

Abstract: The Nonlinear Schrodinger Equation (NS Model)- Zero Curvature Representation- The Riemann Problem- The Hamiltonian Formulation- General Theory of Integrable Evolution Equations- Basic Examples and Their General Properties- Fundamental Continuous Models- Fundamental Models on the Lattice- Lie-Algebraic Approach to the Classification and Analysis of Integrable Models- Conclusion- Conclusion

Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. I. The ground state and the excitation spectrum

Abstract: An integrable generalisation of the XXZ Heisenberg model with arbitrary spin and with light plane type anisotropy is studied. Integral equations describing the thermodynamics of the model are found. The antiferromagnetic ground state, the excitation spectrum, the quantum numbers and scattering amplitudes of the excitations are determined.

Integrable narrow-band model with possible relevance to heavy-fermion systems.

Abstract: A lattice model consisting of a single narrow band is introduced to describe some aspects of heavy electrons. The model excludes double occupancy of the sites and electrons on nearest-neighbor sites interact via a charge interaction and spin exchange. The model is integrable in one dimension for some special values of the coupling constants. These cases are related to the SU(3) invariance. The Bethe-ansatz equations are obtained and ground-state and thermodynamic properties are discussed and solved in some limiting cases.

Inverse scattering method with variable spectral parameter

Abstract: In the traditional scheme of the inverse scattering method, the spectral parameter of the auxiliary linear problem is assumed to be a constant. It is here proposed to regard the parameter as a variable quantity that satisfies an overdetermined system of differential equations which is uniquely determined by the auxiliary linear problem. The nonlinear equations that arise in such an approach contain, as a rule, an explicit dependence on the coordinates. This makes it possible to construct not only the well-known equations (gravitation equation, Heisenberg equation in axial geometry, etc.) but also a number of new integrable equations that have applied significance.

Action-angle maps and scattering theory for some finite-dimensional integrable systems : the pure solution case

Abstract: We construct an action-angle transformation for the Calogero-Moser systems with repulsive potentials, and for relativistic generalizations thereof. This map is shown to be closely related to the wave transformations for a large classl of Hamiltonians, and is shown to have remarkable duality properties. All dynamics inl lead to the same scattering transformation, which is obtained explicitly and exhibits a soliton structure. An auxiliary result concerns the spectral asymptotics of matrices of the formM exp(tD) ast→∞. It pertains to diagonal matricesD whose diagonal elements have pairwise different real parts and to matricesM for which certain principal minors are non-zero.

Integrable two-dimensional generalisation of the sine- and sinh-Gordon equations

Abstract: A two-dimensional generalisation of the sine- and the sinh-Gordon equations, which one refers to as the shine-Gordon equations, is obtained and solved through the inverse spectral transform (IST) method. The Backlund transformation and nonlinear superposition formula are constructed and explicit wave solitons are given. It is shown also that a slightly different procedure furnishes an IST-solvable extension in 2+1 dimensions of the dispersive long-wave equation.

The Topology of Surfaces of Constant Energy in Integrable Hamiltonian Systems, and Obstructions to Integrability

Abstract: The surfaces of constant energy in integrable Hamiltonian systems which possess Bott integrals are classified. A complete topological classification is given of surgery of Liouville tori in general position in integrable Hamiltonian systems. Bibliography: 28 titles.

Lectures on geometric methods in mathematical physics

Abstract: Infinite-Dimensional Hamiltonian Systems Elasticity as a Hamiltonian System Symmetry and Reduction Applications of Reduction Two Completely Integrable Systems Bifurcations of a Forced Beam The Traction Problem in Elastostatics Bifurcations of Momentum Mappings The Space of Solutions of Einstein's Equations Regular and Singular Points.

Birkhoff periodic orbits for small perturbations of completely integrable Hamiltonian systems with convex Hamiltonians

Abstract: Pour des systemes a n degres de liberte, on montre l'existence d'au moins n orbites periodiques distinctes avec un vecteur frequence rationnel admissible proche du tore correspondant du systeme non perturbe sous l'hypothese que le hamiltonien du systeme non perturbe a des surfaces d'energie convexes pour les variables action-angle

On a new class of completely integrable nonlinear wave equations. II. Multi‐Hamiltonian structure

Abstract: The multi‐Hamiltonian structure of a class of nonlinear wave equations governing the propagation of finite amplitude waves is discussed. Infinitely many conservation laws had earlier been obtained for these equations. Starting from a (primary) Hamiltonian formulation of these equations the necessary and sufficient conditions for the existence of bi‐Hamiltonian structure are obtained and it is shown that the second Hamiltonian operator can be constructed solely through a knowledge of the first Hamiltonian function. The recursion operator which first appears at the level of bi‐Hamiltonian structure gives rise to an infinite sequence of conserved Hamiltonians. It is found that in general there exist two different infinite sequences of conserved quantities for these equations. The recursion relation defining higher Hamiltonian structures enables one to obtain the necessary and sufficient conditions for the existence of the (k+1)st Hamiltonian operator which depends on the kth Hamiltonian function. The infinite sequence of conserved Hamiltonians are common to all the higher Hamiltonian structures. The equations of gas dynamics are discussed as an illustration of this formalism and it is shown that in general they admit tri‐Hamiltonian structure with two distinct infinite sets of conserved quantities. The isothermal case of γ=1 is an exceptional one that requires separate treatment. This corresponds to a specialization of the equations governing the expansion of plasma into vacuum which will be shown to be equivalent to Poisson’s equation in nonlinear acoustics.

Lax Pair for the One-Dimensional Hubbard Model

Abstract: A pair of Lax operators which gives a set of fundamental equations in the quantum inverse scattering method is explicitly presented for the one-dimensional Hubbard model. This provides a direct proof that the model is a completely integrable system.

Dual field formulation of quantum integrable models

Abstract: Equal time correlators are studied in completely integrable models. The main example is the quantum non-linear Schrodinger equation. Introduction of an auxiliary Fock space permits us to represent the generating functional of correlators in the form of a determinant of the integral operator.

Exact solution of the integrable XXZ Heisenberg model with arbitrary spin. II. Thermodynamics of the system

Abstract: For ptI see ibid, vol20, p1565 (1987) The thermodynamics of the integrable generalisation of the XXZ Heisenberg model with arbitrary spin is studied The low- and high-temperature heat capacities and magnetic susceptibility are computed

The Lax representation for an integrable class of relativistic dynamical systems

Abstract: We exhibit the Lax pair for the class of relativistic dynamical systems recently introduced by Ruijsenaars and Schneider, whose equations of motion read , whereB is an arbitrary constant and the Weierstrass elliptic function.

On Hamiltonian and Lagrangian Formalisms for the KP-Hierarchy of Integrable Equations

Abstract: has sense. This equation is completely integrable. For each n, we have a hierarchy of equations with rn = 1,2,3, . . , . (The KdV equation corresponds t o n = 2, rn = 3.) Sat0 and other representatives of the Japanese school (see reference 6) noticed that all the hierarchies can be combined into one if the operators L:/\", for all n, are identified. More precisely, if we let L = a + u,a-' + U , x 2 + . . . ,

Generalized solutions to semilinear hyperbolic systems

Abstract: In this article semilinear hyperbolic first order systems in two variables are considered, whose nonlinearity satisfies a global Lipschitz condition. It is shown that these systems admit unique global solutions in the Colombeau algebraG(ℝ2). In particular, this provides unique generalized solutions for arbitrary distributions as initial data. The solution inG(ℝ2) is shown to be consistent with the locally integrable or the distributional solutions, when they exist.

Deformation of algebras and solution of self‐duality equation

Abstract: The solutions of the self‐duality equation depending on a set of functions of three independent variables are constructed in an explicit way. The obtained solutions are shown to be connected with two‐dimensional systems determined by the operators of the Lax pair.

The shadow price of information in continuous time decision problems

Abstract: We formulate a continuous time stochastic control problem and establish the existence of the shadow price of information. This shadow price is the Lagrange multiplier for the constraint that the control be adapted or predictable; it is a stochastic process of integrable variation, and, in one formulation, it is a martingale. The results are applied to problems of security investment, selling an asset, and economic growth. In the last application, it is shown that the existence of the shadow price of information implies the validity of the stochastic maximum principle

Reflection of waves in nonlinear integrable systems

Abstract: Solutions of the boomeron type are found in two nonlinear integrable systems describing the interaction of a long wave with a short wave packet. These solutions follow from two‐soliton solutions if certain additional conditions are imposed on their parameters. The results are relevant to some problems of plasma physics, solid‐state physics, hydrodynamics, etc.

Solitons in Interaction

Abstract: Several new nonlinear systems are given which are completely integrable. These systems can be considered as flows describing the self-interaction of single solitons in multisoliton fields. The construction of action variables, recursion operators, bi-hamiltonian formulation and the like is performed for these nonlinear systems. Furthermore virtual solitons are introduced and it is shown that 2-solitons in general may be understood as the superposition of two pairs of interacting solitons exchanging one virtual soliton and that the interacting soliton itself can be considered as the result of a collision of a wave with a virtual soliton. In a sense, virtual solitons only pop up during the time that solitons interact with each other. In case of the KdV the details of decomposition into interact­ ing and virtual sQlitons are plotted, and a qualitative analysis of interaction is given. A brief discussion is appended, how to describe multisolitons by their "trajectories".

Multisoliton solutions in the scheme for unified description of integrable relativistic massive fields. Non-degenerate sl(2, ℂ) case

Abstract: A scheme allowing systematic construction of integrable two-dimensional models of the Lorentz-invariant Lagrangian massive field theory is presented for the case when the associated linear problem is formulated onsl(2, ℂ) algebra. A natural dressing procedure is developed then for the generic system of two (either scalar or spinor) fields inherent in the scheme and an explicitN-soliton solution on zero background is calculated. Solutions of reduced systems which include both familiar and new equations are extracted from the solution of the generic system, not all of these reductions being related immediately tosl(2, ℂ) real forms. Finally, in the case of scalar equations we present the Miura-type transformations relating solutions with different boundary conditions.

Multi-dimensional integrable systems

Abstract: I began this lecture by posing the question of what the best way is to define integrability. The most promising definition seems to be one based on an associated overdetermined linear system which is “allowable”. The linear systems described in sections 3 and 4 are certainly allowable, and most known integrable equations can be obtained in this way. But there are exceptions, such as the KP equation, so the question is not yet settled. With this sort of definition, it would be difficult to establish whether or not a given equation was integrable. But one could try to classify all the integrable equations which arose from a certain type of linear system.