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V. G. Marikhin

Publications and source records attributed to V. G. Marikhin.

6 recordsLinked to original sources
arXiv2014

On three-dimensional quasi-Stäckel Hamiltonians

A three-dimensional integrable generalization of the Stäckel systems is proposed. The classification of such systems is obtained which results in two families. The first one is the direct sum of the two-dimensional case which is equivalent to the representation of the Schottky--Manakov top in the quasi-Stäckel form and a trivial one-dimensional system. The second family is probably a new three-dimensional system. The system of hydrodynamic type, which we get from this family in a usual way, is a 3-dimensional generalization of the Gibbons--Tsarev system. A generalization of the quasi-Stäckel systems to the case of any dimension is discussed.

nlin.SI↗
arXiv2008

Pairs of commuting Hamiltonians quadratic in momenta

In the case of two degree system the pairs of quadratic in momenta Hamiltonians commuting according the standard Poisson bracket are considered. The new many-parametrical families of such pairs are founded. The universal method of constructing the full solution of Hamilton - Jacobi equation in terms of integrals on some algebraic curve is proposed. For some examples this curve is non-hyperelliptic covering over the elliptic curve.

nlin.SI↗
arXiv2004

Separation of variables on a non-hyperelliptic curve

In the paper we consider several dynamical systems that admit a separation of variables on the algebraic curve of genus 4. The main result of the paper is an explicit formula for the action of these systems. We find it directly from the Hamilton-Jacobi equation. We find the action and a separation of variables for the Kowalewski hyrostat, the Clebsch and the so(4) Schottky-Manakov spinning tops.

nlin.SI↗
arXiv2002

Integrable systems with quadratic nonlinearity in Fourier space

The Lax pair representation in Fourier space is used to obtain a list of integrable scalar evolutionary equations with quadratic nonlinearity. The famous systems of this type such as KdV, intermediate long-wave equation (ILW), Camassa-Holm and A. Degasperis systems are represented in this list. Some new systems are obtained as well. The generalizations on two-dimensional and discrete systems are discussed.

nlin.SI↗
arXiv1999

Self-similar solutions of NLS-type dynamical systems

We study self-similar solutions of NLS-type dynamical systems. Lagrangian approach is used to show that they can be reduced to three canonical forms, which are related by Miura transformations. The fourth Painleve equation (PIV) is central in our consideration - it connects Heisenberg model, Volterra model and Toda model to each other. The connection between the rational solutions of PIV and Coulomb gas in a parabolic potential is established. We discuss also the possibility to obtain an exact solution for optical soliton i.e. of the NLS equation with time-dependent dispersion.

solv-int↗