Explicit Peakon solutions to a family of wave-breaking equations

The singular traveling wave solutions of a general 4-parameter family equation which unifies the Camass-Holm equation, the Degasperis-Procesi equation and the Novikov equation are investigated in this paper. At first, we obtain the explicit peakon solutions for one of its specific case that $a=(p+2)c$, $b=(p+1)c$ and $c=1$, which is referred to a generalized Camassa-Holm-Novikov (CHN) equation, by reducing it to a second-order ordinary differential equation (ODE) and solving its associated first-order integrable ODE. By observing the characteristics of peakon solutions to the CHN equation, we construct the peakon solutions for the general 4-parameter breaking wave equation. It reveals that singularities of the peakon solutions come up only when the solutions attain singular points of the equation, which might be a universal principal for all singular traveling wave solutions for wave breaking equations.     

New traveling waves solutions to generalized Kaup-Kupershmidt and Ito equations

In this paper we consider a special fifth-order KdV equation with constant coefficients and we obtain traveling wave solutions for it, using the projective Riccati equation method. By mean of a scaling, exact solutions to general Kaup-Kupershmidt (KK) equation are obtained. As a particular case, exact solutions to standard KK equation can be derived. Using the same method, we obtain exact solutions to standard Ito equation. By mean of scaling, new exact solutions to general Ito equation are formally derived.     

Doubly periodic solutions of the modified Kawahara equation

Some doubly periodic (Jacobi elliptic function) solutions of the modified Kawahara equation are presented in closed form. Our approach is to introduce a new auxiliary ordinary differential equation and use its Jacobi elliptic function solutions to construct doubly periodic solutions of the modified Kawahara equation. When the module m → 1, these solutions degenerate to the exact solitary wave solutions of the equation. Then we reveal the relation of some exact solutions for the modified Kawahara equation obtained by other authors.     

Solitons solutions for some nonlinear evolution equations

In this paper, we establish new solitary wave solutions to the modified Kawahara equation by the sine-cosine method. Moreover, the periodic solutions and bell-shaped solitons solutions to the generalized fifth-order KdV equation are obtained. The tanh method is used to handle the double sine-Gordon equation and the double sinh-Gordon equation. Families of exact travelling wave solutions are formally derived. The rational triangle sine-cosine method is introduced and to be constructed complex solutions to the modified Degasperis-Procesi (DP) equation and the modified Camassa-Holm (CH) equation.     

(2+1)维modified Zakharov-Kuznetsov方程的复合型新解

给出函数变换,变量分离形式解与第一种椭圆方程相结合的方法,构造了(2+1)维modified Zakharov-Kuznetsov(m ZK)方程的多种复合型新解.步骤一,给出两种函数变换,将(2+1)维m ZK方程转化为能够获得变量分离解的非线性发展方程.步骤二,给出非线性发展方程的变量分离形式解,通过第一种椭圆方程及其相关结论,构造了(2+1)维m ZK方程的双孤子解和双周期解等复合型新解.     

Generalized double Wronskian solutions of the third-order isospectral AKNS equation

The generalized double Wronskian solutions of the third-order isospectral AKNS equation are obtained. Thus we found rational solutions, Matveev solutions, complexitons and interaction solutions. Moreover, rational solutions of the mKdV equation and KdV equation in double Wronskian form are constructed by reduction.     

NEW EXACT TRAVELLING WAVE SOLUTIONS OF NONLINEAR WAVE EQUATIONS USING THE DYNAMICAL SYSTEM METHOD

By the method of dynamical system,we construct the exact travelling wave solutions of a new Hamiltonian amplitude equation and the Ostrovsky equation.Based on this method,the new exact travelling wave solutions of the new Hamiltonian amplitude equation and the Ostrovsky equation,such as solitary wave solutions,kink and anti-kink wave solutions and periodic travelling wave solutions,are obtained,respectively.     

利用F-展开法求解ZK-BBM方程

利用F-展开法求解出了ZK-BBM方程的双周期波解,并在极限形式下得到了ZK-BBM方程的孤波解和单周期波解.从而丰富了该方程解的理论.此方法也可适用求解其它非线性发展方程.     

Generalized Cauchy Matrix Approach for Lattice Boussinesq-Type Equations

The authors generalize the Cauchy matrix approach to get exact solutions to the lattice Boussinesq-type equations:lattice Boussinesq equation,lattice modified Boussinesq equation and lattice Schwarzian...     

New exact kink solutions, solitons and periodic form solutions for a combined KdV and Schwarzian KdV equation

In this short letter, new exact solutions including kink solutions, soliton-like solutions and periodic form solutions for a combined version of the potential KdV equation and the Schwarzian KdV equation are obtained using the generalized Riccati equation mapping method.     

应用改进的简单方程法求非线性数学物理方程的精确解

应用改进的简单方程法求得Cahn-Allen方程和Jimbo-Miwa方程的精确解,这些解包括双曲函数解、三角函数解.当对双曲函数解中的参数取特殊值时,可以得到了孤立波解.当对三角函数解中的参数取特殊值时,可以得到对应的周期波函数解.实践证明,简单方程法对于研究非线性数学物理方程具有非常广泛的应用意义.     

Exact and traveling-wave solutions for convection-diffusion-reaction equation with power-law nonlinearity

Based on the simplest equation method, we propose exact and traveling-wave solutions for a nonlinear convection-diffusion-reaction equation with power law nonlinearity. Such equation can be considered as a generalization of the Fisher equation and other well-known convection-diffusion-reaction equations. Two important cases are considered. The case of density-independent diffusion and the case of density-dependent diffusion. When the parameters of the equation are constant, the Bernoulli equation is used as the simplest equation. This leads to new traveling-wave solutions. Moreover, some wavefront solutions can be derived from the traveling-wave ones. The case of time-dependent velocity in the convection term is studied also. We derive exact solutions of the equations by using the Riccati equation as simplest equation. The exact and traveling-wave solutions presented in this paper can be used to explain many biological and physical phenomena.     

Polygons of differential equations for finding exact solutions

A method for finding exact solutions of nonlinear differential equations is presented. Our method is based on the application of polygons corresponding to nonlinear differential equations. It allows one to express exact solutions of the equation studied through solutions of another equation using properties of the basic equation itself. The ideas of power geometry are used and developed. Our approach has a pictorial interpretation, which is illustrative and effective. The method can be also applied for finding transformations between solutions of differential equations. To demonstrate the method application exact solutions of several equations are found. These equations are: the Korteveg–de Vries–Burgers equation, the generalized Kuramoto–Sivashinsky equation, the fourth-order nonlinear evolution equation, the fifth-order Korteveg–de Vries equation, the fifth-order modified Korteveg–de Vries equation and the sixth-order nonlinear evolution equation describing turbulent processes. Some new exact solutions of nonlinear evolution equations are given.     

Group-invariant solutions, non-group-invariant solutions and conservation laws of Qiao equation

This paper considers a completely integrable nonlinear wave equation which is called Qiao equation. The equation is reduced via Lie symmetry analysis. Two classes of new exact group-invariant solutions are obtained by solving the reduced equations. Specially, a novel technique is proposed for constructing group-invariant solutions and non-group-invariant solutions based on travelling wave solutions. The obtained exact solutions include a set of traveling wave-like solutions with variable amplitude, variable velocity or both. Nonlocal conservation laws of Qiao equation are also obtained with the corresponding infinitesimal generators.     

Self-similar solutions of semilinear wave equation with variable speed of propagation

We investigate the issue of existence of the self-similar solutions of the generalized Tricomi equation in the half-space where the equation is hyperbolic. We look for the self-similar solutions via the Cauchy problem. An integral transformation suggested in [K. Yagdjian, A note on the fundamental solution for the Tricomi-type equation in the hyperbolic domain, J. Differential Equations 206 (2004) 227-252] is used to represent solutions of the Cauchy problem for the linear Tricomi-type equation in terms of fundamental solutions of the classical wave equation. This representation allows us to prove decay estimates for the linear Tricomi-type equation with a source term. Obtained in [K. Yagdjian, The self-similar solutions of the Tricomi-type equations, Z. Angew. Math. Phys., in press, doi:10.1007/s00033-006-5099-2] estimates for the self-similar solutions of the linear Tricomi-type equation are the key tools to prove existence of the self-similar solutions.     

Derivation of the particle dynamics from kinetic equations

The microscopic solutions of the Boltzmann-Enskog equation discovered by Bogolyubov are considered. The fact that the time-irreversible kinetic equation has time-reversible microscopic solutions is rather surprising. We analyze this paradox and show that the reversibility or irreversibility property of the Boltzmann-Enskog equation depends on the considered class of solutions. If the considered solutions have the form of sums of delta-functions, then the equation is reversible. If the considered solutions belong to the class of continuously differentiable functions, then the equation is irreversible. Also, the so called approximate microscopic solutions are constructed. These solutions are continuous and they are reversible on bounded time intervals. This analysis suggests a way to reconcile the time-irreversible kinetic equations with the timereversible particle dynamics. Usually one tries to derive the kinetic equations from the particle dynamics. On the contrary, we postulate the Boltzmann-Enskog equation or another kinetic equation and treat their microscopic solutions as the particle dynamics. So, instead of the derivation of the kinetic equations from the microdynamics we suggest a kind of derivation of the microdynamics from the kinetic equations.     

应用多项式完全判别系统方法求解时空分数阶复Ginzburg-Landau方程

研究了时空分数阶复Ginzburg-Landau方程.首先通过分数阶复变换将时空分数阶复Ginzburg-Landau方程转化为一个常微分方程.然后将常微分方程化为初等积分形式.最后用多项式完全判别系统法求得一系列精确解,其中包含有孤立波解、有理函数解、三角函数周期解、Jacobi椭圆函数双周期解.     

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation

Darboux and Bäcklund transformations of the bidirectional Sawada-Kotera equation are derived with the help of the resulting Riccati equation. As an application, some explicit solutions of the bidirectional Sawada-Kotera equation are obtained, including rational solutions, periodic solutions, and soliton solutions.     

New travelling wave solutions to the Ostrovsky equation

In this work, new travelling wave solutions to the Ostrovsky equation are studied by employing the improved tanh function method. With this method, the Ostrovsky equation is reduced to the nonlinear ordinary differential equation and then the different types of exact solutions are derived based on the solutions of the Riccati equation. We will compare our solutions with those gained by the other methods.     

Degasperis-Procesi方程的类孤子新解

利用辅助方程与函数变换相结合的方法,构造了Degasperis-Procesi(D-P)方程的无穷序列类孤子新解.首先,通过两种函数变换,把D-P方程化为常微分方程组.然后,利用常微分方程组的首次积分,把D-P方程的求解问题化为几种常微分方程的求解问题.最后,利用几种常微分方程的Bcklund变换等相关结论,构造了D-P方程的无穷序列类孤子新解.这里包括由Riemannθ函数、Jacobi椭圆函数、双曲函数、三角函数和有理函数组成的无穷序列光滑孤立子解、尖峰孤立子解和紧孤立子解.