Symmetry Reduction for a System of Nonlinear Evolution Equations

Abstract

In this paper we obtain the maximal Lie symmetry algebra of a system of PDEs. We reduce this system to a system of ODEs, using some rank three subalgebras of the finite-dimensional part of the symmetry algebra. The corresponding invariant solutions of the PDEs are obtained.     

Deformed C λ-Extended Heisenberg Algebra in Noncommutative Phase-Space

We construct a deformed C _λ-extended Heisenberg algebra in two-dimensional space using noncommuting coordinates which close an algebra depends on statistical parameter characterizing exotic particles. The obtained symmetry is nothing but an exotic particles algebra interpolating between bosonic and deformed fermionic algebras. PACS numbers: 03.65.Fd, 02.40.Gh, 05.30.Pr     

The Bohm-Aharonov effect: A seven-dimensional structural group

We realize a nonfaithful representation of a seven-dimensional Lie algebra, the extension of which to its universal enveloping algebra contains most of the observables of the scattering Aharonov-Bohm effect, as essentially self-adjoint operators: the scattering Hamiltonian, the total and kinetic angular momenta, the positions and the kinetic momenta. By restriction, we obtain the model introduced in Lett. Math. Phys. 1 (1976), 155–163.     

Asymptotic symmetry of geometries with Schrödinger isometry

We show that the asymptotic symmetry algebra of geometries with Schrödinger isometry in any dimension is an infinite-dimensional algebra containing one copy of Virasoro algebra. It is compatible with the fact that the corresponding geometries are dual to non-relativistic CFTs whose symmetry algebra is the Schrödinger algebra which admits an extension to an infinite-dimensional symmetry algebra containing a Virasoro subalgebra.     

The Cohomology of the Variational Bicomplex Invariant under the Symmetry Algebra of the Potential Kadomtsev-Petviashvili Equation

Abstract

The variational bicomplex of forms invariant under the symmetry algebra of the potential Kadomtsev-Petviashvili equation is described and the cohomology of the associated Euler-Lagrange complex is computed. The results are applied to a characterization problem of the Kadomtsev-Petviashvili equation by its symmetry algebra originally posed by David, Levi, and Winternitz.     

Invariance Analysis of the (2+1) Dimensional Long Dispersive Wave Equation

Abstract

In this paper, we bring out the Lie symmetries and associated similarity reductions of the recently proposed (2+1) dimensional long dispersive wave equation. We point out that the integrable system admits an infinite-dimensional symmetry algebra along with Kac-Moody-Virasoro-type subalgebras. We also bring out certain physically interesting solutions.     

Minimal representations of supersymmetry and 1<Emphasis Type="Italic">D N</Emphasis>-Extended <Emphasis Type="Italic">σ</Emphasis> models

We discuss the minimal representations of the 1D N-Extended Supersymmetry algebra (the Z ₂-graded symmetry algebra of the Supersymmetric Quantum Mechanics) linearly realized on a finite number of fields depending on a real parameter t, the time. Their knowledge allows to construct onedimensional sigma-models with extended off-shell supersymmetries without using superfields.     

Deformed Schrödinger symmetry on noncommutative space

We construct the deformed generators of Schrödinger symmetry consistent with noncommutative space. The examples of the free particle and the harmonic oscillator, both of which admit Schrödinger symmetry, are discussed in detail. We construct a generalised Galilean algebra where the second central extension exists in all dimensions. This algebra also follows from the Inonu–Wigner contraction of a generalised Poincaré algebra in noncommuting space.     

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.     

Eigenmode scattering relations for plane-stratified gyrotropic media

We consider the 4×4 scattering matrixS for a plane wave incident on a plane-stratified gyrotropic multilayer slab, which is assumed to have a single symmetry axis (the magnetisation vector in a ferrite sheet, or the external magnetic field direction in a plane-stratified magnetoplasma). In the given (original) problem the plane of incidence is at an azimuthal angle ϕ with respect to the magnetic meridian plane (the plane containing the normal to the stratification, and the gyrotropic symmetry axis), and there is a corresponding “conjugate” set of wave fields, in which the plane of incidence is at an azimuthal angle (π−ϕ), with a corresponding conjugate scattering matrixS′. Adjoint wave fields, obtained by reversing the magnetic symmetry vector, are used to yield the eigenmode amplitudes required in the definition of the scattering matrices. At an interface between two adjacent layers the scattering matrices are shown to be uniquely determined by the characteristic wave polarisations, and this is used to prove that the given and conjugate scattering matrices,S andS′, for the overall multilayer system are mutually transposed, i.e. .     

Spin-dependent sum rule for high-energy neutrino reactions

We present a sum rule which related the inclusive cross section for high-energy neutrino scattering on polarized proton targets to the axial-vector coupling constant. The derivation of the sum rule rests on the equal-time algebra of the time-components of the weak currents. The sum rule also provides a discriminatory test of unified models of weak and electromagnetic interactions based on spontaneously-broken gauge symmetry.     

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.     

Exact representation of meson-meson scattering amplitudes consistent with current-algebra constraints

All four-point Green functions connected to the vector and axial-vector currents by Ward identities are constructed, consistent with the constraints of an arbitrary current algebra. The structure of the four-point functions is separated into one-particle reducible and irreducible parts, without approximations. That is, no single-particle approximation is made, nor is it assumed that the strangeness-changing vector currents are conserved. The result is a representation of the meson-meson scattering amplitude that explicitly satisfies all the constraints of the current algebra. In particular, it satisfies: (i) threshold theorems, (ii) crossing symmetry and (iii) reduction to the usual tree-approximation in the single-particle approximation. Application is made to kaon-pion scattering, obtaining a representation of the scattering amplitude that is suitable for the investigation of the constraints of elastic unitarity.     

Dynamical symmetry of monopole scattering

The conserved “Runge-Lenz” vector found by Gibbons and Manton in describing asymptotic scattering of Bogomolny-Prasad-Sommerfield monopoles is shown to generate, together with angular momentum, an o (4) or an o(3,1) symmetry algebra analogous to the one in the Kepler problem. This allows for a group-theoretical derivation of the bound-state spectrum and the scattering cross section.     

A three-parameter Hopf deformation of the algebra of Feynman-like diagrams*

We construct a three-parameter deformation of the Hopf algebra LDIAG. This is the algebra that appears in an expansion in terms of Feynman-like diagrams of the product formula in a simplified version of quantum field theory. This new algebra is a true Hopf deformation which reduces to LDIAG for some parameter values and to the algebra of matrix quasi-symmetric functions (MQSym) for others, and thus relates LDIAG to other Hopf algebras of contemporary physics. Moreover, there is an onto linear mapping preserving products from our algebra to the algebra of Euler–Zagier sums.     

SU(2) andSU(1,1) algebra eigenstates: A unified analytic approach to coherent and intelligent states

We introduce the concept of algebra eigenstates which are defined for an arbitrary Lie group as eigenstates of elements of the corresponding complex Lie algebra. We show that this concept unifies different definitions of coherent states associated with a dynamical symmetry group. On the one hand, algebra eigenstates include different sets of Perelomov's generalized coherent states. On the other hand, intelligent states (which are squeezed states for a system of general symmetry) also form a subset of algebra eigenstates. We develop the general formalism and apply it to theSU(2) andSU(1,1) simple Lie groups. Complete solutions to the general eigenvalue problem are found in both cases by a method that employs analytic representations of the algebra eigenstates. This analytic method also enables us to obtain exact closed expressions for quantum statistical properties of an arbitrary algebra eigenstate. Important special cases such as standard coherent states and intelligent states are examined and relations between them are studied by using their analytic representations.     

O(4,2) dynamical symmetry of the Kaluza-Klein monopole

The o(4)/o(3,1) dynamical symmetry of the Gross-Perry-Sorkin monopole in five-dimensional Kaluza-Klein theory (which also describes the asymptotic scattering of BPS monopoles) is extended, in analogy to the Kepler problem, to the conformal algebra of o(4,2).     

Physical symmetries in a theory of several scalar real fields

Let us consider a theory ofn scalar, real, local, Poincaré covariant quantum fields forming an irreducible set and giving rise to one particle states belonging to the same mass different from zero. The vacuum is unique. It is shown under fairly weak assumptions that every Poincaré and TCP invariant symmetry of the theory, implemented unitarily, which mapps localized elements of the field algebra into operators almost local with respect to the former (such a symmetry we call a physical one) can be defined uniquely in terms of the incoming or outgoing fields and ann-dimensional (real) orthogonal matrix. The symmetry commutes with the scattering matrix. Incidentally we show also that the symmetry groups are compact. A special case of these symmetries are the internal symmetries and symmetries induced by locally conserved currents local with respect to the basic fields and transforming under the same representation of the Poincaré group. We may make linear combinations out the original fields resulting in complex fields and its complex conjugate in a suitable way. The inspection of the representations of the groupsSO(n) and their subgroups sheds some light on the s.c. generalized Carruthers Theorem concerning the self- and pair-conjugate multiplets.     

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.