This paper discusses several algorithmic ways of constructing integrable evolution equations based on Lie algebraic structure. We derive, in a pedagogical style, a large class of two component peakon type dual systems from their two component soliton equations counter part. We study the essential aspects of Hamiltonian flows on coadjoint orbits of the centrally extended semidirect product group...

ABDELOUAHAB KADEM1, DUMITRU BALEANU2,* 1L.M.F.N Mathematics Department, University of Setif, Algeria Email: [email protected] 2Department of Mathematics and Computer Sciences, Faculty of Arts and Sciences, Çankaya University06530, Ankara, Turkey ∗On leave of absence from Institute of Space Sciences, P.O.BOX, MG-23, RO-077125, Magurele-Bucharest, Romania Emails: [email protected], balea...

a modified f-expansion method to find the exact traveling wave solutions of  two-component nonlinear partial differential equations (nlpdes) is discussed. we use this method to construct many new solutions to the nonlinear whitham-broer-kaup system (1+1)-dimensional. the solutions obtained include jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangu...

A modified F-expansion method to find the exact traveling wave solutions of  two-component nonlinear partial differential equations (NLPDEs) is discussed. We use this method to construct many new solutions to the nonlinear Whitham-Broer-Kaup system (1+1)-dimensional. The solutions obtained include Jacobi elliptic periodic wave solutions which exactly degenerate to the soliton solutions, triangu...

It is well known that the Boussinesq equation is the bidirectional equivalent of the celebrated Korteweg-de Vries equation. Here we consider Boussinesq-type versions of two classical unidirectional integrable equations. A procedure is presented for deriving multisoliton solutions of one of these equations – a bidirectional Kaup–Kupershmidt equation. These solitons have the unusual property that...

In the study of nonlinear physics, the search of new exact solutions of nonlinear evolution equations (NEEs) is one of the most important problems. Various methods for obtaining exact solutions to NEEs have been proposed, such as the Lie group method of infinitesimal transformations [1], the nonclassical Lie group method [2], the Clarkson and Kruskal direct method (CK) [3 – 5], the conditional ...

Using the algebraic method of Gardner’s deformations for completely integrable systems, we construct recurrence relations for densities of the Hamiltonians for the Boussinesq and the Kaup–Boussinesq equations. By extending the Magri schemes for these equations, we obtain new integrable systems adjoint with respect to the initial ones and describe their Hamiltonian structures and symmetry proper...

Rational solutions of the fourth order analogue to the Painlevé equations are classified. Special polynomials associated with the rational solutions are introduced. The structure of the polynomials is found. Formulas for their coefficients and degrees are derived. It is shown that special solutions of the Fordy Gibbons, the Caudrey Dodd Gibbon and the Kaup Kupershmidt equations can be expressed...

We propose integrable discretizations of derivative nonlinear Schrödinger (DNLS) equations such as the Kaup–Newell equation, the Chen–Lee–Liu equation and the Gerdjikov–Ivanov equation by constructing Lax pairs. The discrete DNLS systems admit the reduction of complex conjugation between two dependent variables and possess bi-Hamiltonian structure. Through transformations of variables and reduc...

This paper extends the dual-Petrov-Galerkin method proposed by Shen [21], further developed by Yuan, Shen and Wu [27] to general fifth-order KdV type equations with various nonlinear terms. These fifth-order equations arise in modeling different wave phenomena. The method is implemented to compute the multi-soliton solutions of two representative fifth-order KdV equations: the Kaup-Kupershmidt ...