Showing papers on "Integrable system published in 1979"

Integrability of Nonlinear Hamiltonian Systems by Inverse Scattering Method

Abstract: A simple and direct scheme is presented to test the integrability of nonlinear evolution equations by inverse scattering method. The time part of the Lax equation needed for inverse scattering transform is identified with the linearized equation of the original nonlinear Hamiltonian system, while the Lax spectral operator is identified with a recursion operator connecting polynomial solutions of the linearized equation. This spectral operator is obtained by using a perturbative linear mode coupling scheme. A simple example discovered to be integrable by our scheme is shown explicitly to illustrate the detail procedures.

New integrable nonlinear evolution equations

Abstract: A new series of integrable nonlinear evolution equations is presented. The equations are novel in the sense that the nonlinear terms have saturation effects. It is shown that the equations have an infinite number of conservation laws and can be expressed in the Hamiltonian form.

Solitons in Laser Physics

Abstract: We survey some of the applications of soliton theory in laser physics. We briefly treat the theory of optical self-focussing and optical filamentation in neutral dielectrics and plasmas where the governing equation is the non-linear Schrodinger equation or one of its generalisations. We establish the connection of this theory with 1-dimensional Langmuir turbulence in plasmas. We treat the theory of optical self-induced transparency (SIT) at greater length and develop the Maxwell-Bloch (MB), reduced Maxwell-Bloch (RMB), SIT and sine-Gordon (s-G) equations to describe it. An optical three-wave interaction is related to the s-G equation; and reference is made to recent work on solitons in stimulated Raman scattering. The RMB equations are solved by a Zakharov-Shabat-AKNS inverse scattering scheme. The inhomogeneously broadened RMB equations have the unusual feature that ln a is not a constant of the motion. However, the sharp line RMB equations have two infinite sets of conserved densities, and the system constitutes one more example of a completely integrable infinite dimensional Hamiltonian system. In terms of scattering data the Hamiltonian of the RMB equations separates into soliton, breather and `background', that is `radiation', contributions. The sine-Gordon equation and its separable Hamiltonian are found as a special case of the RMB equations and its Hamiltonian. Averaged Lagrangian techniques are independently used to relate the c-number MB, RMB and SIT equations and to analyse the slowly varying phase and envelope approximations by which the SIT equations are derived from the MB or RMB equations. The connection of this Lagrangian theory with the Hamiltonian theory is not established. The Hamiltonian formalism in terms of the scattering data is used to quantise the RMB and s-G equations. The RMB, like the s-G, has the discrete energy level spectrum associated with a quantised breather; but only in the s-G limit is the quantised system easy to interpret. The quantised s-G is used to model a `coarse grained' operator theory of strictly resonant sharp line optical pulse propagation. The validity of such a description is examined. The physics of the c-number RMB equations is also discussed and particularly the relation of the c-number breather solutions to the 2π-pulse solutions of the SIT equations. The c-number RMB breather solutions provide a more general theory of SIT valid (within the 2-level atom model) at all electromagnetic field intensities and restricted only by the low density condition which permits the neglect of back scattering. Finally we look at four problems in resonant non-linear optics from a physical point of view. These are degenerate SIT and singular perturbation theory for it; the collision of oppositely directed resonant optical pulses; the theory of super-radiance; and the theory of optical self-focussing in resonant SIT.

Complete Integrability of General Nonlinear Differential-Difference Equations Solvable by the Inverse Method. I

Abstract: In recent years a large number of nonlinear differential-difference equations which describe discrete systems such as lattices, ladder networks and competition processes have been solved by the method of inverse scattering. The variety of those equations now becomes comparable with that of nonlinear evolution equations which describe continuous systems and are also solvable by the inverse method. For the continuous cases it has been well established that the system governed by an evolution equation to which the inverse method can be applied is nothing but a completely integrable Hamiltonian system.1l~3l The discrete counterpart of this statement is believed to be correct, but so far it has been proved only for two relevant systems; Flaschka and McLaughlin4l .5l have shown that the Toda lattice is a completely integrable Hamiltonian system and the present authors have proved in a previous paper6l that the generalized Volterra system is also completely integrable. In the present and a subsequent papers we want to put into this category two classes of general nonlinear differential-difference equations solvable by the method of inverse scattering. The first class which we here deal with consists of equations generated by a linear eigenvalue equation

Completely integrable systems and groups generated by reflections.

Abstract: We introduce a class of quantum Hamiltonian systems with δ-function potential, related to groups generated by reflections. They generalize the system of equal elastic particles on the line. We show that these systems are completely integrable and we integrate them explicitly. Then we apply our technique to obtain identities for groups generated by reflections.

Lack of screening in the continuous dipole systems

Abstract: We study continuous statistical systems interacting via a regularized dipole potential in the grand canonical ensemble. In the explicitly given region of high temperature (or low density) we show that the effective potential between two parallel dipoles isnot absolutely integrable (it is, however, square integrable), which implies that the effective potential does not fall off faster than |x|−3 in some directions.

Global solutions of the Boltzmann equation on a lattice

Abstract: The nonlinear Boltzmann equation with a discretized spatial variable is studied in a Banach space of absolutely integrable functions of the velocity variables Conservation laws and positivity are utilized to extend weak local solutions to a global solution This is shown to be a strong solution by analytic semigroup techniques

On Systems of Toda Type.

Abstract: : This report presents a group-theoretical interpretation of the structure of certain completely integrable Hamiltonian systems. These systems generalize the nonperiodic Toda Lattice system. Since all of them owe their special properties to the same group-theoretical mechanism, we call them systems of Toda type. This mechanism also provides a foundation for understanding many other completely integrable systems, including the Korteweg-de Vries equations, which will not be discussed here. We give an invariant definition of the class of systems of Toda type, and a method for explicit integration of all systems in this class in terms of rational combinations of linear exponential functions. (Author)

Relations among generalized Korteweg–deVries systems

Abstract: This report presents certain relations among the completely integrable Hamiltonian systems introduced by Gel’fand and Dikii. These relations generalize a formula of A. Lenard linking the higher‐order Korteweg–deVries equations, of which the Gel’fand–Dikii systems are a generalization. The general form of the relations, which connect the various isospectral deformations of linear differential operators, is described, and two examples are given explicitly.

Computer and Solitons

Abstract: The properties of one and more space dimensional solitons are discussed on the basis of the works performed mainly in Dubna at the Joint Institute for Nuclear Research, Laboratory of Computing Techniques and Automation. The conception of near integrable systems is formulated. As an example of interaction of Langmuir solitons it is shown that the Hamiltonian system is able to transform continuously its properties from a deep inelastic state up to the complete integrable one, depending on a physical parameter. The stability and existence problem of many-dimensional solitons is discussed. The results of numerical experiments are given on the dynamics of formation and interaction of many dimensional solitons and pulsons.

An Approximation of Integrable Functions by Step Functions with an Application.

Abstract: : A method of approximating functions which are integrable on (0, infinity) by piecewise constant functions is presented and studied in this paper. The method used and the properties established for it allow one to reduce the study of the convergence of a difference method of theoretical interest for nonlinear time-dependent problems with forcing terms to the simpler study of related problems without forcing terms.

Solutions to the Ginzburg-Landau equations for planar textures in superfluid 3 He

Abstract: The Ginzburg-Landau equations for planar textures of superfluid3He are proved to be equivalent to a completely integrable Hamiltonian system. General solutions to these equations are obtained by means of hyperelliptic integrals.

Stability criterion for Hamiltonian systems

Abstract: The conditions under which the phase space trajectories of a perturbed Hamiltonian system preserve conditional periodicity are investigated. It is shown that the sufficient and necessary condition for the stability of the system is the self-adjointness of the Liouville operator in the Hilbert space of functions absolute square integrable on the energy surface of the system in the phase space.

A special case of integrability in a system with one quasi-cyclic coordinate

Abstract: It is well known that conservative holonomic and scleromic systems with two degrees of freedom which have one cyclic coordinate are ‘integrable’. This means that the solution to the equations of motion can be given analytically in terms of quadratures, due to the existence of the two first integrals: the energy integral and the integral corresponding to the cyclic coordinate. In the present paper it is shown that the system is ‘integrable’ even if it is only holonomic and scleronomic and has one ‘quasi-cyclic’ coordinate, and even if the generalized forces are non-conservative provided the kinetic energy satisfies a certain additional condition.

Numerical treatment of non-integrable dynamical systems

Abstract: A systematic and detailed discussion of the ‘gravitational’ spring-pendulum problem is given for the first time. A procedure is developed for the numerical treatment of non-integrable dynamical systems which possess certain properties in common with the gravitational problem. The technique is important because, in contrast to previous studies, it discloses completely the structure of two-dimensional periodic motion by examining the stability of the one-dimensional periodic motion. Through the parameters of this stability, points have been predicted from which the one-dimensional motion bifurcates into two-dimensional motion. Consequently, families of two-dimensional periodic solutions emanated from these points are studied. These families constitute the generators of the mesh of all the families of periodic solutions of the problem.