Higher-Dimensional Integrable Systems Induced by Motions of Curves in Affine Geometries

We discuss the motions of curves by introducing an extra spatial variable or equivalently, moving surfaces in arffine geometries. It is shown that the 2 ＋1-dimensional breaking soliton equation and a 2 ＋ 1-dimensional nonlinear evolution equation regarded as a generalization to the 1 ＋ 1-dimensional KdV equation arise from such motions.     

A hierarchy of Hamiltonian lattice equations associated with the relativistic Toda type system

By considering a discrete iso-spectral problem, a hierarchy of bi-Hamiltonian relativistic Toda type lattice equations are revisited. After introducing a semi-direct sum Lie algebras of four by four matrices, integrable coupling system associated with the relativistic Toda type lattice are derived. It is shown that the resulting lattice soliton hierarchy possesses Hamiltonian structures and infinitely many common commuting symmetries as well infinitely many conserved functions. The Liouville integrability of the resulting system is then demonstrated.     

Exact solutions to a coupled modified KdV equations with non-uniformity terms

The bilinear form of a coupled modified KdV equations with non-uniformity terms is given and a few soliton solutions are obtained. Furthermore, the multisoliton of the coupled system is expressed by Pfaffian.     

A supersymmetric Sawada-Kotera equation

A new supersymmetric equation is proposed for the Sawada-Kotera equation. The integrability of this equation is shown by the existence of Lax representation and infinite conserved quantities and a recursion operator.     

Grammian solutions and Pfaffianization of the non-isospectral modified Kadomtsev-Petviashvili equation

The Grammian determinant solutions of the non-isospectral modified Kadomtsev-Petviashvili (mKP) equation are presented. Moreover, a new non-isospectral coupled system is constructed by using the Pfaffianization procedure. Furthermore, Gramm-type Pfaffian solutions of the non-isospectral coupled system are obtained.     

From Yang-Baxter Maps to Integrable Recurrences

Starting from known solutions of the functional Yang-Baxter equations, we construct a series of nonautonomous integrable recurrences, “median graphs”, and give their explicit solution.     

Soliton solutions by Darboux transformation for a Hamiltonian lattice system

Starting from a new discrete iso-spectral problem, we derive a hierarchy of Hamiltonian lattice equations. A Darboux transformation is established for the lattice soliton hierarchy. As applications, the soliton solutions of resulted lattice hierarchy are given.     

A simple and direct method for generating travelling wave solutions for nonlinear equations

We propose a simple and direct method for generating travelling wave solutions for nonlinear integrable equations. We illustrate how nontrivial solutions for the KdV, the mKdV and the Boussinesq equations can be obtained from simple solutions of linear equations. We describe how using this method, a soliton solution of the KdV equation can yield soliton solutions for the mKdV as well as the Boussinesq equations. Similarly, starting with cnoidal solutions of the KdV equation, we can obtain the corresponding solutions for the mKdV as well as the Boussinesq equations. Simple solutions of linear equations can also lead to cnoidal solutions of nonlinear systems. Finally, we propose and solve some new families of KdV equations and show how soliton solutions are also obtained for the higher order equations of the KdV hierarchy using this method.     

On soliton structure of isospectral Vakhnenko equation: WKI-eigenvalue problem

In this Letter, an inverse scattering method is developed for the isospectral Vakhnenko equation, and the general N-solution is presented. Using this technique, a typical self-confined solitary wave hereafter named soliton, satisfying some vanishing boundary conditions is elicited. The detail on the scattering behavior of such structures including their phase shifts is outlined. This method is presented to be arguably more simple, tractable and straightforward than that recently investigated by Vakhnenko and Parkes [V.O. Vakhnenko, E.J. Parkes, Chaos Solitons Fractals 13 (2002) 1819] while solving the same equation. As a result, it is shown that when two single soliton solutions with ‘similar’ or ‘dissimilar’ amplitudes collide, there may be two types of features depending on the ratio of the two eigenvalues involved. It is then suggested an existence of some critical value for the ratio of the two eigenvalues at which the collision process changes its characteristic features.     

Full Poncelet Theorem in Minkowski dS and AdS Spaces

We study the reflection of a straight line or a billiard on a plane in an n-dimensional Minkowski space. It is found that the reflection law coincides with that defined with respect to confocal quadratic surfaces in projective geometry. We then establish the full Poncelet theorem which holds in projective geometry in n-dimensional Minkowski space and in their quadratic surfaces including de Sitter and AdS spaces.     

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation

New positon, negaton and complexiton solutions for the Bogoyavlensky-Konoplechenko equation are constructed by means of the Darboux transformation with constant seed solution. The new positon, negaton and complexiton solutions are analytical or singular and given out both analytically and graphically.     

Grammian Solutions to a Variable-Coefficient KP Equation

Solutions in the Grammian form for a variable-coefficient Kadomtsev-Petviashvili （KP） equation which has the Wronskian solutions are derived by means of Pfaffian derivative formulae.     

Soliton-like solutions of certain types of nonlinear diffusion-reaction equations with variable coefficient

With a view to exploring new soliton-like solutions of certain types of nonlinear diffusion-reaction (DR) equations with a variable coefficient, we demonstrate the viability of a method which is the combination of both the symbolic computation technique of Gao and Tian [Y.T. Gao, B. Tian, Comput. Phys. Commun. 133 (2001) 158] and auxiliary equation method of Sirendaoreji [Sirendaoreji, Phys. Lett. A 356 (2006) 124] and used recently for the KdV equation. In particular, the DR equations with quadratic and cubic nonlinearities with a time-dependent velocity in the convective flux term are studied and the existence of soliton-like solutions is shown.     

A Darboux Transformation for the Coupled Kadomtsev--Petviashvili Equation

The coupled Kadomtsev-Petviashvili equation is considered. It is shown that a Darboux transformation can be constructed by means of an elementary approach.     

Chirped and chirpfree soliton solutions of generalized nonlinear Schrödinger equation with distributed coefficients

Using the F-expansion method, we systematically present exact solutions of the generalized nonlinear nonlinear Schrödinger equation with varying intermodal dispersion and nonlinear gain or loss. This approach allows us to obtain large variety of solutions in terms of Jacobi-elliptical and Weierstrass-elliptical functions. The chirped and unchirped spatiotemporal soliton solutions and trigonometric-function solutions have been also obtained as limiting cases. The dynamics of these spatiotemporal soliton is discussed in context of optical fiber communication. To visualize the propagation characteristics of chirp and unchirped dark-bright soliton solutions, few numerical simulations are given. It is found that wave profile of solitons depend on the group velocity dispersion and the gain or loss functions.     

Binary Darboux Transformation for the Modified Kadomtsev--Petviashvili Equation

Via the elementary Darboux transformation (DT) of the modified Kadomtsev--Petviashvili (mKP) equation, a binary Darboux transformation (BDT) of the mKP equation is constructed.     

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system

New positon, negaton and complexiton solutions for the Hirota-Satsuma coupled KdV system are constructed by means of the Darboux transformation with zero seed solution. The new positon, negaton and complexiton solutions are singular and given out both analytically and graphically.     

N-Soliton Solutions of Non-Isospectral Derivative Nonlinear Schrödinger Equation

Bilinear forms of the non-isospectral derivative nonlinear Schrǒdinger equation are derived. The N-soliton solutions of this equation are obtained by Hirota＇s method.