Finite-dimensional Reductions Research Articles

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Comparison of finite-dimensional reductions in smooth variational problems with symmetries

Comparison of finite-dimensional reductions in smooth variational problems with symmetries

Finite-dimensional reductions of the discrete Toda chain

The problem of construction of integrable boundary conditions for the discrete Toda chain is considered. The restricted chains for properly chosen closure conditions are reduced to the well known discrete Painlev\'e equations $dP_{III}$, $dP_{V}$, $dP_{VI}$. Lax representations for these discrete Painlev\'e equations are found.

Finite-Dimensional Discrete Systems Integrated in Quadratures

We consider finite-dimensional reductions (truncations) of discrete systems of the type of the Toda chain with discrete time that retain the integrability. We show that for finite-dimensional chains, in addition to integrals of motion, we can construct a rich family of higher symmetries described by the master symmetry. We reduce the problem of integrating a finite-dimensional system to the implicit function theorem.

Finite-Dimensional Nonlocal Reductions of the Inverse Korteweg–de Vries Dynamical System

We study finite-dimensional Moser-type reductions for the inverse nonlinear Korteweg–de Vries dynamical system and the Liouville integrability of these reductions in quadratures.

Calogero-Françoise flows and periodic peakons

The completely integrable Hamiltonian systems discovered by Calogero and FranCoise contain the finite-dimensional reductions of the Camassa–Holm and Hunter–Saxton equations. We show that the associated spectral problem has the same form as that of the periodic discrete Camassa–Holm equation. The flow is linearized by the Abel map on a hyperelliptic curve. For two-particle systems, which correspond to genus-1 curves, explicit solutions are obtained in terms of the Weierstrass elliptic functions.

Boundary conditions for integrable discrete chains

A method of searching for boundary conditions consistent with the integrability property is suggested for discrete chains. Generating functions of conserved quantities of the corresponding finite-dimensional reductions are given in terms of the L, A pairs.

Finite-Dimensional Reductions of Conservative Dynamical Systems and Numerical Analysis. I

We study infinite-dimensional Liouville–Lax integrable nonlinear dynamical systems. For these systems, we consider the problem of finding an appropriate set of initial conditions leading to typical solutions such as solitons and traveling waves. We develop an approach to the solution of this problem based on the exact reduction of a given nonlinear dynamical system to its finite-dimensional invariant submanifolds and the subsequent investigation of the system of ordinary differential equations obtained by qualitative analysis. The efficiency of the approach proposed is demonstrated by the examples of the Korteweg–de Vries equation, the modified nonlinear Schrodinger equation, and a hydrodynamic model.

Hamiltonian Systems Derived from Constant Mean Curvature Surfaces in Hyperbolic Three-Space

We give a representation formula for surfaces of constant mean curvature in Euclidean or hyperbolic space, which is a natural generalization of Weierstrass-Enneper representation formula. The data (two functions) used in our formula should satisfy a certain system of differential equations. The system can be interpreted as an infinite dimensional Hamiltonian system. We investigate two finite-dimensional reductions in detail.

On some integrable discrete-time systems associated with the Bogoyavlensky lattices

A new class of integrable lattice systems is introduced which are the time-discretisations of the Bogoyavlensky systems. Finite-dimensional reductions of these systems are considered that give rise to integrable mappings. Furthermore, the similarity reduction is shown to lead to higher-order q-difference generalisations of the discrete Painlevé I equation.

Factorization and Poisson correspondences

The Darboux transformation as an example of an integrable infinite-dimensional Poisson correspondence is discussed in the context of the general factorization problem. Generalizations related to energy-dependent Schrodinger operators and to Kac-Moody algebras are considered. We also present the finite-dimensional reductions of the Darboux transformation to stationary flows.

Nonlocal finite dimensional reductions in variational boundary value problems

Nonlocal finite dimensional reductions in variational boundary value problems