Symbolic Computation of Polynomial Conserved Densities, Generalized Symmetries, and Recursion Operators for Nonlinear Differential-Difference Equations

Algorithms for the symbolic computation of polynomial conserved densities, fluxes, generalized symmetries, and recursion operators for systems of nonlinear differential-difference equations are presented. In the algorithms we use discrete versions of the Fréchet and variational derivatives, as well as discrete Euler and homotopy operators. The algorithms are illustrated for prototypical nonlinear polynomial lattices, including the Kac-van Moerbeke (Volterra) and Toda lattices. Results are shown for the modified Volterra and Ablowitz-Ladik lattices

Riccati-parameter solutions of nonlinear second-order ODEs

It has been proven by Rosu and Cornejo-Perez in 2005 that for some nonlinear second-order ODEs it is a very simple task to find one particular solution once the nonlinear equation is factorized with the use of two first-order differential operators. Here, it is shown that an interesting class of parametric solutions is easy to obtain if the proposed factorization has a particular form, which happily turns out to be the case in many problems of physical interest. The method that we exemplify with a few explicitly solved cases consists in using the general solution of the Riccati equation, which contributes with one parameter to this class of parametric solutions. For these nonlinear cases, the Riccati parameter serves as a `growth' parameter from the trivial null solution up to the particular solution found through the factorization procedureComment: 5 pages, 3 figures, change of title and more tex

Gravitating fluids with Lie symmetries

We analyse the underlying nonlinear partial differential equation which arises in the study of gravitating flat fluid plates of embedding class one. Our interest in this equation lies in discussing new solutions that can be found by means of Lie point symmetries. The method utilised reduces the partial differential equation to an ordinary differential equation according to the Lie symmetry admitted. We show that a class of solutions found previously can be characterised by a particular Lie generator. Several new families of solutions are found explicitly. In particular we find the relevant ordinary differential equation for all one-dimensional optimal subgroups; in several cases the ordinary differential equation can be solved in general. We are in a position to characterise particular solutions with a linear barotropic equation of state.Comment: 13 pages, To appear in J. Phys. A: Math. Theo

Conserved Quantities in f(R)f(R) Gravity via Noether Symmetry

This paper is devoted to investigate $f(R)$ gravity using Noether symmetry approach. For this purpose, we consider Friedmann Robertson-Walker (FRW) universe and spherically symmetric spacetimes. The Noether symmetry generators are evaluated for some specific choice of $f(R)$ models in the presence of gauge term. Further, we calculate the corresponding conserved quantities in each case. Moreover, the importance and stability criteria of these models are discussed.Comment: 14 pages, accepted for publication in Chin. Phys. Let

Conformally parametrized surfaces associated with CP^(N-1) sigma models

Two-dimensional conformally parametrized surfaces immersed in the su(N) algebra are investigated. The focus is on surfaces parametrized by solutions of the equations for the CP^(N-1) sigma model. The Lie-point symmetries of the CP^(N-1) model are computed for arbitrary N. The Weierstrass formula for immersion is determined and an explicit formula for a moving frame on a surface is constructed. This allows us to determine the structural equations and geometrical properties of surfaces in R^(N^2-1). The fundamental forms, Gaussian and mean curvatures, Willmore functional and topological charge of surfaces are given explicitly in terms of any holomorphic solution of the CP^2 model. The approach is illustrated through several examples, including surfaces immersed in low-dimensional su(N) algebras.Comment: 32 page

Superposition in nonlinear wave and evolution equations

Real and bounded elliptic solutions suitable for applying the Khare-Sukhatme superposition procedure are presented and used to generate superposition solutions of the generalized modified Kadomtsev-Petviashvili equation (gmKPE) and the nonlinear cubic-quintic Schroedinger equation (NLCQSE).Comment: submitted to International Journal of Theoretical Physics, 23 pages, 2 figures, style change

On the Nonlocal Equations and Nonlocal Charges Associated with the Harry Dym Hierarchy

A large class of nonlocal equations and nonlocal charges for the Harry Dym hierarchy is exhibited. They are obtained from nonlocal Casimirs associated with its bi-Hamiltonian structure. The Lax representation for some of these equations is also given.Comment: to appear in Journal of Mathematical Physics, 17 pages, Late

New Exact Solutions of a Generalized Shallow Water Wave Equation

In this work an extended elliptic function method is proposed and applied to the generalized shallow water wave equation. We systematically investigate to classify new exact travelling wave solutions expressible in terms of quasi-periodic elliptic integral function and doubly-periodic Jacobian elliptic functions. The derived new solutions include rational, periodic, singular and solitary wave solutions. An interesting comparison with the canonical procedure is provided. In some cases the obtained elliptic solution has singularity at certain region in the whole space. For such solutions we have computed the effective region where the obtained solution is free from such a singularity.Comment: A discussion about singularity and some references are added. To appear in Physica Script

Cosymmetries and Nijenhuis recursion operators for difference equations

In this paper we discuss the concept of cosymmetries and co--recursion operators for difference equations and present a co--recursion operator for the Viallet equation. We also discover a new type of factorisation for the recursion operators of difference equations. This factorisation enables us to give an elegant proof that the recursion operator given in arXiv:1004.5346 is indeed a recursion operator for the Viallet equation. Moreover, we show that this operator is Nijenhuis and thus generates infinitely many commuting local symmetries. This recursion operator and its factorisation into Hamiltonian and symplectic operators can be applied to Yamilov's discretisation of the Krichever-Novikov equation