Wave-Breaking and Peakons for a Modified Camassa–Holm Equation

In this paper, we investigate the formation of singularities and the existence of peaked traveling-wave solutions for a modified Camassa-Holm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multi-peakon solutions, of a different character than those of the Camassa-Holm equation. Singularities of the solutions can occur only in the form of wave-breaking, and a new wave-breaking mechanism for solutions with certain initial profiles is described in detail.     

Periodic Semifolded Solitary Waves for （2＋1）-Dimensional Variable Coefficient Broer-Kaup System

Applying the extended mapping method via Riccati equation, many exact variable separation solutions for the （2＆1 ）-dimensional variable coefficient Broer-Kaup equation are obtained. Introducing multiple valued function and Jacobi elliptic function in the seed solution, special types of periodic semifolded solitary waves are derived. In the long wave limit these periodic semifolded solitary wave excitations may degenerate into single semifolded localized soliton structures. The interactions of the periodic semifolded solitary waves and their degenerated single semifolded soliton structures are investigated graphically and found to be completely elastic.     

扩展的(2+1)维浅水波方程的尖峰孤子解及其相互作用

利用改进的 Riccati方程映射法和变量分离法, 得到了扩展的(2+1)维浅水波方程的变量分离解(包括孤波解, 周期波解和有理函数解). 根据得到的孤波解, 构造出了方程的几种不同形状的尖峰孤子结构, 研究了孤子的相互作用.     

非线性波方程尖峰孤子解的一种简便求法及其应用

根据尖峰孤子解的特点,提出了一种待定系数法求非线性波方程尖峰孤子解的思路和方法,并利用该方法求解了5个非线性波方程,即CH（Camassa-Holm)方程、五阶KdV-like 方程、广义Ostrovsky方程、组合KdV-mKdV方程和Klein-Gordon方程,比较简便地得到了这些方程的尖峰孤子解.文献中关于CH方程的结果成为本文结果的特例.通过数值模拟给出了部分解的图像.简要说明了非线性波方程存在尖峰孤子解所须满足的特定条件.该方法也适用于求其他非线性波方程的尖峰孤子解. 关键词： 非线性波方程 尖峰孤子解 待定系数法     

Peaked Traveling Wave Solutions to a Generalized Novikov Equation with Cubic and Quadratic Nonlinearities

The Camassa-Holm equation, Degasperis-Procesi equation and Novikov equation are the three typical integrable evolution equations admitting peaked solitons. In this paper, a generalized Novikov equation with cubic and quadratic nonlinearities is studied, which is regarded as a generalization of these three well-known studied equations. It is shown that this equation admits single peaked traveling wave solutions, periodic peaked traveling wave solutions, and multi-peaked traveling wave solutions.     

Some extensions on the soliton solutions for the Novikov equation with cubic nonlinearity

In this paper, by using the bifurcation method of dynamical systems, we derive the traveling wave solutions of the nonlinear equation UU_τyy ? U_yU_τy + U²U_τ + 3U_y = 0. Based on the relationship of the solutions between the Novikov equation and the nonlinear equation, we present the parametric representations of the smooth and nonsmooth soliton solutions for the Novikov equation with cubic nonlinearity. These solutions contain peaked soliton, smooth soliton, W-shaped soliton and periodic solutions. Our work extends some previous results.     

Dynamics of D’Alembert wave and soliton molecule for a (2+1)-dimensional generalized breaking soliton equation

The D’Alembert solution is an important basic formula in linear partial differential theory due to that it can be considered as a general solution of the wave motion equation. However, the study of the D’Alembert wave is few works in nonlinear partial differential systems. In this paper, one construct the D’Alembert solution of a (2+1)-dimensional generalized breaking soliton equation which possesses the nonlinear terms. This D’Alembert wave has one arbitrary function in the traveling wave variable. We investigate the dynamics of the three soliton molecule, the soliton molecule by bound as an asymmetry soliton and one-soliton, the interaction between the half periodic wave and two-kink, and the interaction among the half periodic wave, one-kink and a kink soliton molecule of the (2+1)-dimensional generalized breaking soliton equation by selecting the appropriate parameters.     

Dispersion-managed soliton in optical fibers with zero average dispersion

A dispersion-managed optical system with stepwise periodic variation of dispersion is studied in the framework of a path-averaged Gabitov-Turitsyn equation. The soliton solution is obtained by means of iterating the path-averaged equation. The dependence of soliton parameters on dispersion map strength is investigated, together with the oscillating tails of the soliton.     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation. Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

Quasi-Periodic Solutions and Asymptotic Properties for the Isospectral BKP Equation

In this paper, based on a Riemann theta function and Hirota's bilinear form, a straightforward way is presented to explicitly construct Riemann theta functions periodic waves solutions of the isospectral BKP equation.Once the bilinear form of an equation obtained, its periodic wave solutions can be directly obtained by means of an unified theta function formula and the way of obtaining the bilinear form is given in this paper. Based on this, the Riemann theta function periodic wave solutions and soliton solutions are presented. The relations between the periodic wave solutions and soliton solutions are strictly established and asymptotic behaviors of the Riemann theta function periodic waves are analyzed by a limiting procedure. The N-soliton solutions of isospectral BKP equation are presented with its detailed proof.     

（2+1）维Sawada-Kotera方程中两个Y周期孤子的相互作用

利用双线性方法给出了2+1维Sawaka-Kotera(SK)方程的N孤子解.将N孤子解中的实参数扩大到复数范围，得到了该方程的呼吸子解，描述线孤子和y周期孤子相互作用的解和两个y周期孤子相互作用的解.从解析和几何两个角度探讨了两个y周期孤子的相互作用.相互作用性质和耦合系数有关.对于SK方程，耦合系数的取值只允许方程中存在弹性的排斥相互作用. 关键词： y周期孤子相互作用 SK方程 双线性方法     

Perturbed soliton-like molecular excitations in a deformed DNA chain

We study the nonlinear dynamics of a deformed Deoxyribonucleic acid (DNA) molecular chain which is governed by a perturbed sine-Gordon equation coupled with a linear wave equation representing the lattice deformation. The DNA chain considered here is assumed to be deformed periodically which is the energetically favourable configuration, and the periodic deformation is due to the repulsive force between base pairs, stress in the helical backbones and due to the elastic strain force in both the strands. A multiple scale soliton perturbation analysis is carried out to solve the perturbed sine-Gordon equation and the resultant perturbed kink and antikink solitons represent open state configuration with small fluctuation. The perturbation due to periodic deformation of the lattice changes the velocity of the soliton. However, the width of the soliton remains unchanged.     

Dispersion-managed soliton in a strong dispersion map limit

A dispersion-managed optical system with stepwise periodic variation of dispersion is studied in a strong dispersion map limit in the framework of the path-averaged Gabitov-Turitsyn equation. The soliton solution is obtained by analytical and numerical iteration of the path-averaged equation. An efficient numerical algorithm for finding a DM soliton shape is developed. The envelope of soliton oscillating tails is found to decay exponentially in time, and the oscillations are described by a quadratic law.     

Breather Dynamics of the Sine-Gordon Equation

This paper studies the adiabatic dynamics of the breather soliton of the sine-Gordon equation. The integrals of motion are found and then used in soliton perturbation theory to derive the differential equation governing the soliton velocity. Time-dependent functions arise and their properties are studied. These functions are found to be bounded and periodic and affect the soliton velocity. The soliton velocity is numerically plotted against time for different combinations of initial velocities and perturbation terms.     

A Series of Variable Separation Solutions and New Soliton Structures of (2+1)-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the (2 1)-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model (24). Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

A Series of Variable Separation Solutions and New Soliton Structures of （2＋1）-Dimensional Korteweg-de Vries Equation

Variable separation approach is introduced to solve the （2＋1）-dimensional KdV equation. A series of variable separation solutions is derived with arbitrary functions in system. We present a new soliton excitation model （24）. Based on this excitation, new soliton structures such as the multi-lump soliton and periodic soliton are revealed by selecting the arbitrary function appropriately.     

一类两维光学格子中稳定的复合孤子

对两维光学格子中非线性薛定谔方程一类新的稳态解做了数值分析，发现其在传播过程中逐渐衰变为一种稳定的复合孤子，孤子的两分量在传播过程中不断交换能量，总能量守恒. 关键词： 稳态解 两维格子 孤子     

一个(3+1)维孤子方程的周期解

利用Hirota方法及Riemann theta函数得到了一个(3+1)维孤子方程的周期解.在极限情况下,该周期解退化为孤子解.另外,利用计算机技术和Mathematica绘制了解的三维曲面图.     

Explicit Soliton and Periodic Solutions to Three-Wave System with Quadratic and Cubic Nonlinearities

Lie group theoretical method and the equation of the Jacobi elliptic function are used to study the three wave system that couples two fundamental frequency (FF) and a single second harmonic (SH) one by competing χ⁽²⁾ (quadratic) andχ⁽³⁾ (cubic) nonlinearities and birefringence. This system shares some of the nice properties of soliton system. On the phase-locked condition, we obtain large families of analytical solutions as the soliton, kink and periodic solutions of this system.     

Asymptotic theory of plane soliton self-focusing in two-dimensional wave media

An asymptotic method is developed to describe a long-term evolution of unstable quasi-plane solitary waves in the Kadomtsev-Petviashvili model for two-dimensional wave media with positive dispersion. An approximate equation is derived for the parameters of soliton transversal modulation and a general solution of this equation is found in an explicit form. It is shown that the development of periodic soliton modulation, in an unstable region, leads to saturation and formation of a two-dimensional stationary wave. This process is accompanied by the radiation of a small-amplitude plane soliton. In a stable region, an amplitude of the modulation is permanently decreasing due to radiation of quasi-harmonic wave packets. The multiperiodic regime of plane soliton self-focusing is also investigated.