16 research outputs found
Tzitzeica solitons versus relativistic Calogero–Moser three-body clusters
We establish a connection between the hyperbolic relativistic Calogero–Moser systems and a class of soliton solutions to the Tzitzeica equation (also called the Dodd–Bullough–Zhiber–Shabat–Mikhailov equation). In the 6N-dimensional phase space Omega of the relativistic systems with 2N particles and N antiparticles, there exists a 2N-dimensional Poincaré-invariant submanifold OmegaP corresponding to N free particles and N bound particle-antiparticle pairs in their ground state. The Tzitzeica N-soliton tau functions under consideration are real valued and obtained via the dual Lax matrix evaluated in points of OmegaP. This correspondence leads to a picture of the soliton as a cluster of two particles and one antiparticle in their lowest internal energy state
On the tau-functions of the Degasperis-Procesi equation
The DP equation is investigated from the point of view of
determinant-pfaffian identities. The reciprocal link between the
Degasperis-Procesi (DP) equation and the pseudo 3-reduction of the
two-dimensional Toda system is used to construct the N-soliton solution of the
DP equation. The N-soliton solution of the DP equation is presented in the form
of pfaffian through a hodograph (reciprocal) transformation. The bilinear
equations, the identities between determinants and pfaffians, and the
-functions of the DP equation are obtained from the pseudo 3-reduction of
the two-dimensional Toda system.Comment: 27 pages, 4 figures, Journal of Physics A: Mathematical and
Theoretical, to be publishe
Laplace transformations of hydrodynamic type systems in Riemann invariants: periodic sequences
The conserved densities of hydrodynamic type system in Riemann invariants
satisfy a system of linear second order partial differential equations. For
linear systems of this type Darboux introduced Laplace transformations,
generalising the classical transformations in the scalar case. It is
demonstrated that Laplace transformations can be pulled back to the
transformations of the corresponding hydrodynamic type systems. We discuss
periodic Laplace sequences of with the emphasize on the simplest nontrivial
case of period 2. For 3-component systems in Riemann invariants a complete
discription of closed quadruples is proposed. They turn to be related to a
special quadratic reduction of the (2+1)-dimensional 3-wave system which can be
reduced to a triple of pairwize commuting Monge-Ampere equations. In terms of
the Lame and rotation coefficients Laplace transformations have a natural
interpretation as the symmetries of the Dirac operator, associated with the
(2+1)-dimensional n-wave system. The 2-component Laplace transformations can be
interpreted also as the symmetries of the (2+1)-dimensional integrable
equations of Davey-Stewartson type. Laplace transformations of hydrodynamic
type systems originate from a canonical geometric correspondence between
systems of conservation laws and line congruences in projective space.Comment: 22 pages, Late
Continuous Symmetries of Difference Equations
Lie group theory was originally created more than 100 years ago as a tool for
solving ordinary and partial differential equations. In this article we review
the results of a much more recent program: the use of Lie groups to study
difference equations. We show that the mismatch between continuous symmetries
and discrete equations can be resolved in at least two manners. One is to use
generalized symmetries acting on solutions of difference equations, but leaving
the lattice invariant. The other is to restrict to point symmetries, but to
allow them to also transform the lattice.Comment: Review articl