12 research outputs found
Separable Hamiltonian equations on Riemann manifolds and related integrable hydrodynamic systems
A systematic construction of St\"{a}ckel systems in separated coordinates and
its relation to bi-Hamiltonian formalism are considered. A general form of
related hydrodynamic systems, integrable by the Hamilton-Jacobi method, is
derived. One Casimir bi-Hamiltonian case is studed in details and in this case,
a systematic construction of related hydrodynamic systems in arbitrary
coordinates is presented, using a cofactor method and soliton symmetry
constraints.Comment: to appear in Journal of Geometry and Physic
Classification of polynomial integrable systems of mixed scalar and vector evolution equations. I
We perform a classification of integrable systems of mixed scalar and vector
evolution equations with respect to higher symmetries. We consider polynomial
systems that are homogeneous under a suitable weighting of variables. This
paper deals with the KdV weighting, the Burgers (or potential KdV or modified
KdV) weighting, the Ibragimov-Shabat weighting and two unfamiliar weightings.
The case of other weightings will be studied in a subsequent paper. Making an
ansatz for undetermined coefficients and using a computer package for solving
bilinear algebraic systems, we give the complete lists of 2nd order systems
with a 3rd order or a 4th order symmetry and 3rd order systems with a 5th order
symmetry. For all but a few systems in the lists, we show that the system (or,
at least a subsystem of it) admits either a Lax representation or a linearizing
transformation. A thorough comparison with recent work of Foursov and Olver is
made.Comment: 60 pages, 6 tables; added one remark in section 4.2.17 (p.33) plus
several minor changes, to appear in J.Phys.
Discretisations of constrained KP hierarchies
We present a discrete analogue of the so-called symmetry reduced or
`constrained' KP hierarchy. As a result we obtain integrable discretisations,
in both space and time, of some well-known continuous integrable systems such
as the nonlinear Schroedinger equation, the Broer-Kaup equation and the
Yajima-Oikawa system, together with their Lax pairs. It will be shown that
these discretisations also give rise to a discrete description of the entire
hierarchy of associated integrable systems. The discretisations of the
Broer-Kaup equation and of the Yajima-Oikawa system are thought to be new.Comment: Accepted for publication in Journal of Mathematical Sciences, The
University of Toky
New developments in Functional and Fractional Differential Equations and in Lie Symmetry
Delay, difference, functional, fractional, and partial differential equations have many applications in science and engineering. In this Special Issue, 29 experts co-authored 10 papers dealing with these subjects. A summary of the main points of these papers follows:Several oscillation conditions for a first-order linear differential equation with non-monotone delay are established in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, whereas a sharp oscillation criterion using the notion of slowly varying functions is established in A Sharp Oscillation Criterion for a Linear Differential Equation with Variable Delay. The approximation of a linear autonomous differential equation with a small delay is considered in Approximation of a Linear Autonomous Differential Equation with Small Delay; the model of infection diseases by Marchuk is studied in Around the Model of Infection Disease: The Cauchy Matrix and Its Properties. Exact solutions to fractional-order Fokker–Planck equations are presented in New Exact Solutions and Conservation Laws to the Fractional-Order Fokker–Planck Equations, and a spectral collocation approach to solving a class of time-fractional stochastic heat equations driven by Brownian motion is constructed in A Collocation Approach for Solving Time-Fractional Stochastic Heat Equation Driven by an Additive Noise. A finite difference approximation method for a space fractional convection-diffusion model with variable coefficients is proposed in Finite Difference Approximation Method for a Space Fractional Convection–Diffusion Equation with Variable Coefficients; existence results for a nonlinear fractional difference equation with delay and impulses are established in On Nonlinear Fractional Difference Equation with Delay and Impulses. A complete Noether symmetry analysis of a generalized coupled Lane–Emden–Klein–Gordon–Fock system with central symmetry is provided in Oscillation Criteria for First Order Differential Equations with Non-Monotone Delays, and new soliton solutions of a fractional Jaulent soliton Miodek system via symmetry analysis are presented in New Soliton Solutions of Fractional Jaulent-Miodek System with Symmetry Analysis
The Algebro-Geometric Solutions for the Ruijsenaars-Toda Hierarchy
We provide a detailed treatment of Ruijsenaars-Toda (RT) hierarchy with
special emphasis on its the theta function representation of all
algebro-geometric solutions. The basic tools involve hyperelliptic curve
associated with the Burchnall-Chaundy polynomial, Dubrovin-type
equations for auxiliary divisors and associated trace formulas. With the help
of a foundamental meromorphic function , Baker-Akhiezer vector on
, the complex-valued algebro-geometric solutions of RT hierarchy
are derived.Comment: 49 pages. arXiv admin note: substantial text overlap with
arXiv:nlin/0702058, arXiv:nlin/0611055 by other author