On a Relation Between the Bach Equation and the Equation of Geometrodynamics

The Bach equation and the equation of geometrodynamics are based on two quite different physical motivations, but in both approaches, the conformal properties of gravitation play the key role. In this paper we present an analysis of the relation between these two equations and show that the solutions of the equation of geometrodynamics are of a more general nature. We show the following non-trivial result: there exists a conformally invariant Lagrangian, whose field equation generalizes the Bach equation and has as solutions those Ricci tensors which are solutions to the equation of break geometrodynamics.     

Solutions of Two Kinds of Non-Isospectral Generalized Nonlinear Schrodinger Equation Related to Bose-Einstein Condensates

Two non-isospectral generalized nonlinear Schrodinger （ONLS） equations, which are two important models of nonlinear excitations of matter waves in Bose-Einstein condensates, are studied. Two novel transformations are constructed such that these two GNLS equations are transformed to the well-known nonlinear Schr6dinger （NLS） equation, which is an isospectral equation. Therefore, once one solution of the NLS equation is provided, we can immediately obtain one solution for two ONLS equations by these transformations. Thus it is unnecessary to solve these two non-isospectral GNLS equations directly. Soliton solutions and periodic solutions are obtained for them by two transformations from the corresponding solutions of the NLS equation, which are generated by Darboux transformation.     

An Integrable Matrix Camassa-Holm Equation

We present an integrable sl(2)-matrix Camassa-Holm(CH) equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.     

New Exact Travelling Wave Solutions to Hirota Equation and (1+1)-Dimensional Dispersive Long Wave Equation

Based on the computerized symbolic Maple, we study two important nonlinear evolution equations, i.e.,the Hirota equation and the (1+1)-dimensional dispersive long wave equation by use of a direct and unified algebraic method named the general projective Riccati equation method to find more exact solutions to nonlinear differential equations. The method is more powerful than most of the existing tanh method. New and more general form solutions are obtained. The properties of the new formal solitary wave solutions are shown by some figures.     

Exact Periodic-Wave Solutions for （2＋1）-Dimensional Boussinesq Equation and（3＋1）-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

Exact Periodic-Wave Solutions for (2+1)-Dimensional Boussinesq Equation and (3+1)-Dimensional KP Equation

The (2 1)-dimensional Boussinesq equation and (3 1)-dimensional KP equation are studied by using the extended Jacobi elliptic-function method. The exact periodic-wave solutions for the two equations are obtained.     

Marchenko Equation for the Derivative Nonlinear SchrSdinger Equation

A simple derivation of the Marchenko equation is given for the derivative nonlinear Schrodinger equation. The kernel of the Marchenko equation is demanded to satisfy the conditions given by the compatibility equations. The soliton solutions to the Marchenko equation are verified. The derivation is not concerned with the revisions of Kaup and Newell.     

Covariant Prolongation Structure of Konno-Asai-Kakuhata Equation

Based upon the covariant prolongation structures theory, we construct the sl(2,R)×R(ρ) prolongation structure for Konno-Asai-Kakuhata equation. By taking two and one-dimensional prolongation spaces, we obtain the inverse scattering equations given by Konno et al. and the corresponding Riccati equation. The Bäcklund transformations are also presented.     

KdV Equation with Self-consistent Sources in Non-uniform Media

Two non-isospectral KdV equations with self-consistent sources are derived. Gauge transformation between the first non-isospectral KdV equation with self-consistent sources （corresponding to λt = -2aA） and its isospectral counterpart is given, from which exact solutions for the first non-isospectral KdV equation with self-consistent sources is easily listed. Besides, the soliton solutions for the two equations are obtained by means of Hirota＇s method and Wronskian technique, respectively. Meanwhile, the dynamical properties for these solutions are investigated.     

The Symmetry Structure of the Heavenly Equation

We show that excitations of physical interest for the heavenly equation are generated by symmetry operators which yield two reduced equations with different characteristics. One equation is of the Liouville type and gives rise to gravitational instantons, including those found by Eguchi–Hanson and Gibbons–Hawking. The second equation appears for the first time in the theory of heavenly spaces and provides meron-like configurations endowed with a fractional topological charge. A link is also established between the heavenly equation and the so-called Schröder equation, which plays a crucial role in the bootstrap model and in renormalization theory.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travellingwave form satisfies some special conditions.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

Equal-time Equation of the Complex Scalar Field

By using the time displacement operators and rearranging the irreducible kernel, the equal-time equation of the twocomponent complex scalar field is established. From the equations of the matrix elements the equation of the usual two composite bosons is separated and a three-dimensional relativistic equation is obtained. The most important feature is that the ghost states disappear in our equation.     

Conditional Stability of Solitary-Wave Solutions for Generalized Compound KdV Equation and Generalized Compound KdV-Burgers Equation

In this paper, we discuss conditional stability of solitary-wave solutions in the sense of Liapunov for the generalized compound KdV equation and the generalized compound KdV-Burgers equations. Linear stability of the exact solitary-wave solutions is proved for the above two types of equations when the small disturbance of travelling wave form satisfies some special conditions.     

Exact Travelling Solutions of Discrete sine-Gordon Equation via Extended Tanh-Function Approach

In this paper, we generalize the extended tanh-function approach, which was used to find new exact travelling wave solutions of nonlinear partial differential equations or coupled nonlinear partial differential equations, to nonlinear differential-difference equations. As illustration, two series of exact travelling wave solutions of the discrete sine-Gordon equation are obtained by means of the extended tanh-function approach.     

Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2+1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1+1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2+1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Hierarchy of Combined TL-RTL Equations and an Associated (2+1)-Dimensional Lattice Equation

A new (2 1)-dimensional lattice equation is presented based upon the first two members in the hierarchy of the combined Toda lattice and relativistic Toda lattice (TL-RTL) equations in (1 1) dimensions. A Darboux transformation for the hierarchy of the combined TL-RTL equations is constructed. Solutions of the first two members in the hierarchy of the combined TL-RTL equations, as well as the new (2 1)-dimensional lattice equation are explicitly obtained by the Darboux transformation.     

Symmetries and Exact Solutions of the Breaking Soliton Equation

With the aid of the classical Lie group method and nonclassical Lie group method, we derive the classical Lie point symmetry and the nonclassical Lie point symmetry of (2+1)-dimensional breaking soliton (BS) equation. Using the symmetries, we find six classical similarity reductions and two nonclassical similarity reductions of the BS equation. Varieties of exact solutions of the BS equation are obtained by solving the reduced equations.     

New Optical Soliton Solutions of Nolinear Evolution Equation Describing Nonlinear Dispersion

In this work, we examine two algorithm schemes, namely, Kudryashov expansion andAuxiliary equation method for obtaining new optical soliton solutions of the discreteelectrical lattice models in nonlinear scheme (Salerno equation). Our solutions obtainedhere are include the hyperbolic, rational, and trigonometric functions. Our two used methods are proved to be effective and powerful methods in obtaining the exact solutions of nonlinearevolution equations (NLEEs).