12 research outputs found
On the structure of the B\"acklund transformations for the relativistic lattices
The B\"acklund transformations for the relativistic lattices of the Toda type
and their discrete analogues can be obtained as the composition of two duality
transformations. The condition of invariance under this composition allows to
distinguish effectively the integrable cases. Iterations of the B\"acklund
transformations can be described in the terms of nonrelativistic lattices of
the Toda type. Several multifield generalizations are presented
Integrability of Dirac reduced bi-Hamiltonian equations
First, we give a brief review of the theory of the Lenard-Magri scheme for a
non-local bi-Poisson structure and of the theory of Dirac reduction. These
theories are used in the remainder of the paper to prove integrability of three
hierarchies of bi-Hamiltonian PDE's, obtained by Dirac reduction from some
generalized Drinfeld-Sokolov hierarchies.Comment: 15 pages. Corrected some typos and added missing equations in Section
5 for g=sl_n, n>
Asymptotic solitons of the Johnson equation
We prove the existence of non-decaying real solutions of the Johnson
equation, vanishing as . We obtain asymptotic formulas as
for the solutions in the form of an infinite series of asymptotic
solitons with curved lines of constant phase and varying amplitude and width
Differential constraints compatible with linearized equations
Differential constraints compatible with the linearized equations of partial
differential equations are examined. Recursion operators are obtained by
integrating the differential constraints
Infinitely many local higher symmetries without recursion operator or master symmetry: integrability of the Foursov--Burgers system revisited
We consider the Burgers-type system studied by Foursov, w_t &=& w_{xx} + 8 w
w_x + (2-4\alpha)z z_x, z_t &=& (1-2\alpha)z_{xx} - 4\alpha z w_x +
(4-8\alpha)w z_x - (4+8\alpha)w^2 z + (-2+4\alpha)z^3, (*) for which no
recursion operator or master symmetry was known so far, and prove that the
system (*) admits infinitely many local generalized symmetries that are
constructed using a nonlocal {\em two-term} recursion relation rather than from
a recursion operator.Comment: 10 pages, LaTeX; minor changes in terminology; some references and
definitions adde