9,115 research outputs found
Creation of the Nonconformal Scalar Particles in Nonstationary Metric
The nonconformal scalar field is considered in N-dimensional space-time with
metric which includes, in particular, the cases of nonhomogeneous spaces and
anisotropic spaces of Bianchi type-I. The modified Hamiltonian is constructed.
Under the diagonalization of it the energy of quasiparticles is equal to the
oscillator frequency of the wave equation. The density of particles created by
nonstationary metric is investigated. It is shown that the densities of
conformal and nonconformal particles created in Friedmann radiative-dominant
Universe coincide.Comment: LaTeX, 4 pages, no figure
Non polynomial conservation law densities generated by the symmetry operators in some hydrodynamical models
New extra series of conserved densities for the polytropic gas model and
nonlinear elasticity equation are obtained without any references to the
recursion operator or to the Lax operator formalism. Our method based on the
utilization of the symmetry operators and allows us to obtain the densities of
arbitrary homogenuity dimensions. The nonpolynomial densities with logarithmics
behaviour are presented as an example. The special attention is paid for the
singular case for which we found new non homogenious solutions
expressed in terms of the elementary functions.Comment: 11 pages, 1 figur
Classification of integrable Vlasov-type equations
Classification of integrable Vlasov-type equations is reduced to a functional
equation for a generating function. A general solution of this functional
equation is found in terms of hypergeometric functions.Comment: latex, 15 pages, to appear in Theoretical and Mathematical Physic
On the bi-Hamiltonian Geometry of WDVV Equations
We consider the WDVV associativity equations in the four dimensional case.
These nonlinear equations of third order can be written as a pair of six
component commuting two-dimensional non-diagonalizable hydrodynamic type
systems. We prove that these systems possess a compatible pair of local
homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and
third order, respectively).Comment: 21 pages, revised published version; exposition substantially
improve
Three Dimensional Reductions of Four-Dimensional Quasilinear Systems
In this paper we show that integrable four dimensional linearly degenerate
equations of second order possess infinitely many three dimensional
hydrodynamic reductions. Furthermore, they are equipped infinitely many
conservation laws and higher commuting flows. We show that the dispersionless
limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking
Soliton equations introduced by O.I. Bogoyavlenski) are one and two component
reductions (respectively) of one of these four dimensional linearly degenerate
equations
Multi-Lagrangians for Integrable Systems
We propose a general scheme to construct multiple Lagrangians for completely
integrable non-linear evolution equations that admit multi- Hamiltonian
structure. The recursion operator plays a fundamental role in this
construction. We use a conserved quantity higher/lower than the Hamiltonian in
the potential part of the new Lagrangian and determine the corresponding
kinetic terms by generating the appropriate momentum map. This leads to some
remarkable new developments. We show that nonlinear evolutionary systems that
admit -fold first order local Hamiltonian structure can be cast into
variational form with Lagrangians which will be local functionals of
Clebsch potentials. This number increases to when the Miura
transformation is invertible. Furthermore we construct a new Lagrangian for
polytropic gas dynamics in dimensions which is a {\it local} functional
of the physical field variables, namely density and velocity, thus dispensing
with the necessity of introducing Clebsch potentials entirely. This is a
consequence of bi-Hamiltonian structure with a compatible pair of first and
third order Hamiltonian operators derived from Sheftel's recursion operator.Comment: typos corrected and a reference adde
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