Double Reduction and Exact Solutions of Zakharov-Kuznetsov Modified Equal width Equation with Power Law Nonlinearity via Conservation Laws

The conservation laws for the (1+2)-dimensional Zakharov-Kuznetsov modified equal width (ZK-MEW) equation with power law nonlinearity are constructed by using Noether's approach through an interesting method of increasing the order of this equation. With the aid of an obtained conservation law, the generalized double reduction theorem is applied to this equation. It can be shown that the reduced equation is a second order nonlinear ODE. Finally, some exact solutions for a particular case of this equation are obtained after solving the reduced equation.     

On symmetries,conservation laws and exact solutions of the nonlinear Schrödinger–Hirota equation

In this paper, conservation laws and exact solution are found for nonlinear Schrödinger–Hirota equation. Conservation theorem is used for finding conservation laws. We get modified conservation laws for given equation. Modified simple equation method is used to obtain the exact solutions of the nonlinear Schrödinger–Hirota equation. It is shown that the suggested method provides a powerful mathematical instrument for solving nonlinear equations in mathematical physics and engineering.     

Breather solutions of a negative order modified Korteweg-de Vries equation and its nonlinear stability

This paper is concerned with a negative order modified Korteweg-de Vries (nmKdV) equation which is in the negative flow of the mKdV hierarchy. We construct the breather solutions by Hirota's bilinear method and derive the infinite conservation laws through the Lax pair of the nmKdV equation. By constructing a new Lyapunov function with the conservation laws, we obtain the nonlinear stability of the breather solutions.     

广义(3+1)维Zakharov-Kuznetsov方程的对称约化、精确解和守恒律

运用李群分析,得到了广义(3+1)维Zakharov-Kuznetsov(ZK)方程的对称及约化方程,结合齐次平衡原理,试探函数法和指数函数法得到了该方程的群不变解和新精确解,包括冲击波解、孤立波解等.进一步给出了广义(3+1)维ZK方程的伴随方程和守恒律.     

一类高阶非线性波方程的李群分析、最优系统、精确解和守恒律

本文运用李群分析的方法研究了一类高阶非线性波方程,得到了五阶非线性波方程的对称以及方程的最优系统,进而运用幂级数的方法,求得了方程的精确幂级数解.最后,给出了五阶非线性波方程的一些守恒律.     

Wild's solution of the nonlinear Boltzmann equation

The relaxation to equilibrium of a spatially uniform pseudo-Maxwellian gas is considered. A modified Wild expansion is defined for solving the nonlinear Boltzmann equation. The positivity and asymptotic conditions, as well as the conservation rules, are maintained at each truncation order. Some particular examples are evaluated. The comparison with exact solutions illustrates the very fast convergence of this method.     

A unified theory of nonlinear Langmuir waves, ion-acousticsolitons, and double layers

Steady-state, nonlinear potential solutions in a two-fluid plasma are formulated in terms of an integral equation that expresses the conservation of total pressure. By appropriately specifying the conservation constants and boundary conditions of the effective potential function, the integral curves yield, respectively, three types of solutions: nonlinear Langmuir waves, ion-acoustic solitons, and double layers. The similarities and differences of the existence conditions for these nonlinear potential structures are discussed     

A Stefan Problem with Surface Tension as the Sharp Interface Limit of a Nonlocal System of Phase-Field Type

A model for the evolution of phase boundaries reminiscent of the phase-field model is considered. The equation related to conservation of thermal energy is diffusive and coupled to an equation for the order parameter, which contains a nonlinear convolution operator, related to the limit of an interacting particle model with Kac-potential. Under diffusive rescaling the solutions converge to solutions of the Stefan problem with kinetic undercooling and surface tension.     

Nonlinear self-adjointness and invariant solutions of a 2D Rossby wave equation

The paper investigates the nonlinear self-adjointness of the nonlinear inviscid barotropic nondivergent vorticity equation in a beta-plane. It is a particular form of Rossby equation which does not possess variational structure and it is studied using a recently method developed by Ibragimov. The conservation laws associated with the infinite-dimensional symmetry Lie algebra models are constructed and analyzed. Based on this Lie algebra, some classes of similarity invariant solutions with nonconstant linear and nonlinear shears are obtained. It is also shown how one of the conservation laws generates a particular wave solution of this equation.     

Spatial optical solitons in fifth order and seventh order weakly nonlocal nonlinear media

We analytically study the (1 + 1)-dimensional spatial optical solitons in weakly nonlocal nonlinear media with cubic–quintic nonlinearity (fifth order nonlinear media) and cubic–quintic–septic nonlinearity (seventh order nonlinear media). Explicit solutions are derived, which include optical bright solitons, singular solutions and singular triangular periodic solution.     

Periodic Solutions for a Class of Nonlinear Partial Differential Equations in Higher Dimension

We prove the existence of periodic solutions in a class of nonlinear partial differential equations, including the nonlinear Schrödinger equation, the nonlinear wave equation, and the nonlinear beam equation, in higher dimension. Our result covers cases of completely resonant equations, where the bifurcation equation is infinite-dimensional, such as the nonlinear Schrödinger equation with zero mass, for which solutions which at leading order are wave packets are shown to exist.     

New extended auxiliary equation method for finding many new Jacobi elliptic function solutions of three nonlinear Schrödinger equations

In this paper, we construct many new types of Jacobi elliptic function solutions of nonlinear evolution equations using the so-called new extended auxiliary equation method. The effectiveness of this method is demonstrated by applications to three higher order nonlinear evolution equations, namely, the higher order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, the higher order dispersive nonlinear Schrödinger equation and the generalized nonlinear Schrödinger equation. The solitary wave solutions and periodic solutions are obtained from the Jacobi elliptic function solutions. Comparing our new results and the well-known results are given.     

一类非线性演化方程的新多级准确解

在Lamé方程和新的Lamé函数的基础上，应用小扰动方法和Jacobi椭圆函数展开法求解一类非线性演化方程（如mKdV方程，非线性Klein-Gordon方程Ⅱ等），获得多种新的多级准确解 .这些多级准确解对应着不同形式的周期波解.这些解在极限条件下可以退化为多种形式的孤 立波解，如带状孤立子、钟形孤立子等. 关键词： Jacobi椭圆函数 Lam函数 多级准确解 非线性演化方程 扰动方法     

Analysis of solitonic phenomenon for a two-mode KdV equation

The two-mode Korteweg-de Vries (TKdV) equation is used as a model to describe the wave propagation in one space dimension as a weakly nonlinear and weakly dispersive system. Theoretical and numerical investigations have been performed to search for multi-soliton solutions. A spectral collocation scheme for the equation is developed to obtain numerical soliton solutions. In the theoretical approach, the Lagrange density and some conservation laws for the equation are derived. In addition, the Hamiltonian structure presented here helps to identify the partial integrability of the equation.     

一类非线性发展方程的复合型双孤子新解

通过下列步骤,构造了一类非线性发展方程的无穷序列复合型双孤子新解: 步骤一, 给出两种函数变换,把一类非线性发展方程化为二阶非线性常微分方程; 步骤二, 再通过函数变换, 二阶非线性常微分方程转化为一阶非线性常微分方程组,并获得了该方程组的首次积分; 步骤三, 利用首次积分与两种椭圆方程的新解与Bäcklund 变换, 构造了一类非线性发展方程的无穷序列复合型双孤子新解.     

Modulation instability,rogue waves and conservation laws in higher-order nonlinear Schrödinger equation

In this paper, the modulation instability(MI), rogue waves(RWs) and conservation laws of the coupled higher-order nonlinear Schr?dinger equation are investigated. According to MI and the2?×?2 Lax pair, Darboux-dressing transformation with an asymptotic expansion method, the existence and properties of the one-, second-, and third-order RWs for the higher-order nonlinear Schr?dinger equation are constructed. In addition, the main characteristics of these solutions are discussed through some graphics, which are draw widespread attention in a variety of complex systems such as optics, Bose–Einstein condensates, capillary flow, superfluidity, fluid dynamics,and finance. In addition, infinitely-many conservation laws are established.     

Invariance Analysis and Variational Conservation Laws for the Wave Equation on Some Manifolds

In this paper we discuss symmetries of a nonlinear wave equation that arises as a consequence of some Riemannian metrics of signature −2. The objective of this study is to show how geometry can be responsible in giving rise to a nonlinear inhomogeneous wave equation rather than assuming nonlinearities in the wave equation from physical considerations. We find Lie point symmetries of the corresponding wave equations and give their solutions in two cases. Some interesting physical conclusions relating to conservation laws such as energy, linear and angular momenta are also determined.     

Generation of a Quadphoton via Seventh‐Order Nonlinear Susceptibility

With a wide range of potential applications, the generation of nonclassical multiphoton and number states has attracted renewed interest recently. Here, a correlated quadphoton with seventh‐order nonlinear susceptibility is reported in which all four photons are simultaneously created via an eight‐wave mixing (EWM) process in an atomic medium. The efficiency of the pure EWM process is low, which results from the small order of magnitude of the seventh‐order nonlinear susceptibility (the quadphoton correlation with four periods corresponding to 8 channels and 12 resonance positions). But the EWM process has a strong dressing effect. It not only increases the nonlinear susceptibility, but also produces 15 or 16 resonance positions in the dressing state picture. In addition, with the energy conservation of these positions, 12 coherent channels can be generated in this system. According to the interference between multichannels of the quadphoton, the quadphoton coincidence count appears as a damped Rabi oscillation, which has eight or ten oscillation periods. If only the nonlinear optical response is considered, the coincidence counts of the quadphoton behave as a damped Rabi oscillation with periods that can be controlled by the dressing field. These outcomes may contribute to a new promising method for quantum communication.     

(2+1)维广义Calogero-Bogoyavlenskii-Schiff方程的无穷序列类孤子解

为了构造高维非线性发展方程的无穷序列类孤子新解, 研究了二阶常系数齐次线性常微分方程, 获得了新结论. 步骤一, 给出一种函数变换把二阶常系数齐次线性常微分方程的求解问题转化为一元二次方 程和Riccati方程的求解问题. 在此基础上, 利用Riccati方程解的非线性叠加公式, 获得了二阶常系数齐次线性常微分方程的无穷序列新解. 步骤二, 利用以上得到的结论与符号计算系统Mathematica, 构造了(2+1)维广义Calogero-Bogoyavlenskii-Schiff (GCBS)方程的无穷序列类孤子新解. 关键词： 常微分方程 非线性叠加公式 高维非线性发展方程 无穷序列类孤子新解     

Nonlinear waves in media with fifth order dispersion

Nonlinear waves described of the fifth order dispersive nonlinear evolution equation are numerically investigated. The numerical method for boundary value problem for this equation is proposed. Exact solutions to nonlinear evolution equation of the fifth order are given. The numerical method was tested using some exact solutions. The influence of the fifth order dispersion on the propagation of nonlinear waves and formation of the periodic structures is studied.