Maxim V. Pavlov

Integrable shallow-water equations and undular bores

On the basis of the integrable Kaup-Boussinesq version of the shallow-water equations, an analytical theory of undular bores is constructed. A complete classification for the problem of the decay of an initial discontinuity is made.

The Kupershmidt hydrodynamic chains and lattices

This paper is devoted to the very important class of hydrodynamic chains (see [9], [23], [24]) first derived by B. Kupershmidt in [14], later re-discovered by M. Blaszak in [4] (see also [21]). An infinite set of local Hamiltonian structures, hydrodynamic reductions parameterized by the hypergeometric function and reciprocal transformations for the Kupershmidt hydrodynamic chains are described. In honour of Boris Kupershmidt

Extending Hamiltonian operators to get bi-Hamiltonian coupled KdV systems

Abstract An analysis of the extension of Hamiltonian operators from lower order to higher order of the matrix paves the way for constructing Hamiltonian pairs which may result in hereditary operators. Based on a specific choice of Hamiltonian operators of lower order, new local bi-Hamiltonian coupled KdV systems are proposed. As a consequence of the bi-Hamiltonian structure, they all possess infinitely many symmetries and infinitely many conserved densities.

Kinetic Equation for a Soliton Gas and Its Hydrodynamic Reductions

We introduce and study a new class of kinetic equations, which arise in the description of nonequilibrium macroscopic dynamics of soliton gases with elastic collisions between solitons. These equations represent nonlinear integro-differential systems and have a novel structure, which we investigate by studying in detail the class of N-component ‘cold-gas’ hydrodynamic reductions. We prove that these reductions represent integrable linearly degenerate hydrodynamic type systems for arbitrary N which is a strong evidence in favour of integrability of the full kinetic equation. We derive compact explicit representations for the Riemann invariants and characteristic velocities of the hydrodynamic reductions in terms of the ‘cold-gas’ component densities and construct a number of exact solutions having special properties (quasiperiodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed light on the structure of a continuum limit for a large class of integrable systems of hydrodynamic type and are also relevant to the description of turbulent motion in conservative compressible flows.

The Calogero equation and Liouville type equations

We present a two-component generalization of the C-integrable Calogero equation that is also C-integrable. We show that the Calogero equation and its two-component generalization are solvable by a reciprocal transformation to ODEs.

RELATIONSHIPS BETWEEN DIFFERENTIAL SUBSTITUTIONS AND HAMILTONIAN STRUCTURES OF THE KORTEWEG-DE VRIES EQUATION

Abstract A new regular method for (a) classification of integrable equations, (b) constructing an infinite set of differential substitutions, and (c) reducing every nonlocal Hamiltonian structure into canonical form is presented. An explanation of the origin of these differential substitutions is given via a relationship between the spectral problem and a Miura transformation. A relationship of these differential substitutions with a generating function of conservation law densities is found. Every Hamiltonian structure of the Korteweg-de Vries equation possesses a transformation to the canonical “ d dx ”- type by a combination of some differential substitutions and reciprocal transformations. Some well-known equations are embedded into a unified chain.

On generating functions in the AKNS hierarchy

It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leads to other hierarchies of integrable equations. In particular, it is shown that stimulated Raman scattering equations generate the hierarchy of flows which include the Heisenberg model equations. This observation reveals some new relationships between known integrable equations and permits one to construct their new physically important combinations. Reductions of the AKNS hierarchy to ones with complex conjugate and real dependent variables are also discussed and the corresponding generating functions of positive and negative flows are found. Generating function of Whitham modulation equations in the AKNS hierarchy is obtained.It is shown that the self-induced transparency equations can be interpreted as a generating function for as positive so negative flows in the AKNS hierarchy. Mutual commutativity of these flows leads to other hierarchies of integrable equations. In particular, it is shown that stimulated Raman scattering equations generate the hierarchy of flows which include the Heisenberg model equations. This observation reveals some new relationships between known integrable equations and permits one to construct their new physically important combinations. Reductions of the AKNS hierarchy to ones with complex conjugate and real dependent variables are also discussed and the corresponding generating functions of positive and negative flows are found. Generating function of Whitham modulation equations in the AKNS hierarchy is obtained.

The constant astigmatism equation. New exact solution

In this paper we present a new solution for the constant astigmatism equation. This solution is parameterized by an arbitrary function of a single variable.

Projective-geometric aspects of homogeneous third-order Hamiltonian operators

Abstract We investigate homogeneous third-order Hamiltonian operators of differential-geometric type. Based on the correspondence with quadratic line complexes, a complete list of such operators with n ≤ 3 components is obtained.

Lagrangian and Hamiltonian structures for the constant astigmatism equation

In this paper we found a Lagrangian representation and corresponding Hamiltonian structure for the constant astigmatism equation. Utilizing this Hamiltonian structure and extra conservation law densities we construct a first evolution commuting flow of the third order. We also apply the recursion operator and present a second Hamiltonian structure. This bi-Hamiltonian structure allows us to replicate infinitely many local commuting flows and corresponding local conservation law densities.