arXiv subjects

M. V. Pavlov

Publications and source records attributed to M. V. Pavlov.

At least 19 recordsLinked to original sources

arXiv2026

On a class of 3D second-order integrable Lagrangians and their dispersive deformations

We investigate integrability of Euler-Lagrange equations associated with 3D second-order Lagrangians of the form \begin{equation*} \int f(u_{xy},u_{xt},u_{yt})\ \text{d}x\text{d}y\text{d}t. \end{equation*} It is demonstrated that there are exactly four different types of such Lagrangian densities $f$: the first one is given by the formula $f=\sqrt{u_{xy}u_{xt}u_{yt}}$, the second and the third are more complicated (although still representable in elementary functions), whereas the most generic fourth one is expressible in terms of the Lobachevsky function, revealing unexpected links to spherical/hyperbolic trigonometry and Schläfly-type formulas. Dispersionless Lax pairs and integrable dispersive deformations of the corresponding Euler-Lagrange equations are also constructed.Remarkably, dispersive deformation of the first Lagrangian density coincides with the Lagrangian of the classical Darboux system arising in the theory of triply-orthogonal coordinate systems in $\mathbb{R}^3$. Dispersive deformations of the three other cases provide Lagrangian formulation of semi-discrete and fully discrete versions of the Darboux system, with one, two and three discrete variables, respectively.

nlin.SI↗

arXiv2026

Lagrangian formulation of the Darboux system

The classical Darboux system governing rotation coefficients of three-dimensional metrics of diagonal curvature possesses an equivalent formulation as a sixth-order PDE for a scalar potential (related to the corresponding $τ$-function). We demonstrate that this PDE is Lagrangian and can be viewed as an explicit scalar form of the `generating PDE of the KP hierarchy' as discussed recently in Nijhoff [arXiv:2406.13423] in the Lagrangian multiform approach to the Darboux and KP hierarchies. Scalar Lagrangian formulations for differential-difference and fully discrete versions of the Darboux system are also constructed. In the first three cases (continuous and differential-difference with one and two discrete variables), the corresponding Lagrangians are expressible via elementary functions (logarithms), whereas the fully discrete case requires special functions (dilogarithms). Remarkably, dispersionless limits of the above Lagrangians provide a complete list of 3D second-order integrable Lagrangians of the form $\int f(u_{xy}, u_{xt}, u_{yt})\, dxdydt$.

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arXiv2021

Kinetic equation for soliton gas: integrable reductions

Macroscopic dynamics of soliton gases can be analytically described by the thermodynamic limit of the Whitham equations, yielding an integro-differential kinetic equation for the density of states. Under a delta-functional ansatz, the kinetic equation for soliton gas reduces to a non-diagonalisable system of hydrodynamic type whose matrix consists of several 2x2 Jordan blocks. Here we demonstrate the integrability of this system by showing that it possesses a hierarchy of commuting hydrodynamic flows and can be solved by an extension of the generalised hodograph method. Our approach is a generalisation of Tsarev's theory of diagonalisable systems of hydrodynamic type to quasilinear systems with non-trivial Jordan block structure.

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arXiv2021

Integrable systems of the intermediate long wave type in 2+1 dimensions

We classify 2+1 dimensional integrable systems with nonlocality of the intermediate long wave type. Links to the 2+1 dimensional waterbag system are established. Dimensional reductions of integrable systems constructed in this paper provide dispersive regularisations of hydrodynamic equations governing propagation of long nonlinear waves in a shear flow with piecewise linear velocity profile (for special values of vorticities).

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arXiv2020

Recursion operators and the hierarchies of MKdV equations related to $D_4^{(1)}$, $D_4^{(2)}$ and $D_4^{(3)}$ Kac-Moody algebras

We constructed the three nonequivalent gradings in the algebra $D_4 \simeq so(8)$. The first one is the standard one obtained with the Coxeter automorphism $C_1=S_{α_2} S_{α_1}S_{α_3}S_{α_4}$ using its dihedral realization. In the second one we use $C_2 = C_1R$ where $R$ is the mirror automorphism. The third one is $C_3 = S_{α_2}S_{α_1}T$ where $T$ is the external automorphism of order 3. For each of these gradings we constructed the basis in the corresponding linear subspaces $\mathfrak{g}^{(k)}$, the orbits of the Coxeter automorphisms and the related Lax pairs generating the corresponding mKdV hierarchies. We found compact expressions for each of the hierarchies in terms of the recursion operators. At the end we wrote explicitly the first nontrivial mKdV equations and their Hamiltonians. For $D_4^{(1)}$ these are in fact two mKdV systems, due to the fact that in this case the exponent $3$ has multiplicity 2. Each of these mKdV systems consist of 4 equations of third order with respect to $\partial_x$. For $D_4^{(2)}$ this is a system of three equations of third order with respect to $\partial_x$. Finally, for $D_4^{(3)}$ this is a system of two equations of fifth order with respect to $\partial_x$.

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arXiv2018

Bi-Hamiltonian structure of the Oriented Associativity Equation

The Oriented Associativity equation plays a fundamental role in the theory of Integrable Systems. In this paper we prove that the equation, besides being Hamiltonian with respect to a first-order Hamiltonian operator, has a third-order non-local homogeneous Hamiltonian operator belonging to a class which has been recently studied, thus providing a highly non-trivial example in that class and showing intriguing connections with algebraic geometry.

math-ph↗

arXiv2018

Six-dimensional heavenly equation. Dressing scheme and the hierarchy

We consider six-dimensional heavenly equation as a reduction in the framework of general six-dimensional linearly degenerate dispersionless hierarchy. We characterise the reduction in terms of wave functions, introduce generating relation, Lax-Sato equations and develop the dressing scheme for the reduced hierarchy. Using the dressing scheme, we construct a class of solutions for six-dimensional heavenly equation in terms of implicit functions.

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arXiv2018

On a class of third-order nonlocal Hamiltonian operators

Based on the theory of Poisson vertex algebras we calculate skew-symmetry conditions and Jacobi identities for a class of third-order nonlocal operators of differential-geometric type. Hamiltonian operators within this class are defined by a Monge metric and a skew-symmetric two-form satisfying a number of differential-geometric constraints. Complete classification results in the 2-component and 3-component cases are obtained.

math-ph↗

arXiv2017

Systems of conservation laws with third-order Hamiltonian structures

We investigate $n$-component systems of conservation laws that possess third-order Hamiltonian structures of differential-geometric type. The classification of such systems is reduced to the projective classification of linear congruences of lines in $\mathbb{P}^{n+2}$ satisfying additional geometric constraints. Algebraically, the problem can be reformulated as follows: for a vector space $W$ of dimension $n+2$, classify $n$-tuples of skew-symmetric 2-forms $A^α \in Λ^2(W)$ such that \[ ϕ_{βγ}A^β\wedge A^γ=0, \] for some non-degenerate symmetric $ϕ$.

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arXiv2016

Nonlinear wave interaction problems in three dimensional case

Three dimensional nonlinear wave interactions have been analytically described. The procedure under interest can be applied to three dimensional quasilinear systems of first order, whose hydrodynamic reductions are homogeneous semi-Hamiltonian hydrodynamic type systems (i.e. possess a diagonal form and infinitely many conservation laws). The interaction of N waves was studied. In particular we prove that they behave like simple waves and they distort after the collision region. The amount of the distortion can be analytically computed.

nlin.SI↗

arXiv2016

Remarks on the Lagrangian representation of bi-Hamiltonian equations

The Lagrangian representation of multi-Hamiltonian PDEs has been introduced by Y. Nutku and one of us (MVP). In this paper we focus on systems which are (at least) bi-Hamiltonian by a pair $A_1$, $A_2$, where $A_1$ is a hydrodynamic-type Hamiltonian operator. We prove that finding the Lagrangian representation is equivalent to finding a generalized vector field $τ$ such that $A_2=L_τA_1$. We use this result in order to find the Lagrangian representation when $A_2$ is a homogeneous third-order Hamiltonian operator, although the method that we use can be applied to any other homogeneous Hamiltonian operator. As an example we provide the Lagrangian representation of a WDVV hydrodynamic-type system in $3$ components.

math-ph↗

arXiv2016

Linearly degenerate hierarchies of quasiclassical SDYM type

We demonstrate that SDYM equations for the Lie algebra of one-dimensional vector fields represent a natural reduction in the framework of general linearly degenerate dispersionless hierarchy. We define the reduction in terms of wave functions, introduce generating relation, Lax-Sato equations and the dressing scheme for the reduced hierarchy. Multidimensional case is also discussed.

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arXiv2016

On exact solutions of nonlinear acoustic equations

Solutions of nonlinear acoustic equations describing propagation of strong sound pulses with account of curvature of wave fronts in multi-dimensional geometry are obtained from simple physical considerations. The form of these solutions suggests ansatz suitable for finding solutions of much more general equations of Khokhlov-Zabolotskaya type. General method is illustrated by an example of nonlinear sound pulse focused in one transverse direction and defocused in the other direction.

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arXiv2015

Towards the classification of homogeneous third-order Hamiltonian operators

Let $V$ be a vector space of dimension $n+1$. We demonstrate that $n$-component third-order Hamiltonian operators of differential-geometric type are parametrised by the algebraic variety of elements of rank $n$ in $S^2(Λ^2V)$ that lie in the kernel of the natural map $S^2(Λ^2V)\to Λ^4V$. Non-equivalent operators correspond to different orbits of the natural action of $SL(n+1)$. Based on this result, we obtain a classification of such operators for $n\leq 4$.

math-ph↗

arXiv2015

On the bi-Hamiltonian Geometry of WDVV Equations

We consider the WDVV associativity equations in the four dimensional case. These nonlinear equations of third order can be written as a pair of six component commuting two-dimensional non-diagonalizable hydrodynamic type systems. We prove that these systems possess a compatible pair of local homogeneous Hamiltonian structures of Dubrovin--Novikov type (of first and third order, respectively).

math-ph↗

arXiv2014

Linearly Degenerate Hamiltonian PDEs and a New Class of Solutions to the WDVV Associativity Equations

We define a new class of solutions to the WDVV associativity equations. This class is determined by the property that one of the commuting PDEs associated with such a WDVV solution is linearly degenerate. We reduce the problem of classifying such solutions of the WDVV equations to the particular case of the so-called algebraic Riccati equation and, in this way, arrive at a complete classification of irreducible solutions.

nlin.SI↗

arXiv2012

Waves in the Skyrme--Faddeev model and integrable reductions

In the present article we show that the Skyrme--Faddeev model possesses nonlinear wave solutions, which can be expressed in terms of elliptic functions. The Whitham averaging method has been exploited in order to describe slow deformation of periodic wave states, leading to a quasi-linear system. The reduction to general hydrodynamic systems have been considered and it is compared with other integrable reductions of the system.

nlin.SI↗

arXiv2010

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 (quasi-periodic, self-similar). Hydrodynamic symmetries are then derived and investigated. The obtained results shed the 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.

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