Infinite Number Of Conservation Laws Research Articles

Essential Conservation Laws for Equations with Infinite Symmetries

We consider partial differential equations of variational problems with infinite symmetry groups. We study local conservation laws associated with arbitrary functions of one variable in the group generators. We show that only symmetries with arbitrary functions of dependent variables lead to an infinite number of conservation laws. We also calculate local conservation laws for the potential Zabolotskaya-Khokhlov equation for one of its infinite subgroups.

Hamiltonian structure and infinite number of conservation laws for the coupled discrete KdV equations

A new discrete isospectral problem is introduced, from which the coupled discrete KdV hierarchy is deduced and is written in its Hamiltonian form by means of the trace identity. It is shown that each equation in the resulting hierarchy is Liouville integrable. Furthermore, an infinite number of conservation laws are shown explicitly by direct computation.

Multi-component WKI equations and their conservation laws

In this Letter, a two-component WKI equation is obtained by using the fact that when curvature and torsion of a space curve satisfy the vector modified KdV equation, a graph of the curve satisfies the two-component WKI equation, which is a natural generalization to the WKI equation. It is shown that the two-component WKI equation can be solved in terms of the extended WKI scheme, and it admits an infinite number of conservation laws. In the same vein, a n-component generalization to the WKI equation is proposed.

Painlevé property and conservation laws of multi-component mKdV equations

Abstract It is shown that two classes of n-component mKdV equations are Painleve integrable, and the resonances occur, respectively, at −1,3,4, 0,…,0 n−1 , 1,…,1 n−1 , 5,…,5 n−1 for the geometric n-component mKdV equation which arises from curve motions in Euclidean space, and −1,3,4, 0,…,0 n−1 , 2,…,2 n−1 , 4,…,4 n−1 for another n-component mKdV equation which has an infinite number of Lie Backlund symmetries. It is also shown that the two-component geometric mKdV equation admits an infinite number of conservation laws. Conservation laws for several systems are constructed.

A Cauchy problem for the nonlocal nonlinear Schrödinger equation

A Cauchy problem for the nonlocal nonlinear Schrödinger equation is formulated by meansof the inverse scattering transform (IST) method. A closed system of IST equations for theJost functions is derived through the analysis of the direct and inverse scatteringproblems. Then, the orthogonality and completeness relations are establishedfor the Jost functions. An infinite number of conservation laws is also obtained,together with their expressions in terms of the scattering data. Finally, an exactlysolvable potential is presented which yields the discrete eigenvalues in explicit forms.

Negatons, positons, rational-like solutions and conservation laws of the Korteweg–de Vries equation with loss and non-uniformity terms

Solitons, negatons, positons, rational-like solutions and mixed solutions of a non-isospectral equation, the Korteweg–de Vries equation with loss and non-uniformity terms, are obtained through the Wronskian technique. The non-isospectral characteristics of the motion behaviours of some solutions are described with some figures made by using Mathematica. We also derive an infinite number of conservation laws of the equation.

Hamiltonian flows for a reduced Maxwell–Bloch system with permanent dipole

We place a reduced Maxwell–Bloch (rMB) system with permanent dipole in a Lie algebraic framework that enables us to reveal a hierarchy of systems in involution, the first of which is the rMB model. This is done in the context of the Adler–Kostant–Symes theorem. We provide the Hamiltonian functions for the commuting flows and establish an infinite number of conservation laws.

Lax pairs and conservation laws for two differential-difference systems

A coupled extended Lotka–Volterra lattice and a special Toda lattice are derived from the existing bilinear equations. Starting from the corresponding bilinear Bäcklund transformation, Lax pairs for these two differential-difference systems are obtained. Furthermore, an infinite number of conservation laws for the differential-difference equations are deduced from the Lax pairs in a systematic way.

The conservation laws of some discrete soliton systems

A systematic approach to constructing an infinite number of conservation laws for discrete soliton systems is proposed, and three examples are given.

Characteristics, conservation laws, and symmetries of the kinetic equations of motion of bubbes in a fluid

Generalized characteristics and Riemann invariants that are preserved along the characteristics are found for a kinetic model of motion of bubbles in a fluid. Conditions that ensure the hyperbolicity of a set of equations of a bubbly flow are obtained. It is shown that the set of equations of motion has an infinite number of conservation laws. An infinite series of generalized symmetries admitted by the equations is constructed. Solutions that are invariant under the generalized symmetries of solution and describe the propagation of running and simple waves in a bubbly fluid are found.

The Coupled Modified Korteweg-de Vries Equations

Generalization of the modified KdV equation to a multi-component system, that is expressed by \(\frac{\partial u_i}{\partial t} + 6 \bigl( \sum_{j,k=0}^{M-1} C_{jk} u_j u_k \bigr) \frac{\partial u_i}{\partial x} + \frac{\partial^3 u_{i}}{\partial x^3} =0\), i =0, 1, …, M -1, is studied. We apply a new extended version of the inverse scattering method to this system. It is shown that this system has an infinite number of conservation laws and multi-soliton solutions. Further, the initial value problem of the model is solved.

Scaling properties of passive scalars in one dimension

This paper is a set of notes about work in progress. Since the work is directed toward scaling, it seems quite appropriate to report this in a context devoted to Ben Widom's scientific contributions. Model equations for the motion of a passive scalar in one dimension are set up and some scaling properties of their solutions are derived. All these models follow the approach set up by Kraichnan in which the driving velocities have Gaussian correlations with scaling properties in space, but zero-range correlations in time. The simplest and most natural versions of the model fail to satisfy an incompressibility condition. In another version, the model does have an incompressible flow. To permit the incompressibility, there are two pipes — each of which is described by temperature and velocity fields which are functions of a single coordinate, x, and time. Flow from one pipe to the other transfers both mass and heat. This model has two conserved quantities in the limit of zero dissipation. In still another version, fluid elements are interchanged in position so that there are an infinite number of conservation laws in the no-dissipation limit. Scaling properties of the resulting two-point functions are derived.

Equations with an infinite number of explicit conservation laws

A large class of first-order partial nonlinear differential equations in two independent variables which possess an infinite set of polynomial conservation laws derived from an explicit generating function is constructed. The conserved charge densities are all homogeneous polynomials in the unknown functions which satisfy the differential equations in question. The simplest member of the class of equations is related to the Born–Infeld Equation in two dimensions. It is observed that some members of this class possess identical charge densities. This enables the construction of a set of multivariable equations with an infinite number of conservation laws.

Quantum-liquid regimes for spin chains coupled to phonons: Phonon density wave versus magnetic order.

We consider a chain of localized spins, coupled to phonons. Recently this problem has been solved exactly for a ``basic model,'' a family of spin-phonon Hamiltonians ${\mathit{H}}_{\mathrm{BM}}$ characterized by one parameter (coupling constant K), and a zero-T first-order phase transition from the magnetic (ferro or antiferro) state at low couplings to the nonmagnetic state with a phonon density wave at high couplings was found. Here we probe the general case, constructing an effective Hamiltonian H for low-energy degrees of freedom by means of regular expansion in deviations \ensuremath{\delta}H=H-${\mathit{H}}_{\mathrm{BM}}$ of the general Hamiltonian H from that of the basic model. In linear approximation in \ensuremath{\delta}H the problem appears to be exactly solvable as well, due to an infinite number of conservation laws. If K is far enough from the critical value ${\mathit{K}}_{\mathit{c}}$, then the character of the basic model solution is not altered. In the vicinity of ${\mathit{K}}_{\mathit{c}}$ the magnetic state is dramatically reconstructed: Here the ground state is a gapless magnetic quantum liquid, consisting of mobile singlet spin-phonon complexes and unbound spins. The fraction of singlets increases gradually upon approaching ${\mathit{K}}_{\mathit{c}}$, and the magnetic order parameter gradually vanishes. Thus we have here a partial screening of spins by phonons without formation of a phonon density wave. The latter appears only at K=${\mathit{K}}_{\mathit{c}}$ in the first-order phase transition. Corrections, quadratic in \ensuremath{\delta}H, destroy the integrability of the system, but outside a narrow critical region around ${\mathit{K}}_{\mathit{c}}$ they only lead to an opening of a small gap in the spectrum of the quantum liquid. The behavior of the system within the critical region is an open question. Most likely the continuous magnetic phase transition at K=${\mathit{K}}_{\mathit{c}}$ becomes a first-order one, but close to second order. The relevance of our results for three-dimensional systems and possible applications to compounds with anomalously weak magnetism are briefly discussed. \textcopyright{} 1996 The American Physical Society.

A spectral transform for the intermediate nonlinear Schrödinger equation

A new spectral transform system is introduced to solve the initial-value problem for the intermediate nonlinear Schrödinger (INLS) equation describing envelope waves in a deep stratified fluid. The spectral system is a combination of the Zakharov–Shabat linear system and a local Riemann–Hilbert problem in a strip of the complex plane. The inverse scattering transform technique is developed and the Bäcklund–Darboux transformation, soliton solutions and an infinite number of conservation laws are constructed. It is shown that all these properties of the INLS equation are closely related to those of the intermediate long-wave equation.

Quantum field theory with null-fronted metrics.

There is a large class of classical null-fronted metrics in which a free scalar field has an infinite number of conservation laws. In particular, if the scalar field is quantized, the number of particles is conserved. However, with more general null-fronted metrics, field quantization cannot be interpreted in terms of particle creation and annihilation operators, and the physical meaning of the theory becomes obscure.

Currents, charges, and canonical structure of pseudodual chiral models.

We discuss the pseudodual chiral model to illustrate a class of two-dimensional theories which have an infinite number of conservation laws but allow particle production, at variance with naive expectations. We describe the symmetries of the pseudodual model, both local and nonlocal, as transmutations of the symmetries of the usual chiral model. We refine the conventional algorithm to more efficiently produce the nonlocal symmetries of the model, and we discuss the complete local current algebra for the pseudodual theory. We also exhibit the canonical transformation which connects the usual chiral model to its fully equivalent dual, further distinguishing the pseudodual theory.

Turbulence with an infinite number of conservation laws.

Turbulence with an infinite number of conservation laws.

Q-Deformed Dressing Operators And Modified Integrable Hierarchies

A q-deformation of the dressing operator introduced by Sato is suggested. It is shown that it produces q-deformation of known integrable heirarchies, with the infinite number of conservation laws. A modification introduced by Kupershmidt when incorporated leads to both modified and deformed integrable systems.

Solitary waves generated by instabilities

The traditional soliton theory deals with integrable dynamical systems possessing an infinite number of conservation laws. However, one observes solitary waves also in physical systems that are not conservative. Such "solitons" can be generated spontaneously by an instability of the homogeneous state. In some cases, this phenomenon takes place in nearly integrable systems (e.g., described by slightly perturbed Korteweg-de Vries equation), but solitary waves can exist even in essentially non-conservative systems. The present paper gives a review of some recent investigations of solitary waves and localized states in unstable systems. The main attention is granted to the instabilities in fluids produced by buoyancy and thermocapillary effects. The case of the perturbed Korteweg-de Vries equation is considered in detail.