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Maxim V. Pavlov

Publications and source records attributed to Maxim V. Pavlov.

At least 37 records · Page 2Linked to original sources
arXiv2013

Classical Mechanical Systems with one-and-a-half Degrees of Freedom and Vlasov Kinetic Equation

We consider non-stationary dynamical systems with one-and-a-half degrees of freedom. We are interested in algorithmic construction of rich classes of Hamilton's equations with the Hamiltonian H=p^2/2+V(x,t) which are Liouville integrable. For this purpose we use the method of hydrodynamic reductions of the corresponding one-dimensional Vlasov kinetic equation. Also we present several examples of such systems with first integrals with non-polynomial dependencies w.r.t. to momentum. The constructed in this paper classes of potential functions {$V(x,t)$} which give integrable systems with one-and-a-half degrees of freedom are parameterized by arbitrary number of constants.

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arXiv2013

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. Also, we apply the recursion operator and present a second Hamiltonian structure. This bi-Hamiltonian structure allows to replicate infinitely many local commuting flows and corresponding local conservation law densities.

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arXiv2011

Generalised hydrodynamic reductions of the kinetic equation for soliton gas

We derive generalised multi-flow hydrodynamic reductions of the nonlocal kinetic equation for a soliton gas and investigate their structure. These reductions not only provide further insight into the properties of the new kinetic equation but also could prove to be representatives of a novel class of integrable systems of hydrodynamic type, beyond the conventional semi-Hamiltonian framework.

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arXiv2010

Integrable hydrodynamic chains associated with Dorfman Poisson brackets

This paper is devoted to a description of integrable Hamiltonian hydrodynamic chains associated with Dorfman Poisson brackets. Three main classes of these hydrodynamic chains are selected. Generating functions of conservation laws and commuting flows are found. Hierarchies of these Hamiltonian hydrodynamic chains are extended on negative moments and negative time variables. Corresponding three dimensional quasilinear equations of the second order are presented.

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arXiv2010

On a Nonlocal Ostrovsky-Whitham Type Dynamical System, Its Riemann Type Inhomogeneous Regularizations and Their Integrability

Short-wave perturbations in a relaxing medium, governed by a special reduction of the Ostrovsky evolution equation, and later derived by Whitham, are studied using the gradient-holonomic integrability algorithm. The bi-Hamiltonicity and complete integrability of the corresponding dynamical system is stated and an infinite hierarchy of commuting to each other conservation laws of dispersive type are found. The well defined regularization of the model is constructed and its Lax type integrability is discussed. A generalized hydrodynamical Riemann type system is considered, infinite hierarchies of conservation laws, related compatible Poisson structures and a Lax type representation for the special case N=3 are constructed.

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arXiv2009

Classification of conservative hydrodynamic chains. Vlasov type kinetic equation, Riemann mapping and the method of symmetric hydrodynamic reductions

A complete classification of integrable conservative hydrodynamic chains is presented. These hydrodynamic chains are written via special coordinates -- moments, such that right hand sides of these infinite component systems depend linearly on a discrete independent variable $k$. All variable coefficients of these hydrodynamic chains can be expressed via modular forms with respect to moment $A^{0}$, via hypergeometric functions with respect to moment $A^{1}$; they depend polynomially on moment $A^{2}$ and linearly on all other higher moments $A^{k}$. A dispersionless Lax representation is found. Corresponding collisionless Boltzmann (Vlasov like kinetic) equation is derived. A Riemann mapping is constructed. A generating function of conservation laws and commuting flows is presented.

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arXiv2009

Integrability of the Manakov--Santini hierarchy

The first example of the so-called "coupled" integrable hydrodynamic chain is presented. Infinitely many commuting flows are derived. Compatibility conditions of the first two of them lead to the remarkable Manakov--Santini system. Integrability of this four component three dimensional quasilinear system of the first order as well as the coupled hydrodynamic chain is proved by the method of hydrodynamic reductions. In comparision with a general case considered by E.V. Ferapontov and K.R. Khusnutdinova, in this degenerate case N component hydrodynamic reductions are parameterized by N+M arbitrary functions of a single variable, where M is a number of branch points of corresponding Riemann surface. These hydrodynamic reductions also are written as symmetric hydrodynamic type systems. New classes of particular solutions are found.

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arXiv2006

New Hamiltonian formalism and Lagrangian representations for integrable hydrodynamic type systems

New Hamiltonian formalism based on the theory of conjugate curvilinear coordinate nets is established. All formulas are ``mirrored'' to corresponding formulas in the Hamiltonian formalism constructed by B.A. Dubrovin and S.P. Novikov (in a flat case) and E.V. Ferapontov (in a non-flat case). In the ``mirrored-flat'' case the Lagrangian formulation is found. Multi-Hamiltonian examples are presented. In particular Egorov's case, generalizations of local Nutku--Olver's Hamiltonian structure and corresponding Sheftel--Teshukov's recursion operator are presented. An number of Hamiltonian structures of all odd orders is found.

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arXiv2006

The Kupershmidt hydrodynamic chains and lattices

This paper is devoted to the very important class of hydrodynamic chains first derived by B. Kupershmidt and later re-discovered by M. Blaszak. An infinite set of local Hamiltonian structures, hydrodynamic reductions parameterized by the hypergeometric function and reciprocal transformations for the Kupershmidt hydrodynamic chains are described.

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