arXiv subjects

E. V. Ferapontov

Publications and source records attributed to E. V. Ferapontov.

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 $\int f(u_{xy}, u_{xt}, u_{yt})\, dxdydt$.

nlin.SI↗

arXiv2024

Higher-order reductions of the Mikhalev system

We consider the 3D Mikhalev system, $$ u_t=w_x, \quad u_y= w_t-u w_x+w u_x, $$ which has first appeared in the context of KdV-type hierarchies. Under the reduction $w=f(u)$, one obtains a pair of commuting first-order equations, $$ u_t=f'u_x, \quad u_y=(f'^2-uf'+f)u_x, $$ which govern simple wave solutions of the Mikhalev system. In this paper we study {\it higher-order} reductions of the form $$ w=f(u)+εa(u)u_x+ε^2[b_1(u)u_{xx}+b_2(u)u_x^2]+..., $$ which turn the Mikhalev system into a pair of commuting higher-order equations. Here the terms at $ε^n$ are assumed to be differential polynomials of degree $n$ in the $x$-derivatives of $u$. We will view $w$ as an (infinite) formal series in the deformation parameter $ε$. It turns out that for such a reduction to be non-trivial, the function $f(u)$ must be quadratic, $f(u)=λu^2$, furthermore, the value of the parameter $λ$ (which has a natural interpretation as an eigenvalue of a certain second-order operator acting on an infinite jet space), is quantised. There are only two positive allowed eigenvalues, $λ=1$ and $λ=3/2$, as well as infinitely many negative rational eigenvalues. Two-component reductions of the Mikhalev system are also discussed. We emphasise that the existence of higher-order reductions of this kind is a reflection of {\it linear degeneracy} of the Mikhalev system, in particular, such reductions do not exist for most of the known 3D dispersionless integrable systems such as the dispersionless KP and Toda equations.

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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.

nlin.SI↗

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).

nlin.SI↗

arXiv2021

On a class of integrable Hamiltonian equations in 2+1 dimensions

We classify integrable Hamiltonian equations in 3D with the Hamiltonian operator d/dx, where the Hamiltonian density h(u, w) is a function of two variables: dependent variable u and the non-locality w such that w_x=u_y. Based on the method of hydrodynamic reductions, the integrability conditions are derived (in the form of an involutive PDE system for the Hamiltonian density h). We show that the generic integrable density is expressed in terms of the Weierstrass elliptic functions. Dispersionless Lax pairs, commuting flows and dispersive deformations of the resulting equations are also discussed.

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arXiv2020

On a class of 2D integrable lattice equations

We develop a new approach to the classification of integrable equations of the form $$ u_{xy}=f(u, u_x, u_y, \triangle_z u \triangle_{\bar z}u, \triangle_{z\bar z}u), $$ where $\triangle_{ z}$ and $\triangle_{\bar z}$ are the forward/backward discrete derivatives. The following 2-step classification procedure is proposed: (1) First we require that the dispersionless limit of the equation is integrable, that is, its characteristic variety defines a conformal structure which is Einstein-Weyl on every solution. (2) Secondly, to the candidate equations selected at the previous step we apply the test of Darboux integrability of reductions obtained by imposing suitable cut-off conditions.

nlin.SI↗

arXiv2020

Quasilinear systems of Jordan block type and the mKP hierarchy

We demonstrate that commuting quasilinear systems of Jordan block type are parametrised by solutions of the modified KP hierarchy. Systems of this form naturally occur as hydrodynamic reductions of multi-dimensional linearly degenerate dispersionless integrable PDEs.

nlin.SI↗

arXiv2018

Dispersionless integrable hierarchies and GL(2,R) geometry

Paraconformal or $GL(2)$ geometry on an $n$-dimensional manifold $M$ is defined by a field of rational normal curves of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P} T^*M$. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov-Wilczynski) invariants. In this paper we discuss yet another natural source of $GL(2)$ structures, namely dispersionless integrable hierarchies of PDEs (for instance the dKP hierarchy). In the latter context, $GL(2)$ structures coincide with the characteristic variety (principal symbol) of the hierarchy. Dispersionless hierarchies provide explicit examples of various particularly interesting classes of $GL(2)$ structures studied in the literature. Thus, we obtain torsion-free $GL(2)$ structures of Bryant that appeared in the context of exotic holonomy in dimension four, as well as totally geodesic $GL(2)$ structures of Krynski. The latter, also known as involutive $GL(2)$ structures, possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesic $α$-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein-Weyl geometry. Our main result states that involutive $GL(2)$ structures are governed by a dispersionless integrable system. This establishes integrability of the system of Wünschmann conditions.

nlin.SI↗

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↗

arXiv2018

Integrability of dispersionless Hirota type equations in 4D and the symplectic Monge-Ampere property

We prove that integrability of a dispersionless Hirota type equation implies the symplectic Monge-Ampere property in any dimension $\geq 4$. In 4D this yields a complete classification of integrable dispersionless PDEs of Hirota type through a list of heavenly type equations arising in self-dual gravity. As a by-product of our approach we derive an involutive system of relations characterising symplectic Monge-Ampere equations in any dimension. Moreover, we demonstrate that in 4D the requirement of integrability is equivalent to self-duality of the conformal structure defined by the characteristic variety of the equation on every solution, which is in turn equivalent to the existence of a dispersionless Lax pair. We also give a criterion of linerisability of a Hirota type equation via flatness of the corresponding conformal structure, and study symmetry properties of integrable equations.

nlin.SI↗

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 $ϕ$.

nlin.SI↗

arXiv2016

Quasilinear systems with linearizable characteristic webs

We classify quasilinear systems in Riemann invariants whose characteristic webs are linearizable on every solution. Although the linearizability of an individual web is a rather nontrivial differential constraint, the requirement of linearizability of characteristic webs on all solutions imposes simple second-order constraints for the characteristic speeds of the system. It is demonstrated that every such system with n>3 components can be transformed by a reciprocal transformation to n uncoupled Hopf equations. All our considerations are local.

math.DG↗

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↗

arXiv2013

Characteristic integrals in 3D and linear degeneracy

Conservation laws vanishing along characteristic directions of a given system of PDEs are known as characteristic conservation laws, or characteristic integrals. In 2D, they play an important role in the theory of Darboux-integrable equations. In this paper we discuss characteristic integrals in 3D and demonstrate that, for a class of second-order linearly degenerate dispersionless integrable PDEs, the corresponding characteristic integrals are parametrised by points on the Veronese variety.

nlin.SI↗

arXiv2013

On the classification of discrete Hirota-type equations in 3D

In the series of recent publications we have proposed a novel approach to the classification of integrable differential/difference equations in 3D based on the requirement that hydrodynamic reductions of the corresponding dispersionless limits are `inherited' by the dispersive equations. In this paper we extend this to the fully discrete case. Our only constraint is that the initial ansatz possesses a non-degenerate dispersionless limit (this is the case for all known Hirota-type equations). Based on the method of deformations of hydrodynamic reductions, we classify discrete 3D integrable Hirota-type equations within various particularly interesting subclasses. Our method can be viewed as an alternative to the conventional multi-dimensional consistency approach.

nlin.SI↗

arXiv2013

Hamiltonian operators of Dubrovin-Novikov type in 2D

First order Hamiltonian operators of differential-geometric type were introduced by Dubrovin and Novikov in 1983, and thoroughly investigated by Mokhov. In 2D, they are generated by a pair of compatible flat metrics $g$ and $\tilde g$ which satisfy a set of additional constraints coming from the skew-symmetry condition and the Jacobi identity. We demonstrate that these constraints are equivalent to the requirement that $\tilde g$ is a linear Killing tensor of $g$ with zero Nijenhuis torsion. This allowed us to obtain a complete classification of $n$-component operators with $n\leq 4$ (for $n=1, 2$ this was done before). For 2D operators the Darboux theorem does not hold: the operator may not be reducible to constant coefficient form. All interesting (non-constant) examples correspond to the case when the flat pencil $g, \tilde g$ is not semisimple, that is, the affinor $\tilde g g^{-1}$ has non-trivial Jordan block structure. In the case of a direct sum of Jordan blocks with distinct eigenvalues we obtain a complete classification of Hamiltonian operators for any number of components $n$, revealing a remarkable correspondence with the class of trivial Frobenius manifolds modelled on $H^*(CP^{n-1})$.

math.DG↗

arXiv2013

Towards the classification of integrable differential-difference equations in 2 + 1 dimensions

We address the problem of classification of integrable differential-difference equations in 2+1 dimensions with one/two discrete variables. Our approach is based on the method of hydrodynamic reductions and its generalisation to dispersive equations. We obtain a number of classification results of scalar integrable equations including that of the intermediate long wave and Toda type.

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