The Hamilton Cartan formalism for rth-order Lagrangians and the integrability of the KdV and modified KdV equations

The Hamilton Cartan formalism for rth order Lagrangians is presented in a form suited to dealing with higher-order conserved currents. Noether's Theorem and its converse are stated and Poisson brackets are defined for conserved charges. An isomorphism between the Lie algebra of conserved currents and a Lie algebra of infinitesimal symmetries of the Cartan form is established. This isomorphism, together with the commutativity of the Bäcklund transformations for the KdV and modified KdV equations, allows a simple geometric proof that the infinite collections of conserved charges for these equations are in involution with respect to the Poisson bracket determined by their Lagrangians. Thus, the formal complete integrability of these equations appears as a consequence of the properties of their Bäcklund transformations.It is noted that the Hamilton Cartan formalism determines a symplectic structure on the space of functionals determined by conserved charges and that, in the case of the KdV equation, the structure is the same as that given by Miura et al. [5].     

Lie superalgebraic approach to super Toda lattice and generalized super KdV equations

We propose a super Lax type equation based on a certain class of Lie superalgebra as a supersymmetric extension of generalized (modified) KdV hierarchy. We are able to construct an infinite set of conservation laws and the consistent time evolution generators for generalized modified super KdV equations. Thefirst few of the conserved currents, the (modified) super KdV equation and the super Miura transformation are worked out explicitly in the case of twisted affine Lie superalgebraOSp(2/2)⁽²⁾.Partially supported by the Grant-in-Aid for Scientific Research from the Ministry of Education, Science and Culture (#01540246 and #01790203).RIFP will be known as Yukawa Institute for Theoretical Physics from June 8, 1990     

Two expanding forms of a Lie algebra and their application

With the help of a known Lie algebra,two new high order Lie algebras are constructed.It is remarkable that they have different constructing approaches.The first Lie algebra is constructed by the definition of integrable couplings.the second one by an extension of Lie algebra,Then by making use of Tu scheme,a generalized AKNS hierarchy and another new hierarchy are obtained.As a reduction case of the first hierarchy,a kind of coupled KdV equation is presented.As a reduction case of the second one,a new coupled Schroedinger equation is given.     

On Unique Symmetry of Two Nonlinear Generalizations of the Schrödinger Equation

Abstract

We prove that two nonlinear generalizations of the nonlinear Schrödinger equation are invariant with respect to a Lie algebra that coincides with the invariance algebra of the Hamilton-Jacobi equation.     

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.     

The higher order Hamiltonian structures for the modified classical Yang-Baxter equation

We consider constructing the higher order Hamiltonian structures on the dual of the Lie algebra from the first Hamiltonian structure of the coadjoint orbit method. For this purpose we show that the structure of the Lie algebrag is inherited to the algebra of vector fields ong ^* through the solution of the Modified Classical Yang-Baxter equation (Classicalr matrix). We study the algebra that generates the compatible Poisson brackets.This work was supported by Grant Aid for Scientific Research, the Ministry of Education.     

Symmetry and general symmetry groups of the coupled Kadomtsev--Petviashvili equation

In this paper, the Lie symmetry algebra of the coupled Kadomtsev--Petviashvili (cKP) equation is obtained by the classical Lie group method and this algebra is shown to have a Kac--Moody--Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et al。 From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.     

Lie Symmetries of Einstein’s Vacuum Equations in N Dimensions

Abstract

We investigate Lie symmetries of Einstein’s vacuum equations in N dimensions, with a cosmological term. For this purpose, we first write down the second prolongation of the symmetry generating vector fields, and compute its action on Einstein’s equations. Instead of setting to zero the coefficients of all independent partial derivatives (which involves a very complicated substitution of Einstein’s equations), we set to zero the coefficients of derivatives that do not appear in Einstein’s equations. This considerably constrains the coefficients of symmetry generating vector fields. Using the Lie algebra property of generators of symmetries and the fact that general coordinate transformations are symmetries of Einstein’s equations, we are then able to obtain all the Lie symmetries. The method we have used can likely be applied to other types of equations.     

NEW SYMMETRIES AND LIE ALGEBRAS OF THE GENERAL KdV EQUATION

The strong symmetry of the general KdV equation is factorized to a simple form and then the inverse strong symmetry is obtained explicitly. Acting a strong symmetry of the general KdV equation on the trivial symmetry and the known re symmetry, we obtain four new sets of symmetries of the general KdV equation. All these sets of symmetries constitute an infinite dimensional Lie algebra.     

Prolongation theory. A new nonlinear Schrödinger equation

We discuss a new kind of nonlinear Schrödinger equation from the viewpoint of prolongation theory. It is shown that the equation possess a Lax pair with a 3 × 3 matrix structure. It is further demonstrated that by a multiple scale perturbation of Zakharovet al. it can be reduced to the usual KdV equation.     

Quantum Lie Algebras, Their Existence, Uniqueness and q-Antisymmetry

Quantum Lie algebras are generalizations of Lie algebras which have the quantum parameter h built into their structure. They have been defined concretely as certain submodules of the quantized enveloping algebras . On them the quantum Lie product is given by the quantum adjoint action. Here we define for any finite-dimensional simple complex Lie algebra an abstract quantum Lie algebra independent of any concrete realization. Its h-dependent structure constants are given in terms of inverse quantum Clebsch-Gordan coefficients. We then show that all concrete quantum Lie algebras are isomorphic to an abstract quantum Lie algebra . In this way we prove two important properties of quantum Lie algebras: 1) all quantum Lie algebras associated to the same are isomorphic, 2) the quantum Lie product of any is q-antisymmetric. We also describe a construction of which establishes their existence. Received: 23 May 1996 / Accepted: 17 October 1996     

Generalized Virasoro algebra: left-symmetry and related algebraic and hydrodynamic properties

Motivated by the work of Kupershmidt (J. Nonlin. Math. Phys. 6 (1998), 222 –245) we discuss the occurrence of left symmetry in a generalized Virasoro algebra. The multiplication rule is defined, which is necessary and sufficient for this algebra to be quasi-associative. Its link to geometry and nonlinear systems of hydrodynamic type is also recalled. Further, the criteria of skew-symmetry, derivation and Jacobi identity making this algebra into a Lie algebra are derived. The coboundary operators are defined and discussed. We deduce the hereditary operator and its generalization to the corresponding 3–ary bracket. Further, we derive the so-called ρ–compatibility equation and perform a phase-space extension. Finally, concrete relevant particular cases are investigated.     

Jacobi’s Last Multiplier and the Complete Symmetry Group of the Ermakov-Pinney Equation

Abstract

The Ermakov-Pinney equation possesses three Lie point symmetries with the algebra sl(2, R). This algebra does not provide a representation of the complete symmetry group of the Ermakov-Pinney equation. We show how the representation of the group can be obtained with the use of the method described in Nucci, J. Nonlin. Math. Phys. 12 (2005) (this issue), which is based on the properties of Jacobi’s last multiplier (Bianchi L, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918), the method of reduction of order (Nucci,J. Math. Phys 37 (1996), 1772–1775) and an interactive code for calculating symmetries (Nucci, Interactive REDUCE programs for calcuating classical, non-classical and Lie-Bäcklund symmetries for differential equations (preprint: Georgia Institute of Technology, Math 062090-051, 1990, and CRC Handbook of Lie Group Analysis of Differential Equations. Vol. 3: New Trends in Theoretical Developments and Computational Methods, Editor: Ibragimov N H, CRC Press, Boca Raton, 1996, 415–481).     

Conservation Laws and Non-Lie Symmetries for Linear PDEs

Abstract

We introduce a method to construct conservation laws for a large class of linear partial differential equations. In contrast to the classical result of Noether, the conserved currents are generated by any symmetry of the operator, including those of the non-Lie type. An explicit example is made of the Dirac equation were we use our construction to find a class of conservation laws associated with a 64 dimensional Lie algebra of discrete symmetries that includes CPT.     

Decomposition of symmetric tensor fields in the presence of a flat contact projective structure

Abstract

Let M be an odd-dimensional Euclidean space endowed with a contact 1-form α. We investigate the space of symmetric contravariant tensor fields over M as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up of those vector fields that preserve the contact structure defined by a. If we consider symmetric tensor fields with coefficients in tensor densities (also called symbols), the vertical cotangent lift of the contact form a defines a contact invariant operator. We also extend the classical contact Hamiltonian to the space of symbols. This generalized Hamiltonian operator on the space of symbols is invariant with respect to the action of the projective contact algebra sp(2n+2) the algebra of vector fields which preserve both the contact structure and the projective structure of the Euclidean space. These two operators lead to a decomposition of the space of symbols, except for some critical density weights, which generalizes a splitting proposed by V. Ovsienko in [18].     

On Relativistic Mass Spectra of a Two-Particle System

Abstract

A relativistic two-particle system with time-asymmetric scalar and vector interactions in the two-dimensional space-time is considered within the frame of the front form of dynamics using the dynamical symmetry approach. The mass-shell equation may be represented in terms of the nonlinear canonical realization of the Lie algebra of the group SO(2, 1). This allows us to quantize the system and to obtain a closed form for the mass spectrum.     

Wronski Brackets and the Ferris Wheel

We connect the Bayesian order on classical states to a certain Lie algebra on . This special Lie algebra structure, made precise by an idea we introduce called a Wronski bracket, suggests new phenomena the Bayesian order naturally models. We then study Wronski brackets on associative algebras, and in the commutative case, discover the beautiful result that they are equivalent to derivations. PACS: 03.67.a, 02.20.Sv.     

Algebraical comparison of classical and quantum polynomial observables

The symplectic vector spaceE of theq andp's of classical mechanics allows a basis free definition of the Poisson bracket in the symmetric algebra overE. Thus the symmetric algebra overE becomes a Lie algebra, which can be compared with the quantum mechanical Weyl algebra with its commutator Lie structure. The universality of the Weyl algebra is used to study the well-known ‘classical’ Moyal realisation of the Weyl algebra in the symmetric algebra. Quantisations are defined as linear mappings of the underlying vector spaces of the two algebras. It is shown that the classical Lie algebra is −2 graded, whereas the quantum Lie algebra is not. This proves that they are not isomorphic, and hence there is no Dirac quantisation.     

An Analogue of Holstein-Primakoff and Dyson Realizations for Lie Superalgebras. The Lie Superalgebra sl(1/n)

Abstract

An analogue of the Holstein-Primakoff and Dyson realizations for the Lie superalgebra sl(1/n) is written down. Expressions are the same as for the Lie algebra sl(n + 1), however in the latter, Bose operators have to be replaced with Fermi operators.