Showing papers on "Integrable system published in 1981"
TL;DR: In this article, the problem of reduction for systems of nonlinear equations integrable by the inverse scattering method is discussed and an infinite set of conservation laws is constructed for the system of equations for a two-dimensional Toda chain, the inverse problem is solved and exact N-soliton solutions are found.
486 citations
TL;DR: In this paper, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique, which is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length.
Abstract: The properties of exactly integrable two-dimensional quantum systems are reviewed and discussed. The nature of exact integrability as a physical phenomenon and various aspects of the mathematical formalism are explored by discussing several examples, including detailed treatments of the nonlinear Schr\"odinger (delta-function gas) model, the massive Thirring model, and the six-vertex (ice) model. The diagonalization of a Hamiltonian by Bethe's Ansatz is illustrated for the nonlinear Schr\"od\'{\i}nger model, and the integral equation method of Lieb for obtaining the spectrum of the many-body system from periodic boundary conditions is reviewed. Similar methods are applied to the massive Thirring model, where the fermion-antifermion and bound-state spectrum are obtained explicitly by the integral equation method. After a brief review of the classical inverse scattering method, the quantum inverse method for the nonlinear Schr\"odinger model is introduced and shown to be an algebraization of the Bethe Ansatz technique. In the quantum inverse method, an auxiliary linear problem is used to define nonlocal operators which are functionals of the original local field on a fixed-time string of arbitrary length. The particular operators for which the string is infinitely long (free boundary conditions) or forms a closed loop around a cylinder (periodic boundary conditions) correspond to the quantized scattering data and have a special significance. One of them creates the Bethe eigenstates, while the other is the generating function for an infinite number of conservation laws. The analogous operators on a lattice are constructed for the symmetric six-vertex model, where the object which corresponds to a solution of the auxiliary linear problem is a string of vertices contracted over horizontal links (arrows). The relationship between the quantum inverse method and the transfer matrix formalism is exhibited. The inverse Gel'fand-Levitan transform which expresses the local field operator as a functional of the quantized scattering data is formulated for the nonlinear Schr\"odinger equation, and some interesting properties of this transformation are noted, including its reduction to a Jordan-Wigner transformation in the limit of infinitely repulsive coupling.
430 citations
404 citations
Book•
01 Jan 1981
TL;DR: In this article, the Fermi-Pasta-Ulam problem is used to find an Integrable Lattice, which is a generalization of the Kac-Moerbeke System.
Abstract: 1. Introduction.- 1.1 The Fermi-Pasta-Ulam Problem.- 1.2 Henon-Heiles Calculation.- 1.3 Discovery of Solitons.- 1.4 Dual Systems.- 2. The Lattice with Exponential Interaction.- 2.1 Finding of an Integrable Lattice.- 2.2 The Lattice with Exponential Interaction.- 2.3 Periodic Solutions.- 2.4 Solitary Waves.- 2.5 Two-Soliton Solutions.- 2.6 Hard-Sphere Limit.- 2.7 Continuum Approximation and Recurrence Time.- 2.8 Applications and Extensions.- 2.9 Poincare Mapping.- 2.10 Conserved Quantities.- 3. The Spectrum and Construction of Solutions.- 3.1 Matrix Formalism.- 3.2 Infinite Lattice.- 3.3 Scattering and Bound States.- 3.4 The Gel'fand-Levitan Equation.- 3.5 The Initial Value Problem.- 3.6 Soliton Solutions.- 3.7 The Relationship Between the Conserved Quantities and the Transmission Coefficient.- 3.8 Extensions of the Equations of Motion and the Kac-Moerbeke System.- 3.9 The Backlund Transformation.- 3.10 A Finite Lattice.- 3.11 Continuum Approximation.- 4. Periodic Systems.- 4.1 Discrete Hill's Equation.- 4.2 Auxiliary Spectrum.- 4.3 Relation Between ?j (k) and ?j (0).- 4.4 Related Integrals on the Riemann Surface.- 4.5 Solution to the Inverse Problem.- 4.6 Time Evolution.- 4.7 A Simple Example (A Cnoidal Wave).- 4.8 Periodic System of Three-Particles.- 5. Application of the Hamilton-Jacobi Theory.- 5.1 Canonically Conjugate Variables.- 5.2 Action Variables.- 6. Recent Advances in the Theory of Nonlinear Lattices.- 6.1 The KdV Equation as a Limit of the TL Equation.- 6.2 Interacting Soliton Equations.- 6.3 Integrability.- 6.4 Generalization of the TL Equation.- 6.5 Two-Dimensional TL.- 6.6 Bethe Ansatz.- 6.7 The Thermodynamic Limit.- 6.8 Hierarchy of Nonlinear Equations.- 6.9 Some Numerical Results.- Appendices.- Simplified Answers to Main Problems.- References.- List of Authors Cited in Text.
218 citations
TL;DR: In this article, the fundamental theorem of homogeneous equations is interpreted as follows: "the fibers of the classical solutions of the homogeneous equation can be approximated and constancy on the fibers".
Abstract: page Introduction .387 1. Basic notation and ingredients.389 2. Approximation and constancy on the fibers of the classical solutions of the homogeneous equations . 393 3. First consequences and interpretation of the fundamental theorem. .398 4. Distribution solutions of the homogeneous equations. 402 5. Regularity of the solutions of the homogeneous equations .......409 6. Example of a system that is analytic hypo-elliptic but not Co hypo-elliptic.413 References 420
171 citations
TL;DR: A collisionless Boltzmann equation, describing long waves in a dense gas of particles interacting via short-range forces, is shown to be equivalent to the Benney equations, which describe long wave in a perfect two-dimensional fluid with a free surface as mentioned in this paper.
162 citations
TL;DR: In this paper, a method for the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G is presented.
Abstract: A method is exposed which allows the construction of families of first integrals in involution for Hamiltonian systems which are invariant under the Hamiltonian action of a Lie group G. This is applied to invariant Hamiltonian systems on the tangent bundles of certain homogeneous spaces M = G/K. It is proved, for example, that every such invariant Hamiltonian system is completely integrable if M is a real or complex Grassmannian manifold or SU(n + 1)/SO(n + 1) or a distance sphere in ℂPn+1. In particular, the geodesic flows of these homogeneous spaces are integrable.
113 citations
TL;DR: The class of exactly integrable non-linear evolution equations related to the general first order n × n linear problem is studied in this paper, where the set of independent scattering data J is determined and trace identities are obtained.
89 citations
TL;DR: The integrable statistical physics model on the rectangular two-dimensional lattice which we call "the L-model" was constructed in this article, which generated the integrably quantum sine-Gordon model.
Abstract: The integrable statistical physics model on the rectangular two-dimensional lattice which we call ‘the L-model’ is constructed. This model generates the integrable quantum sine-Gordon model on the one-dimensional lattice in the same way as the ice model generates the XXZ model.
83 citations
TL;DR: In this article, the self-duality equations in the specific case of potentials independent of one of the coordinates are reduced to a relativistic-invariant system in the (2-1)-dimensional space-time.
Abstract: In the space-time with signature (2,2) the self-duality equations in the specific case of potentials independent of one of the coordinates are reduced to a relativistic-invariant system in the (2-1)-dimensional space-time. A general solution of this system is constructed by means of IST. A soliton solution, finite in all directions, is discussed. It is found that there is no classical scattering of both solitons and continuous spectrum waves.
68 citations
TL;DR: In this article, the XYZ Heisenberg model is considered from the stand-point of the quantum inverse problem method and it is shown that this model is a completely integrable quantum system Algebraic generalization of the Bethe Ansatz for finding eigenvectors and eigenvalues of the Hamiltonian of the XZ model.
TL;DR: In this article, a one parameter family of piecewise linear measure preserving transformations of a torus which can be viewed as a perturbation of the twist mapping is introduced, and theorems on their ergodic properties for an infinite set of parameters are proved.
Abstract: A one parameter family of piecewise linear measure preserving transformations of a torus which can be viewed as a perturbation of the twist mapping is introduced. Theorems on their ergodic properties for an infinite set of parameters are proved. For some parameters coexistence of stochastic and integrable behaviour is obtained.
TL;DR: In this paper, the authors derived nonlinear equations which are integrable through a discrete Gel'fand-Levitan 'integral' equation in two (resp one, resp zero) continuous and one (resp two, resp three) discrete variables.
Abstract: Following the work of Zakharov and Shabat (1974), the authors derive nonlinear equations which are integrable through a discrete Gel'fand-Levitan 'integral' equation in two (resp one, resp zero) continuous and one (resp two, resp three) discrete variables.
TL;DR: In this article, a simplified perturbational approach appropriate for systems of solitons governed by the perturbed integrable equations is described, and some applications of this method are reviewed.
TL;DR: In this paper, it was shown that every higher Korteveg-de Vries equation can be included in a one-parameter family of integrable equations, which is a family of Kortegeg-De Vries equations.
Abstract: It is shown that every higher Korteveg‐de Vries equation can be included in a one‐parameter family of integrable equations.
TL;DR: In this article, a symplectic structure for stationary Lax equations of the type [L, P]=0 is constructed, where L is a matrix differential operator of the first order.
Abstract: A symplectic structure for stationary Lax equations of the type [L, P]=0 is constructed, whereL is a matrix differential operator of the first order. It is shown that the equation has a sufficient for the complete integrability amount of first integrals in involution. The well-known linearization of the equation by the Abelian mapping is obtained in a natural manner in consequent exercising of Liouville's procedure.
TL;DR: In this article, exactly integrable field theoretic models are constructed which are gauge equivalent to the n−component or m⋅n component nonlinear Schrodinger equations and to the O(n) nonlinear σ•model.
Abstract: Exactly integrable field theoretic models are constructed which are gauge equivalent to the n‐component or m⋅n component nonlinear Schrodinger equations and to the O(n) nonlinear σ‐model. We obtain the CPn‐ Heisenberg model or the Grassmann–Heisenberg model and the generalized sine‐Gordon model respectively. Consequences for the conserved quantities are discussed.
TL;DR: The generalization of the AKNS method to the matrix polynomial spectral problem of arbitrary order is given in this article, where both the general form of the integrable partial differential equations and their Backlund transformations are described.
Abstract: The generalisation of the AKNS method (Ablowitz et al. 1974) to the matrix polynomial spectral problem of arbitrary order is given. Both the general form of the integrable partial differential equations and their Backlund transformations are described.
TL;DR: In this article, a new class of solutions to integrable nonlinear evolution equations which are thought to have a deep connection with the n-point correlation functions of exactly solvable models in statistical mechanics is introduced.
TL;DR: In this article, it was shown that equilibrium states of classical systems of point particles are translation invariant whenever they have integrable clustering, and that jellium systems (with uniform charged background) are also translation invariants in dimension greater than one.
TL;DR: Using an ansatz that allows for propagation of spin excitations, the Landau-Lifshitz equation for 2-D Heisenberg ferromagnets is reduced to a non-linear system of the integrable type as mentioned in this paper.
01 May 1981
TL;DR: In this article, a factorized S matrix is obtained for particles of opposite parities in 1+1 dimensions, which can be expressed in terms of elliptic functions and contains two free parameters, which is used for the construction of a commuting family of transfer matrices for a 21-vertex lattice model in statistical mechanics.
Abstract: A factorized S matrix is obtained for particles of opposite parities in 1+1 dimensions. The solution of the factorization equation in dimension three, which can be expressed in terms of elliptic functions and contains two free parameters, is utilized for the construction of a commuting family of transfer matrices for a 21-vertex lattice model in statistical mechanics, as well as construction of a quantum Hamiltonian for an integrable chain of spin-1 particles with anisotropic interaction.
TL;DR: In this article, a family of complete integrable three-dimensional N-body quantum systems is introduced and completely solved by the dynamical algebras O(3N+1,2) and their representations.
Abstract: A family of complete integrable three-dimensional N-body quantum systems is introduced and completely solved by the dynamical algebras O(3N+1,2) and their representations. In a particular realisation of these algebras the particles interact by N-body 'Coulomb-type' potentials. A complete set of commuting integrals of motions, their spectra, the energy levels for both discrete and continuous spectra and their degeneracy have been explicitly determined. Relativistic generalisations and applications are briefly discussed.
TL;DR: In this article, a new family of completely integrable infinite-dimensional Hamiltonian systems is found and two canonical maps are deduced which transform this family into the families of Hamiltonian system associated with the generalized Zakharov-Shabat and energy-dependent Schrodinger operators.
Abstract: A new family of completely integrable infinite‐dimensional Hamiltonian systems is found. Two canonical maps are deduced which transform this family into the families of Hamiltonian systems associated with the generalized Zakharov–Shabat and energy‐dependent Schrodinger operators. Through these maps several relevant Hamiltonian structures are connected. In particular, a simple explanation of the second Hamiltonian structure for AKNS equations is given.
TL;DR: In this paper, a symplectic structure is constructed and the Liouville integration carried out for a stationary Lax equation [L, P]=0, where L is a scalar differential operator of an arbitrary ordernth order matrix operator.
Abstract: A symplectic structure is constructed and the Liouville integration carried out for a stationary Lax equation [L, P]=0, whereL is a scalar differential operator of an arbitrary ordernth order operators are included into the variety of first-order matrix operators, and properties of this inclusion are studied
01 Jan 1981
TL;DR: Theorem 3.1 as discussed by the authors shows that generalized integrals are -integrable on with base and -primitive functions, where is a continuous function of bounded variation on.
Abstract: A number of properties of generalized integrals are proved. The main result isTheorem 3. Suppose that is -integrable on with base and -primitive function , and , where is a continuous function of bounded variation on . Then the product is -integrable on with base , and Theorem 3 can be used to prove that if is finite everywhere on , then for .Bibliography: 10 titles.
TL;DR: In this article, the authors derived the Kirkwood-Salsburg and the Mayer-Montroll equations for an arbitrary stable interaction for the case of an exponentially integrable external potential (of which a finite volume is a special case).
Abstract: The Kirkwood-Salsburg and the Mayer-Montroll equations for an arbitrary stable interaction are derived for the case of an exponentially integrable external potential (of which a finite volume is a special case). It is shown that the Mayer-Montroll equation has at least one solution (the equilibrium state) if the activity z is such that the grand canonical partition function Ξ( z ) is nonzero; also, there is at least one eigenvector with eigenvalue 1 for z such that Ξ( z ) = 0. The difference with the Kirkwood-Salsburg equation lies in the possibility of more solutions, as is shown by an example.