5,937 research outputs found
Non-Hamiltonian generalizations of the dispersionless 2DTL hierarchy
We consider two-component integrable generalizations of the dispersionless
2DTL hierarchy connected with non-Hamiltonian vector fields, similar to the
Manakov-Santini hierarchy generalizing the dKP hierarchy. They form a
one-parametric family connected by hodograph type transformations. Generating
equations and Lax-Sato equations are introduced, a dressing scheme based on the
vector nonlinear Riemann problem is formulated. The simplest two-component
generalization of the dispersionless 2DTL equation is derived, its differential
reduction analogous to the Dunajski interpolating system is presented. A
symmetric two-component generalization of the dispersionless elliptic 2DTL
equation is also constructed.Comment: 10 pages, the text of the talk at NEEDS 09. Notations clarified,
references adde
Internal Time Peculiarities as a Cause of Bifurcations Arising in Classical Trajectory Problem and Quantum Chaos Creation in Three-Body System
A new formulation of the theory of quantum mechanical multichannel scattering
for three-body collinear systems is proposed. It is shown, that in this simple
case the principle of quantum determinism in the general case breaks down and
we have a micro-irreversible quantum mechanics. The first principle
calculations of the quantum chaos (wave chaos) were pursued on the example of
an elementary chemical reaction Li+(FH)->(LiFH)*->(LiF)+H.Comment: 7 pages, 3 figures, accepted for publication in Int. J. of
Bifurcation & Chao
Dunajski generalization of the second heavenly equation: dressing method and the hierarchy
Dunajski generalization of the second heavenly equation is studied. A
dressing scheme applicable to Dunajski equation is developed, an example of
constructing solutions in terms of implicit functions is considered. Dunajski
equation hierarchy is described, its Lax-Sato form is presented. Dunajsky
equation hierarchy is characterized by conservation of three-dimensional volume
form, in which a spectral variable is taken into account.Comment: 13 page
Grassmannians Gr(N-1,N+1), closed differential N-1 forms and N-dimensional integrable systems
Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the
requirement of closedness of the differential N-1 forms of rank
N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these
flows, given by the systems of the N-1 quasi-linear differential equations,
describe coisotropic deformations of (N-1)-dimensional linear subspaces. For
the class of solutions which are Laurent polynomials in one variable these
systems coincide with N-dimensional integrable systems such as Liouville
equation (N=2), dispersionless Kadomtsev-Petviashvili equation (N=3),
dispersionless Toda equation (N=3), Plebanski second heavenly equation (N=4)
and others. Gauge invariant part of the forms provides us with
the compact form of the corresponding hierarchies. Dual quasi-linear systems
associated with the projectively dual Grassmannians Gr(2,N+1) are defined via
the requirement of the closedness of the dual forms . It
is shown that at N=3 the self-dual quasi-linear system, which is associated
with the harmonic (closed and co-closed) form , coincides with the
Maxwell equations for orthogonal electric and magnetic fields.Comment: 26 pages, references adde
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