arXiv subjects
Publications and source records attributed to Maxim V. Pavlov.
General and particular solutions of the so called semi-Hamiltonian hydrodynamic type systems can be obtained by the Tsarev Generalized Hodograph Method. Here we show that a natural extension of this approach applied to dispersive integrable systems is determined by isomonodromic deformations.
In this paper we consider a new class of Hamiltonian hydrodynamic type systems, whose conservation laws are polynomial with respect to one of field variables.
The aim of this article is to classify pairs of first-order Hamiltonian operators of Dubrovin-Novikov type such that one of them has a non-local part defined by an isometry of its leading coefficient. An example of such bi-Hamiltonian pair was recently found for the constant astigmatism equation. We obtain a classification in the case of 2 dependent variables, and a significant new example that is an extension of a hydrodynamic type system obtained from a particular solution of the WDVV equations.
We investigate integrability of Euler-Lagrange equations associated with 2D second-order Lagrangians of the form \begin{equation*} \int f(u_{xx},u_{xy},u_{yy})\ dxdy. \end{equation*} By deriving integrability conditions for the Lagrangian density $f$, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.
In this paper we construct multi-phase solutions for integrable dispersive chains associated with the three-dimensional linearly degenerate Mikhalev system of first order. These solutions are parameterized by infinitely many arbitrary parameters. As byproduct we describe multi-phase solutions for finite component dispersive reductions of these integrable dispersive chains.
In this paper we consider non-diagonalisable hydrodynamic type systems integrable by the Extended Hodograph Method. We restrict our consideration to non-diagonalisable hydrodynamic reductions of the Mikhalev equation. We show that families of these hydrodynamic type systems are reducible to the Heat hierarchy. Then we construct new particular explicit solutions for the Mikhalev equation.
We construct a point transformation between two integrable systems, the multi-component Harry Dym equation and the multi-component extended Harry Dym equation, that does not preserve the class of multi-phase solutions. As a consequence we obtain a new type of wave-like solutions, generalising the~multi-phase solutions of the multi-component extended Harry Dym equation. Our construction is easily transferable to other integrable systems with analogous properties.
In this paper we introduce a new property of two-dimensional integrable systems -- existence of infinitely many local three-dimensional conservation laws for pairs of integrable two-dimensional commuting flows. Infinitely many three-dimensional local conservation laws for the Korteweg de Vries pair of commuting flows and for the Benney commuting hydrodynamic chains are constructed. As a by-product we established a new method for computation of local conservation laws for three-dimensional integrable systems. The Mikhalev equation and the dispersionless limit of the Kadomtsev--Petviashvili equation are investigated. All known local and infinitely many new quasi-local three-dimensional conservation laws are presented. Also four-dimensional conservation laws are considered for couples of three-dimensional integrable quasilinear systems and for triples of corresponding hydrodynamic chains.
Starting from a homogeneous polynomial in momenta of arbitrary order we extract multi-component hydrodynamic-type systems which describe 2-dimensional geodesic flows admitting the initial polynomial as integral. All these hydrodynamic-type systems are semi-Hamiltonian, thus implying that they are integrable according to the generalized hodograph method. Moreover, they are integrable in a constructive sense as polynomial first integrals allow to construct generating equations of conservation laws. According to the multiplicity of the roots of the polynomial integral, we separate integrable particular cases.
We show that four Lax-integrable 3D differential equations are related via Bäcklund transformations.
In this paper we show that integrable four dimensional linearly degenerate equations of second order possess infinitely many three dimensional hydrodynamic reductions. Furthermore, they are equipped infinitely many conservation laws and higher commuting flows. We show that the dispersionless limits of nonlocal KdV and nonlocal NLS equations (the so-called Breaking Soliton equations introduced by O.I. Bogoyavlenski) are one and two component reductions (respectively) of one of these four dimensional linearly degenerate equations.
In this paper we construct multiparametric families of two dimensional metrics with polynomial first integral. Such integrable geodesic flows are described by solutions of some semi-Hamiltonian hydrodynamic type system. We find infinitely many conservation laws and commuting flows for this system. This procedure allows us to present infinitely many particular metrics by the generalized hodograph method.
B.A. Dubrovin proved that remarkable WDVV associativity equations are integrable systems. In a simplest nontrivial three-component case these equations can be written as a nondiagonalizable hydrodynamic type system equivalent to a symmetric reduction of the three wave interaction and to the matrix Hopf equation. Then E.V. Ferapontov and O.I. Mokhov found a local Hamiltonian structure. Finally E.V. Ferapontov, C.A.P. Galvão, O.I. Mokhov, Ya. Nutku found a second local Hamiltonian structure. Both local Hamiltonian structure are homogeneous of first and third order (respectively) of Dubrovin--Novikov type. In our paper we suggest a special scaling procedure for independent variables applicable for homogeneous nonlinear PDE's, which allows to incorporate an auxiliary parameter $ε$, such that a corresponding \textquotedblleft intermediate\textquotedblright\ system possesses two remarkable limits: a high-frequency limit ($ε\rightarrow \infty $) back to the original system and a dispersionless limit ($ε\rightarrow 0$) which yields diagonalizable integrable hydrodynamic type system. This means that our procedure allows to transform a homogeneous third order local Hamiltonian structure to non-homogeneous of third order. Thus we create an integrable hierarchy equipped by a pair of local Hamiltonian structures, which (both of them) possess a dispersionless limit. Also we show that this bi-Hamiltonian diagonalizable hydrodynamic type system possesses at least two different dispersive integrable extensions (in a framework of B.A. Dubrovin's approach)
The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to some known integrable systems.
In this paper the two-dimensional Benney system describing long wave propagation of a finite depth fluid motion and the multi-dimensional Russo--Smereka kinetic equation describing a bubbly flow are considered. The Hamiltonian approach established by J. Gibbons for one-dimensional Vlasov kinetic equation is extended to a multi-dimensional case. A local Hamiltonian structure associated with the hydrodynamic lattice of moments derived by D.J. Benney is constructed. A relationship between this hydrodynamic lattice of moments and the two-dimensional Vlasov kinetic equation is found. In the two-dimensional case a Hamiltonian hydrodynamic lattice for the Russo--Smereka kinetic model is constructed. Simple hydrodynamic reductions are presented.
In this paper we consider integrable dispersive chains associated with the so called Energy Dependent Schrodinger operator. In a general case multi component reductions of these dispersive chains are new integrable systems, which are characterised by two arbitrary natural numbers. Also we show that integrable three dimensional linearly degenerate quasilinear equations of a second order possess infinitely many differential constraints. Corresponding dispersive reductions are integrable systems associated with the Energy Dependent Schrodinger operator.
The one-dimension Russo--Smereka kinetic equation describing the propagation of nonlinear concentration waves in a rarefied bubbly fluid is considered. Reductions of the model to finite component systems are derived. Stability of the bubbly flow in terms of hyperbolicity of the kinetic equation is studied. Conservation form of the model is proposed and numerical solution of the Cauchy problem with discontinuous initial data is obtained.
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 for two and three components is obtained.