Showing papers on "Integrable system published in 1980"
01 Jan 1980
TL;DR: In this article, integrable Hamiltonian systems are defined as nonlinear systems of ordinary differential equations described by a Hamiltonian function and possessing sufficiently many integrals (or conserved quantities) so that they are more or less explicitly solvable by quadrature.
Abstract: In this paper we are concerned with integrable Hamiltonian systems. This concept goes back to classical analytical dynamics of the last century. Briefly these are nonlinear systems of ordinary differential equations described by a Hamiltonian function and possessing sufficiently many integrals (or conserved quantities) so that they are more or less explicitly solvable by quadrature. Therefore these systems played a crucial role in the last century before more qualitative methods for differential equations were developed at the turn of the century. Subsequently interest in these systems decreased, partly due to the realization that the existence of global integrals can be established only for exceptional Hamiltonian systems.
272 citations
TL;DR: In this paper, the principal chiral fields on the symplectic, unitary and orthogonal Lie groups are shown to be integrable by means of the inverse scattering problem method.
Abstract: Well known classical spinor relativistic-invariant two-dimensional field theory models, including the Gross-Neveu, Vaks-Larkin-Nambu-Jona-Lasinio and some other models, are shown to be integrable by means of the inverse scattering problem method. These models are shown to be naturally connected with the principal chiral fields on the symplectic, unitary and orthogonal Lie groups. The respective technique for construction of the soliton-like solutions is developed.
253 citations
TL;DR: In this paper, a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform is presented, and a Backlund transformation and superposition formula for the general system is presented.
Abstract: We present a class of nonlinear Klein-Gordon systems which are soluble by means of a scattering transform. More specifically, for eachN≧2 we present a system of (N−1) nonlinear Klein-Gordon equations, together with the correspondingN ×N matrix scattering problem which can be used to solve it. We illustrate these with some special examples. The general system is shown to be closely related to the equations of the periodic Toda lattice. We present a Backlund transformation and superposition formula for the general system.
204 citations
01 Jan 1980
TL;DR: In this paper, a systematic exposition of different methods of obtaining equations which are integrable by the inverse scattering method is presented, starting with elementary methods and concluding with the method of dressing operator families.
Abstract: The present article is devoted to a systematic exposition of different methods of obtaining equations which are integrable by the inverse scattering method. The exposition begins with elementary methods and concludes with the method of dressing operator families. Many results (this refers both to the elementary part and in particular to the method of dressing) are original and published for the first time.
169 citations
TL;DR: It is shown that any Bäcklund transformation of a nonlinear differential equation integrable by the multichannel Schrödinger eigenvalue problem can be written in the form V(x) = U'V - VU.
Abstract: It is shown that any Backlund transformation of a nonlinear differential equation integrable by the multichannel Schrodinger eigenvalue problem can be written in the form Vx = U′V - VU. This allows us to interpret the Backlund transformation formally as a nonlinear differential difference equation for which we can immediately construct the soliton solutions.
138 citations
TL;DR: The infinite-volume partition function for the planar Feynman diagrams with massless scalar propagators was shown to be a completely integrable lattice statistical system in this article.
137 citations
TL;DR: In this paper, the permanent profile solutions of the continuous classical Heisenberg chain are reviewed and the author expounds on the application of the inverse scattering method, and exhibits the 'diagonal' action angle representation of the model.
Abstract: The permanent profile solutions of the continuous classical Heisenberg chain is reviewed and the author expounds on the application of the inverse scattering method. Extending and amplifying the work of Takhtajan (1977) he exhibits the 'diagonal' action angle representation of the model. The spectrum of the Hamiltonian is exhausted by a magnon band and a soliton band. The magnons have no internal degrees of freedom and can be characterised by the dispersion law E=p2. Like the sine Gordon doublet, the solitons have internal structure, they carry a continuous angular momentum m, and are characterised by the dispersion law E=16 sin2(p/4)/ mod m mod , in accordance with Tjon and Wright (1977). The continuous Heisenberg chain is a completely integrable Hamiltonian system possessing an infinite number of constants of motion. The recursive procedure for the determination of the conserved integrated densities is established.
133 citations
TL;DR: In this paper, the authors developed a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrodinger spectral problems whose potential functions depend on the spectral parameter.
Abstract: We develop a method to derive infinite families of completely integrable nonlinear Hamiltonian evolution equations associated with Schrodinger spectral problems whose potential functions depend on the spectral parameter.
131 citations
TL;DR: In this paper, the theory of Hamiltonian systems with symmetry is applied to dynamical systems like the nonperiodic Toda Lattice of Moser to yield striking information about their trajectories.
113 citations
TL;DR: In this paper, the authors used Poincare sections to show that the advection of a passive marker by three vortices displays chaos, and formally reduced the four-vortex problem to a two degrees of freedom hamiltonian.
97 citations
TL;DR: The integrable generalised nonlinear Schrodinger equation with linearly x-dependent coefficients was shown to be equivalent to the equation of motion of a generalised Heisenberg ferromagnet in the continuum limit.
TL;DR: In this paper, the time evolution of wavefunctions Φ( t ) composed of superpositions of energy eigenstates are compared when the wave functions are initiated in (1) the non-integrable and (2) the integrable regimes in the phase space of the Henon - Heiles system.
TL;DR: In this paper, the Bethe hypothesis was used to prove the integrability of the resonant level model equivalent to the Kondo problem, which is completely integrable.
TL;DR: The general form of the differential equations which are integrable by the general linear spectral matrix problem of the order N x N and the corresponding Backlund transformations are found in this paper, where the Backlund transformation is used to solve the problem.
01 Aug 1980
TL;DR: In this paper, a solution of the factorization equations in dimension 3 is obtained, which corresponds to a factorized S-matrix with U(1) symmetry and allows one to construct a commuting family of transfer matrices for a 19-vertex lattice statistical system.
Abstract: A solution of the factorization equations in dimension 3 is obtained. The solution is expressed in terms of trigonometric functions and contains one free parameter. It corresponds to a factorized S-matrix with U(1) symmetry. The solution also allows one to construct a commuting family of transfer matrices for a 19-vertex lattice statistical system and the quantum Hamiltonian of an integrable anisotropic spin-1 Heisenberg chain.
01 Jan 1980
TL;DR: In this paper, the history of the soliton has been reviewed, since the first recorded observation of the great solitary wave by Russell in 1834, as a means of developing the mathematical properties of a large class of solvable nonlinear evolution equations.
Abstract: We review the history of the soliton, since the first recorded observation of the ‘great solitary wave’ by Russell in 1834, as a means of developing the mathematical properties of a large class of solvable nonlinear evolution equations. This class embraces, amongst others, the Korteweg-de Vries, sine-Gordon, and nonlinear Schrodinger equations. Solitary waves, solitons, Backlund transformations, conserved quantities and integrable evolution equations as completely integrable Hamiltonian systems are all introduced this way and form a basis for the more detailed discussions which follow in the remaining chapters. The differential geometry of one large class of nonlinear evolution equations is described. Some connections with nonlinear field theories and with solvable many-body problems are established. The short biography of John Scott Russell which forms much of the first section is continued as an appendix at the back of this volume.
TL;DR: In this paper, a completely integrable SU(n)-invariant Heisenberg spin chain was derived from an underlying U(n−1) invariant non-linear Schrodinger equation.
01 Jan 1980
TL;DR: In this paper, the authors introduced the Riemann-Hilbert monodromy problem and showed that all the one-dimensional classical completely integrable systems, connected with commuting matrix differential operators, can be represented as simplified Schlesinger systems.
Abstract: These lectures serve as an introduction into the Riemann-Hilbert monodromy problem. Our main aim is to relate known completely integrable systems with isomonodromy deformation. We describe Schlesinger isomonodromy deformation equations for Fuchsian linear differential equations and their connection with Painleve transcendents. Moreover it is shown that all the one-dimensional classical completely integrable systems, connected with commuting matrix differential operators, can be represented as simplified Schlesinger systems. We generalize isomonodromy deformation equations for the case of two-dimensional systems. This way arise Painleve type equations in one space and one time situation. The last part of the lectures is devoted to Riemann boundary value problem. It is explained, how using boundary problem for analytic function on a Riemann surface of finite genus one can solve classical multidimensional isospectral and isomonodromy deformation equations. Examples of so called Bakes’s functions are given.
TL;DR: In this paper, it was shown that almost all solutions to the linearized variational equations derived from bounded, integrable hamiltonian systems exhibit an average linear growth with time, becoming unbounded at t → ∞.
TL;DR: In this article, a family of Hamiltonians on the dual space to a Lie algebra of triangular matrices for which the Euler equations are completely integrable in the sense of Liouville on orbits in general position was constructed.
Abstract: In this paper there is constructed a family of Hamiltonians on the dual space to a Lie algebra of triangular matrices for which the Euler equations are completely integrable in the sense of Liouville on orbits in general position. Bibliography: 4 titles.
01 Jan 1980
TL;DR: In this paper, the authors discuss another aspect of the theory of nonlinear evolution equations which are integrable by the inverse scattering method, and they show that these equations are infinite-dimensional Hamiltonian systems, and their explicit solvability has the following interpretation in the language of Hamiltonian system: a transform from the initial Cauchy data to the scattering data which underlies the inverse scatter method represents a nonlinear canonical transformation to variables of the action-angle type.
Abstract: In this chapter we shall discuss another aspect of the theory of nonlinear evolution equations which are integrable by the inverse scattering method. Those having applications of physical interest prove to be infinite-dimensional Hamiltonian systems. Their explicit solvability has the following interpretation in the language of Hamiltonian systems: a transform from the initial Cauchy data to the scattering data which underlies the inverse scattering method represents a nonlinear canonical transformation to variables of the action-angle type. This interpretation was originally suggested by ZAKHAROV and this author [11.1] for the case of the Korteweg-de Vries equation. Its most interesting applications are in the quantization problem for nonlinear equations. It also played an important heuristic role in clarifying the slow stochastization of an oscillator lattice [11.2] and in deriving the integrability of finite-dimensional systems of stationary points for the higher conservation laws [11.3]. In this connection we note that the N-dimensional rigid body equations have recently been shown to be integrable by MANAKOV [11.4].
TL;DR: In this article, a survey of systems with known S-operators and an approach for solving them rigorously are presented, and several problems and conjectures are formulated, and an invariance property of the wave and scattering operators is discovered and argued to hold at the classical level.
TL;DR: The integrable evolution equations imbeddable in SU (2) are shown to have two gauge equivalent forms; the AKNS form, and a spin form for which the field is a threedimensional vector of unit length as discussed by the authors.
Abstract: The integrable evolution equations imbeddable in SU (2) are shown to have two gauge equivalent forms; the AKNS form, and a spin form for which the field is a three‐dimensional vector of unit length. These equations are the compatibility conditions for the existence of a bilocal Lie group in two distinct frames of reference. These frames are associated with moving bases on surfaces formed by the motion of the strings introduced by Lamb. Both forms of the evolution equation are derivable from a locality assumption for the generators of the bilocal Lie group. The assumption is sufficient to distinguish between integrable and nonintegrable systems imbedded in SU (2).
01 Jan 1980
TL;DR: In this paper, the geometrical theory of nonlinear evolution equations (NEEs) solvable by the AKNS1-generalised Zakharov-Shabat2 scattering problem was reviewed.
Abstract: We review the geometrical theory of nonlinear evolution equations (NEEs) solvable by the AKNS1-generalised Zakharov-Shabat2 scattering problem introduced by one of us previously3,4. We show how the theory contains within it the canonical structure known to be associated with integrable NEEs. We exploit the “gaugerd transformations of the geometric theory to derive an infinite set of non-local Hamiltonian densities for the sine-Gordon equation. We show that it is from these that the hierarchy of Lax-type sine-Gordon equations can be derived. We summarise the relation between the geometric theory and the theory of prolungation structures due to Wahlquist and Estabrook.
TL;DR: This paper showed that the generalized backlund transformations of Calogero and Degasperis form the group of canonical transformations that keep integrable nonlinear evolution equations in 1 + 1 dimensions invariant.
TL;DR: In this paper, numerical examples show that the degree of stochasticity does not always increase monotonically with increasing energy, and that at high energies the diagonal anharmonicities (nonlinearities) dominate the off-diagonal (coupling) terms.
TL;DR: In this article, the authors derive necessary and sufficient conditions for the exact controllability of a linear system to a linear subspace in the framework of square integrable control and causal information structure.
Abstract: We derive a necessary and a sufficient condition for the exact controllability of a linear system to a linear subspace in the framework of square integrable controls and causal information structure. We also give some results in the framework of absolutely integrable controls, relating the latter to the former.