42 research outputs found
On the classification of scalar evolutionary integrable equations in dimensions
We consider evolutionary equations of the form where
is the nonlocality, and the right hand side is polynomial
in the derivatives of and . The recent paper \cite{FMN} provides a
complete list of integrable third order equations of this kind. Here we extend
the classification to fifth order equations. Besides the known examples of
Kadomtsev-Petviashvili (KP), Veselov-Novikov (VN) and Harry Dym (HD) equations,
as well as fifth order analogues and modifications thereof, our list contains a
number of equations which are apparently new. We conjecture that our examples
exhaust the list of scalar polynomial integrable equations with the nonlocality
. The classification procedure consists of two steps. First, we classify
quasilinear systems which may (potentially) occur as dispersionless limits of
integrable scalar evolutionary equations. After that we reconstruct dispersive
terms based on the requirement of the inheritance of hydrodynamic reductions of
the dispersionless limit by the full dispersive equation
Analytic-bilinear approach to integrable hierarchies. II. Multicomponent KP and 2D Toda lattice hierarchies
Analytic-bilinear approach for construction and study of integrable
hierarchies is discussed. Generalized multicomponent KP and 2D Toda lattice
hierarchies are considered. This approach allows to represent generalized
hierarchies of integrable equations in a condensed form of finite functional
equations. Generalized hierarchy incorporates basic hierarchy, modified
hierarchy, singularity manifold equation hierarchy and corresponding linear
problems. Different levels of generalized hierarchy are connected via
invariants of Combescure symmetry transformation. Resolution of functional
equations also leads to the -function and addition formulae to it.Comment: 43 pages, Late
Stability for an inverse problem for a two speed hyperbolic pde in one space dimension
We prove stability for a coefficient determination problem for a two velocity
2x2 system of hyperbolic PDEs in one space dimension.Comment: Revised Version. Give more detail and correct the proof of
Proposition 4 regarding the existence and regularity of the forward problem.
No changes to the proof of the stability of the inverse problem. To appear in
Inverse Problem
Recursion operator for stationary Nizhnik--Veselov--Novikov equation
We present a new general construction of recursion operator from zero
curvature representation. Using it, we find a recursion operator for the
stationary Nizhnik--Veselov--Novikov equation and present a few low order
symmetries generated with the help of this operator.Comment: 6 pages, LaTeX 2
Propagation of transient electromagnetic waves in time-varying media-direct and inverse scattering problems
Relation between hyperbolic Nizhnik-Novikov-Veselov equation and stationary Davey-Stewartson II equation
A Lax system in three variables is presented, two equations of which form the
Lax pair of the stationary Davey-Stewartson II equation. With certain nonlinear
constraints, the full integrability condition of this Lax system contains the
hyperbolic Nizhnik-Novikov-Veselov equation and its standard Lax pair. The
Darboux transformation for the Davey-Stewartson II equation is used to solve
the hyperbolic Nizhnik-Novikov-Veselov equation. Using Darboux transformation,
global -soliton solutions are obtained. It is proved that each -soliton
solution approaches zero uniformly and exponentially at spatial infinity and is
asymptotic to lumps of peaks at temporal infinity.Comment: 25 pages, 5 figure
Gauge-invariant description of some (2+1)-dimensional integrable nonlinear evolution equations
New manifestly gauge-invariant forms of two-dimensional generalized
dispersive long-wave and Nizhnik-Veselov-Novikov systems of integrable
nonlinear equations are presented. It is shown how in different gauges from
such forms famous two-dimensional generalization of dispersive long-wave system
of equations, Nizhnik-Veselov-Novikov and modified Nizhnik-Veselov-Novikov
equations and other known and new integrable nonlinear equations arise.
Miura-type transformations between nonlinear equations in different gauges are
considered.Comment: 13 pages, LaTeX, no figure
Gauge-invariant description of several (2+1)-dimensional integrable nonlinear evolution equations
We obtain new gauge-invariant forms of two-dimensional integrable systems of
nonlinear equations: the Sawada-Kotera and Kaup-Kuperschmidt system, the
generalized system of dispersive long waves, and the Nizhnik-Veselov-Novikov
system. We show how these forms imply both new and well-known two-dimensional
integrable nonlinear equations: the Sawada-Kotera equation, Kaup-Kuperschmidt
equation, dispersive long-wave system, Nizhnik-Veselov-Novikov equation, and
modified Nizhnik-Veselov-Novikov equation. We consider Miura-type
transformations between nonlinear equations in different gauges.Comment: Talk given at the Workshop "Nonlinear Physics: Theory and Experiment.
V", Gallipoli (Lecce, Italy), 12-21 June, 200
The Cauchy problem for the (2+1) integrable nonlinear Schr\"odinger equation
We study the Cauchy problem for the (2+1) integrable nonlinear Schr\"odinger
equation by the inverse scattering transform (IST) method. This Cauchy problem
with given initial data and boundary data at infinity is reduced by IST to the
Cauchy problem for the linear Schr\"odinger equation, in which the potential is
expressed in terms of boundary data. The results on direct and inverse
scattering problems for a two-dimensional Dirac system with special potentials
are used and refined. The Cauchy problem admits an explicit solution if the IST
of the solution is an integral operator of rank 1. We give one such solution