Peakon Solutions to Supersymmetric Camassa-Holm Equation and Degasperis-Procesi Equation

In this paper,the supersymmetric Camassa-Holm equation and Degasperis-Procesi equation are derived from a general superfield equations by choosing different parameters.Their peakon-type solutions are shown in weak sense.At the same time,the dynamic behaviors are analyzed particularly when the two peakons collide elastically,and some results are compared with each other between the two equations.     

A method for solving the dispersionless KP equation and its exact solutions

We present a systematic method to produce a class of exact solutions of the dispersionless KP equation, using the conservation equations derived from the semi-classical limit of the KP theory. These exact solutions include rarefaction waves (global solutions) and shock waves (breaking solutions in finite time).     

Comparative study of travelling-wave and numerical solutions for the coupled short pulse （CSP） equation

The Lie symmetry analysis is performed for the coupled short plus (CSP) equation. We derive the infinitesimals that admit the classical symmetry group. Five types arise depending on the nature of the Lie symmetry generator. In all types, we find reductions in terms of system of ordinary differential equations, and exact solutions of the CSP equation are derived, which are compared with numerical solutions using the classical fourth-order Runge-Kutta scheme.     

Integrability analysis of a conformal equation in relativity

In 1987 C. C. Dyer, G. C. McVittie, and L. M. Oattes derived the (two) field equations for shear-free, spherically symmetric perfect fluid spacetimes which admit a conformai symmetry. We use the techniques of the Lie and Painlevé analyses of differential equations to find solutions of these equations. The concept of a pseudo-partial Painlevé property is introduced for the first time which could assist in finding solutions to equations that do not possess the Painlevé property. The pseudo-partial Painlevé property throws light on the distinction between the classes of solutions found independently by P. Havas and M. Wyman. We find a solution for all values of a particular parameter for the first field equation and link it to the solution of the second equation. We indicate why we believe that the first field equation cannot be solved in general. Both techniques produce similar results and demonstrate the close relationship between the Lie and Painlevé analyses. We also show that both of the field equations of Dyeret al. may be reduced to the same Emden-Fowler equation of index two.     

The auxiliary elliptic-like equation and the exp-function method

The auxiliary equation method is very useful for finding the exact solutions of the nonlinear evolution equations. In this paper, a new idea of finding the exact solutions of the nonlinear evolution equations is introduced. The idea is that the exact solutions of the auxiliary elliptic-like equation are derived using exp-function method, and then the exact solutions of the nonlinear evolution equations are derived with the aid of auxiliary elliptic-like equation. As examples, the RKL models, the high-order nonlinear Schrödinger equation, the Hamilton amplitude equation, the generalized Hirota-Satsuma coupled KdV system and the generalized ZK-BBM equation are investigated and the exact solutions are presented using this method.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations. As concrete examples of its application, we apply this method to the (2 1)-dimensional modified Broer-Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Special Conditional Similarity Reduction Solutions for Two Nonlinear Partial Differential Equations

We present a method of special conditional similarity reduction solutions for nonlinear partial differential equations, As concrete examples of its application, we apply this method to the （2＋1）-dimensional modified Broer- Kaup equations and the variable coefficient KdV-mKdV equation, which have extensive physics backgrounds, and obtain abundant exact solutions derived from some reduction equations.     

Classification of the similarity solutions of free Kramers equation

We obtain, a complete classification of all possible nontrivial similarity solutions of the free Kramers equations, together with a necessary and sufficient condition for each type to be reducible to the heat equation. A confluent hypergeometric solution of the free Kramers equation is derived for some classes of similarity solutions.     

A connection between deterministic motion and the steady-state probability distribution

We start with a given deterministic equation of motion for a set of macrovariables. Suppose that we can construct a corresponding master equation which has the following properties. In the limit of large systems the deterministic equation can be derived from the master equation for not too long times and the steady-state probability distribution is the exponential of an extensive quantity. Then, using a theorem concerning the solutions of master equations, we can show that the solutions of the deterministic equations evolve into the direction of a nondecreasing steady-state probability.     

非线性波方程求解的新方法

从Legendre椭圆积分和Jacobi椭圆函数的定义出发，得到了新的变换，并把它用于非线性演化方程的求解.用三个具体的例子，如非线性Klein-Gordon方程、Boussinesq方程和耦合的mKdV方程组，说明了具体的求解步骤.比较方便地得到非线性演化方程或方程组的新解析解，如周期解、孤子解等. 关键词： Jacobi椭圆函数 非线性方程 周期解 孤子解     

Three types of generalized Kadomtsev－Petviashvili equations arising from baroclinic potential vorticity equation

By means of the reductive perturbation method, three types of generalized (2+1)-dimensional Kadomtsev--Petviashvili (KP) equations are derived from the baroclinic potential vorticity (BPV) equation, including the modified KP (mKP) equation, standard KP equation and cylindrical KP (cKP) equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.     

Non-local equations for general relativity

The field equations for two non-local variables, equivalent to the Einstein vacuum equations, are presented. These variables are the holonomy operator associated with special paths and the light cone cut function.

Starting from these equations, one shows via a perturbation argument that a single, fourth-order equation for the cut function can be derived. This single equation encodes the entire conformal structure of a vacuum space—time. The same perturbation technique yields, via quadratures, solutions to the vacuum Einstein equations to any order.     

Collective-Coordinate Analysis of Inhomogeneous Nonlinear Klein–Gordon Field Theory

Two different sets of collective-coordinate equations for solitary solutions of nonlinear Klein–Gordon (NKG) model are introduced. The collective-coordinate equations are derived using different approaches for adding the inhomogeneities as external potentials to the soliton equation of motion. The interaction of the NKG field with a local inhomogeneity like a delta function potential wall or a delta-function potential well is investigated using the presented collective-coordinate equations, and the results of the two different models are compared. Most of the characters of the interaction are derived analytically. The analytical results are also compared to the results of numerical simulations.     

Negative-order KdV equations in (3+1) dimensions by using the KdV recursion operator

We develop a variety of negative-order Korteweg-de Vries (KdV) equations in (3+1)-dimensions. The recursion operator of the KdV equation is used to derive these higher dimensional models. The new equations give distinct solitons structures and distinct dispersion relations as well. We also determine multiple soliton solutions for each derived model.     

The Hirota bilinear method for the coupled Burgers equation and the high-order Boussinesq–Burgers equation

This paper studies the coupled Burgers equation and the high-order Boussinesq–Burgers equation. The Hirota bilinear method is applied to show that the two equations are completely integrable. Multiple-kink (soliton) solutions and multiple-singular-kink (soliton) solutions are derived for the two equations.     

Solutions and conservation laws of Benjamin–Bona–Mahony–Peregrine equation with power-law and dual power-law nonlinearities

In this paper, exact solutions of Benjamin–Bona–Mahony–Peregrine equation are obtained with power-law and dual power-law nonlinearities. The Lie group analysis as well as the simplest equation method are used to carry out the integration of these equations. The solutions obtained are cnoidal waves, periodic solutions and soliton solutions. Subsequently, the conservation laws are derived for the underlying equations.     

New exact solutions to the space–time fractional nonlinear wave equation obtained by the ansatz and functional variable methods

In this paper, the ansatz method and the functional variable method are employed to find new analytic solutions for the space–time nonlinear fractional wave equation, the space–time fractional Kadomtsev–Petviashvili–Benjamin–Bona–Mahony equation and the space–time fractional modified Korteweg–de Vries–Zakharov–Kuznetsov equation. As a result, some exact solutions are obtained in terms of hyperbolic and periodic functions. It is shown that the proposed methods provide a more powerful mathematical tool for constructing exact solutions for many other nonlinear fractional differential equations occurring in nonlinear physical phenomena. We have also presented the numerical simulations for these equations by means of three dimensional plots.     

Optical analogue of relativistic Dirac solitons in binary waveguide arrays

We study analytically and numerically an optical analogue of Dirac solitons in binary waveguide arrays in the presence of Kerr nonlinearity. Pseudo-relativistic soliton solutions of the coupled-mode equations describing dynamics in the array are analytically derived. We demonstrate that with the found soliton solutions, the coupled mode equations can be converted into the nonlinear relativistic 1D Dirac equation. This paves the way for using binary waveguide arrays as a classical simulator of quantum nonlinear effects arising from the Dirac equation, something that is thought to be impossible to achieve in conventional (i.e. linear) quantum field theory.