arXiv subjects

Evgeny Ferapontov

Publications and source records attributed to Evgeny Ferapontov.

3 recordsLinked to original sources

arXiv2025

Involutive scroll structures on solutions of 4D dispersionless integrable hierarchies

A rational normal scroll structure on an $(n+1)$-dimensional manifold $M$ is defined as a field of rational normal scrolls of degree $n-1$ in the projectivised cotangent bundle $\mathbb{P}T^*M$. We show that geometry of this kind naturally arises on solutions of various 4D dispersionless integrable hierarchies of heavenly type equations. In this context, rational normal scrolls coincide with the characteristic varieties (principal symbols) of the hierarchy. Furthermore, such structures automatically satisfy an additional property of involutivity. Our main result states that involutive scroll structures are themselves governed by a dispersionless integrable hierarchy, namely, the hierarchy of conformal self-duality equations.

nlin.SI↗

arXiv2023

Defining differential equations for modular forms and Jacobi forms

It is well known that every modular form~$f$ on a discrete subgroup $Γ\leqslant \textrm{SL}(2, \mathbb R)$ satisfies a third-order nonlinear ODE that expresses algebraic dependence of the functions~$f$, $f'$, $f''$ and~$f'''$. These ODEs are automatically invariant under the Lie group $\textrm{SL}(2, \mathbb R)$, which acts on the solution spaces thereof with an open orbit (and the discrete stabiliser~$Γ$ of a generic solution). Similarly, every modular form satisfies a fourth-order nonlinear ODE that is invariant under the Lie group $\textrm{GL}(2, \mathbb R)$ acting on its solution space with an open orbit. ODEs for modular forms can be compactly expressed in terms of the differential invariants of these actions. The invariant forms of both ODEs define plane algebraic curves naturally associated with every modular form; the corresponding ODEs can be seen as modular parametrisations of the associated curves. After reviewing examples of nonlinear ODEs satisfied by classical modular forms (such as Eisenstein series, modular forms on congruence subgroups of level two and three, theta constants, and some newforms of weight two), we generalise these results to Jacobi forms; these satisfy involutive third-order PDE systems that are invariant under the Lie group $\textrm{SL}(2, \mathbb R)\ltimes H$ where $H$ is the Heisenberg group.

nlin.SI↗

arXiv2022

Linearisable Abel equations and the Gurevich--Pitaevskii problem

Applying symmetry reduction to a class of $\mathrm{SL}(2,\mathbb R)$-invariant third-order ODEs, we obtain Abel equations whose general solution can be parametrised by hypergeometric functions. Particular case of this construction provides a general parametric solution to the Kudashev equation, an ODE arising in the Gurevich--Pitaevskii problem, thus giving the first term of a large-time asymptotic expansion of its solution in the oscillatory (Whitham) zone.

math.CA↗