On the classification of scalar evolutionary integrable equations in 2+12+1 dimensions

We consider evolutionary equations of the form $u_t=F(u, w)$ where $w=D_x^{-1}D_yu$ is the nonlocality, and the right hand side $F$ is polynomial in the derivatives of $u$ and $w$. 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 $w$. 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 $\tau$-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

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 $n$-soliton solutions are obtained. It is proved that each $n$-soliton solution approaches zero uniformly and exponentially at spatial infinity and is asymptotic to $n^2$ 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