Infinite Number Of Conservation Laws Research Articles

An integrable shallow water equation with peaked solitons.

We derive a new completely integrable dispersive shallow water equation that is bi-Hamiltonian and thus possesses an infinite number of conservation laws in involution. The equation is obtained by using an asymptotic expansion directly in the Hamiltonian for Euler's equations in the shallow water regime. The soliton solution for this equation has a limiting form that has a discontinuity in the first derivative at its peak.

Conservation laws in stochastic deposition-evaporation models in one dimension

An infinite number of conservation laws is identified for a stochastic model of deposition and evaporation of trimers on a linear chain. These laws can be encoded into a single nonlocal invariant, the irreducible string, which uniquely lables an exponentially large number of kinetically disconnected sectors of phase space. This enables the number and sizes of sectors to be determined. The effects of conservation laws on some thermodynamic properties are studied.

Conservation laws of a class of differential equations II

It is proved that unless a 1 a 2 is a multiple root of order three of the algebraic equation a 3 − a 4 λ + a 5 λ 2 − a 6 λ 3 = 0, the class of nonlinear evolution equation u t + u x + a 1 uu x + a 2 uu t + a 3 u xxx + a 4 u xxt + a 5 u xtt + a 6 u ttt = 0 (where a i ϵ R>, i = 1, 2, …, 6) has finite number of conservation laws; otherwise, the class has infinite number of conservation laws.

Multidimensional pattern formation has an infinite number of constants of motion.

Extending our previous work on 2D growth for the Laplace equation we study here {\it multidimensional} growth for {\it arbitrary elliptic} equations, describing inhomogeneous and anisotropic pattern formations processes. We find that these nonlinear processes are governed by an infinite number of conservation laws. Moreover, in many cases {\it all dynamics of the interface can be reduced to the linear time--dependence of only one ``moment" $M_0$} which corresponds to the changing volume while {\it all higher moments, $M_l$, are constant in time. These moments have a purely geometrical nature}, and thus carry information about the moving shape. These conserved quantities (eqs.~(7) and (8) of this article) are interpreted as coefficients of the multipole expansion of the Newtonian potential created by the mass uniformly occupying the domain enclosing the moving interface. Thus the question of how to recover the moving shape using these conserved quantities is reduced to the classical inverse potential problem of reconstructing the shape of a body from its exterior gravitational potential. Our results also suggest the possibility of controlling a moving interface by appropriate varying the location and strength of sources and sinks.

A new integrable equation in the (2+1)-dimension-dispersionless limit of the BKP equation

A new integrable equation is obtained as the dispersionless limit of the BKP equation. It is then studied in the framework of Kodama's approach (1988). The new equation also turns out to be completely integrable with an infinite number of conservation laws. The hodograph transformation method is applied to obtain exact solutions in various cases.

Statistical mechanics, Euler's equation, and Jupiter's Red Spot.

A statistical-mechanical treatment of equilibrium flows in the two-dimensional Euler fluid is constructed which respects all conservation laws. The vorticity field is fundamental, and its long-range Coulomb interactions lead to an exact set of nonlinear mean-field equations for the equilibrium state. The equations depend on an infinite set of parameters, in one-to-one correspondence with the infinite set of conserved variables. The equations contain all previous approximations as special limiting cases: for example the Kraichnan energy-enstrophy theory, and the Lundgren and Pointin point vortex mean-field theories are rederived. The techniques may be generalized to a number of other Coulomb-like Hamiltonian systems with an infinite number of conservation laws, including some in higher dimensions. For example, we rederive Lynden-Bell’s theory of stellar-cluster formation, as well as the Debye-Huckel theory of electrolytes. Our results may also be applicable to cylindrically bound guiding-center plasmas, which under idealized conditions provide another realization of 2-d Euler flow. Finally, a phenomenological theory of the weakly driven, weakly damped Euler fluid, based on weak perturbations of the equilibrium state, is presented. A very simple two-parameter model is used to illustrate the principal ideas.

Bäcklund transformation, conservation laws, and inverse scattering transform of a model integrodifferential equation for water waves

The Bäcklund transformation (BT), an infinite number of conservation laws, and the inverse scattering transform (IST) of a model integrodifferential equation for water waves in fluids of finite depth [Y. Matsuno, J. Math. Phys. 29, 49(1989)] are constructed by employing the bilinear transformation method. The model equation is also shown to pass the Painlevé test. These facts prove the complete integrability of the equation. Both the deep- and shallow-water limits of various results thus obtained are then investigated in detail. In addition, a new method to evaluate conserved quantities for pure N-soliton is developed by utilizing actively the time part of the BT. It is found that the structure of conservation laws exhibits peculiar characteristics in comparison with those of usual water wave equations such as the Benjamin–Ono and the Korteweg–de Vries equations. The most important problem left open in this paper is to solve various IST equations.

Statistical mechanics of Euler equations in two dimensions.

We formulate the statistical mechanics of a two-dimensional inviscid incompressible fluid in a manner which, for the first time, respects all conservation laws. For a special case, we demonstrate that a mean-field theory is exact. A consequence of our arguments is that, in an inviscid fluid evolving from initial conditions to statistical equilibrium, only the energy and certain one-body integrals appear to be conserved. Our methods may be applied to a variety of Hamiltonian systems possessing an infinite number of conservation laws.

Quantum integrability of perturbed Virasoro-degenerate conformal models

Off-critical conservation laws of a class of irrational conformal models are examined. It has been conjectured that the massive theories that arise under perturbation with the energy (respectively spin-density) operator possess an infinite number of conservation laws, constructable in the Hilbert space of the CFT. A constructive proof of this conjecture is given by means of an algorithmic Fock space procedure.

Further results for infinite conservation laws in the one-dimensional Hubbard model.

It is found that a new two-dimensional (2D) lattice model of two coupled asymmetric six-vertex models provides a one-parameter family of transfer matrices commuting with the Hamiltonian of the 1D Hubbard model with a chemical potential term, for a special choice of the coupling constants. This new statistical model can be regarded as a four-state model in two dimensions, since the six-vertex model can be viewed as a two-state one. It is conjectured that this covering model satisfies the Yang-Baxter equations, i.e., it possesses a commuting family of row-to-row transfer matrices. Moreover, an alternative generating functional for an infinite number of conservation laws in the 1D Hubbard model is written down. This may be useful in understanding the completeness of the set of local conserved currents.

ON A QUANTUM VERSION OF CONSERVATION LAWS FOR DERIVATIVE NONLINEAR SCHRÖDINGER EQUATION

We have derived the quantum mechanical versions of infinite number of conservation laws associated with Derivative Nonlinear Schrödinger equation with the help of a methodology used in string theory. The renormalised version of the conserved quantities are obtained with explicit forms of the counter terms.

Supersymmetric extension of the Korteweg–de Vries equation

It is shown that among a one-parameter family of supersymmetric extensions of the Korteweg–de Vries equation, there is a special system that has an infinite number of conservation laws, which can be formulated in the second Hamiltonian structure, and which has a nontrivial Lax representation. Its modified version is also discussed.

COMPUTER SIMULATION OF THREE-DIMENSIONAL VORTEX DYNAMICS

The discrete model, approximating with exponential accuracy the set of interacting closed vortex lines in an ideal fluid, is proposed and investigated by means of the computer. The vortex lines move in their own velocity field according to the Biot-Savart law. This is a generalized Hamiltonian system possessing in addition an infinite number of conservation laws. Nevertheless, the motion becomes stochastic for certain initial conditions, and may be interpreted as marking the onset of turbulence.

Quantum mechanics as an integrable system

Basic dynamical equations of quantum mechanics, including the Schrödinger, Pauli, Dirac, and Klein-Gordon equations, are bi-hamiltonian systems with an infinite number of conservation laws.

An infinite lie group of symmetry of one-dimensional gas flow, for a class of entropy distributions

We have shown in earlier works the existence of three previously unknown symmetries of the equations of one-dimensional gas dynamics, with arbitrary entropy distribution and arbitrary polytropic index γ. These symmetries are seen here to form a group whenever the equation of state is of the form P = ϱ 3( a 0 + a 1 M + a 2 M 2) −2 where M = ∝ ϱd r is the Lagrangian mass coordinate. Introducing the remaining symmetry of space-translation enlarges the group into a Lie group of symmetry of infinite order, from which an infinite number of conservation laws can be deduced by application of Noether's theorem. The Lie group has a finite sub-algebra of order eight, which has SU3 structure; the list of associated conservation laws includes each of the six ones that are derivable from general physical principles, namely: the energy, momentum and the center-of-mass integrals, two integrals expressing scale invariance, and one associated with the virial theorem; the remaining two integrals of the octet are of a new type. Such a situation reminds us of the case of the Korteweg-de Vries equation in the soliton problem, where the symmetries and infinite number of conservation laws arise as a result of the possibility to linearize through the inverse-scattering method. Thus the question is raised of whether the inverse-scattering method also applies to gas-dynamical equations (with the above equation of state), or else whether another method of linearization may be found.

SU(3) symmetry of the equations of unidimensional gas flow, with arbitrary entropy distribution

We have shown in an earlier work that, assuming a particular class of equations of state, the Euler equations of one-dimensional gas flow are invariant under an SU(3) group of transformations, and in fact admit of a Lie group of symmetry of infinite order; they, therefore, possess an infinite number of conservation laws. We show in the present work that the SU(3) symmetrical formalism still brings about tremendous simplification and analytical order in the most general case where the equation of state is arbitrary. The six characteristic equations assume a vector form and relate two conjugate, three-dimensional vectors U and X. The SU(3) symmetry is only broken to a minor extent through the occurrence of a multiplicative factor Γ in the equations. The conservation laws take the form of the Cauchy integrability condition for the elements of a traceless second rank tensor εij and, taken all together, form an SU(3) octet; in the most general case, however, there exist four conservation laws only (five if the gas is monatomic) as a result of the breaking of symmetry. Application of these results to the theory of self-similar flow is also discussed. Finally, we show the invariance of the equations of monatomic gas flow under Lorentz transformations in a three-dimensional Minkowski space; that raises the question of whether a geometrical relation may exist between the Minkowski light cones and characteristics.

Generalized internal long waves equations: Construction, Hamiltonian structure, and conservation laws

A general class of the ILW type equations is constructed. We introduce a Hamiltonian structure and construct an infinite number of conservation laws.

A variational approach to bäcklund and infinitesimal bäcklund transformation for Kadomtsev-Petviashvili equation

A new form of the Bäcklund transformation for the Kadomtsev-Petviashvili equation is obtained through the variational formalism. At no stage we use the equation of motion for the deduction of the Bäcklund transformation. So we follow the method of Steudel for the derivation of the infinite number of conservation laws using Noether's theorem in three dimensions and the corresponding infinitesimal Bäcklund transformation.

Solution of Toda systems

The Toda field is a multicomponent field in two space-time dimensions, satisfying a generalisation of the Liouville equation ∂ 2 φ + exp φ = 0. We define and solve non-perturbatively a quantum Toda field theory and its classical limit. Our technique generates an infinite number of conservation laws. By restricting the fields to one dimension we also solve the quantum mechanics and the classical mechanics of the Toda lattice.

Critical exponents of the Ashkin-Teller model

We show that the time-continuous Hamiltonian of the two-dimensional Ashkin-Teller model is that of a massive Thirring model. This implies the existence of an infinite number of conservation laws in the critical region and enables us to determine the critical exponents of the energy, crossover, and polarization operators.