25,228 research outputs found
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
Geometry of jet spaces and integrable systems
An overview of some recent results on the geometry of partial differential
equations in application to integrable systems is given. Lagrangian and
Hamiltonian formalism both in the free case (on the space of infinite jets) and
with constraints (on a PDE) are discussed. Analogs of tangent and cotangent
bundles to a differential equation are introduced and the variational Schouten
bracket is defined. General theoretical constructions are illustrated by a
series of examples.Comment: 54 pages; v2-v6 : minor correction
Psi-floor diagrams and a Caporaso-Harris type recursion
Floor diagrams are combinatorial objects which organize the count of tropical
plane curves satisfying point conditions. In this paper we introduce Psi-floor
diagrams which count tropical curves satisfying not only point conditions but
also conditions given by Psi-classes (together with points). We then generalize
our definition to relative Psi-floor diagrams and prove a Caporaso-Harris type
formula for the corresponding numbers. This formula is shown to coincide with
the classical Caporaso-Harris formula for relative plane descendant
Gromov-Witten invariants. As a consequence, we can conclude that in our case
relative descendant Gromov-Witten invariants equal their tropical counterparts.Comment: minor changes to match the published versio
Integrable dispersionless PDE in 4D, their symmetry pseudogroups and deformations
We study integrable non-degenerate Monge-Ampere equations of Hirota type in
4D and demonstrate that their symmetry algebras have a distinguished graded
structure, uniquely determining the equations. This is used to deform these
heavenly type equations into new integrable PDE of the second order with large
symmetry pseudogroups. We classify the obtained symmetric deformations and
discuss self-dual hyper-Hermitian geometry of their solutions, which encode
integrability via the twistor theory.Comment: This version is updated with an appendix about multi-component
extensions of the integrable equations. Our deformations can be considered as
reductions of such extensions (as they are reductions of the self-duality
equation), but we stress that second order deformations carry the natural
geometry which encodes integrability. We also expanded the introduction a bi
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