46 research outputs found
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
Investigation of dynamical systems using tools of the theory of invariants and projective geometry
The investigation of nonlinear dynamical systems of the type
by means of reduction to
some ordinary differential equations of the second order in the form
is done. The main
backbone of this investigation was provided by the theory of invariants
developed by S. Lie, R. Liouville and A. Tresse at the end of the 19th century
and the projective geometry of E. Cartan. In our work two, in some sense
supplementary, systems are considered: the Lorenz system and the R\"o\ss ler system
. The invarinats for the ordinary
differential equations, which correspond to the systems mentioned abouve, are
evaluated. The connection of values of the invariants with characteristics of
dynamical systems is established.Comment: 18 pages, Latex, to appear in J. of Applied Mathematics (ZAMP
Heat operator with pure soliton potential: properties of Jost and dual Jost solutions
Properties of Jost and dual Jost solutions of the heat equation,
and , in the case of a pure solitonic potential are studied in
detail. We describe their analytical properties on the spectral parameter
and their asymptotic behavior on the -plane and we show that the values of
and the residua of at special discrete
values of are bounded functions of in a polygonal region of the
-plane. Correspondingly, we deduce that the extended version of the
heat operator with a pure solitonic potential has left and right annihilators
for belonging to these polygonal regions.Comment: 26 pages, 3 figure
Lattice and q-difference Darboux-Zakharov-Manakov systems via -dressing method
A general scheme is proposed for introduction of lattice and q-difference
variables to integrable hierarchies in frame of -dressing
method . Using this scheme, lattice and q-difference Darboux-Zakharov-Manakov
systems of equations are derived. Darboux, B\"acklund and Combescure
transformations and exact solutions for these systems are studied.Comment: 8 pages, LaTeX, to be published in J Phys A, Letters
On the central quadric ansatz: integrable models and Painleve reductions
It was observed by Tod and later by Dunajski and Tod that the Boyer-Finley
(BF) and the dispersionless Kadomtsev-Petviashvili (dKP) equations possess
solutions whose level surfaces are central quadrics in the space of independent
variables (the so-called central quadric ansatz). It was demonstrated that
generic solutions of this type are described by Painleve equations PIII and
PII, respectively. The aim of our paper is threefold:
-- Based on the method of hydrodynamic reductions, we classify integrable
models possessing the central quadric ansatz. This leads to the five canonical
forms (including BF and dKP).
-- Applying the central quadric ansatz to each of the five canonical forms,
we obtain all Painleve equations PI - PVI, with PVI corresponding to the
generic case of our classification.
-- We argue that solutions coming from the central quadric ansatz constitute
a subclass of two-phase solutions provided by the method of hydrodynamic
reductions.Comment: 12 page
On a class of reductions of Manakov-Santini hierarchy connected with the interpolating system
Using Lax-Sato formulation of Manakov-Santini hierarchy, we introduce a class
of reductions, such that zero order reduction of this class corresponds to dKP
hierarchy, and the first order reduction gives the hierarchy associated with
the interpolating system introduced by Dunajski. We present Lax-Sato form of
reduced hierarchy for the interpolating system and also for the reduction of
arbitrary order. Similar to dKP hierarchy, Lax-Sato equations for (Lax
fuction) due to the reduction split from Lax-Sato equations for (Orlov
function), and the reduced hierarchy for arbitrary order of reduction is
defined by Lax-Sato equations for only. Characterization of the class of
reductions in terms of the dressing data is given. We also consider a waterbag
reduction of the interpolating system hierarchy, which defines
(1+1)-dimensional systems of hydrodynamic type.Comment: 15 pages, revised and extended, characterization of the class of
reductions in terms of the dressing data is give
Solutions of the Kpi Equation with Smooth Initial Data
The solution of the Kadomtsev--Petviashvili I (KPI) equation with
given initial data belonging to the Schwartz space is considered. No
additional special constraints, usually considered in literature, as
are required to be satisfied by the initial data. The
problem is completely solved in the framework of the spectral transform theory
and it is shown that satisfies a special evolution version of the
KPI equation and that, in general, has different left and
right limits at the initial time . The conditions of the type
, and so on (first, second,
etc. `constraints') are dynamically generated by the evolution equation for
. On the other side with prescribed
order of integrations is not necessarily equal to zero and gives a nontrivial
integral of motion.Comment: 17 pages, 23 June 1993, LaTex fil