Solutions to a Novel Casimir Equation for the Ito System

We discuss two classes of solutions to a novel Casimir equation associated with the Ito system, a coupled nonlinear wave equation. Both travelling wave
solutions and separable self-similar solutions are discussed. In a number of cases, explicit exact solutions are found. Such results, particularly the exact solutions, are useful in that they provide us a baseline of comparison to any numerical simulations.Besides, such solutions provide us a glimpse of the behavior of the Ito system,and hence the behavior of a type of nonlinear wave equation, for certain parameter regimes.     

A model of an open universe as an oscillator with losses

Problems of defining exact cosmological solutions describing an open universe with consideration of volume viscosity are considered. A method which permits reduction of open universe modeling to the problem of mechanical motion of a particle in a specified force field is utilized. The behavior of the equation of state, which changes during the evolution process, is studied. New exact solutions of simple form are found.Krasnoyarsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 104–109, September, 1994.     

Modulation instability analysis,optical and other solutions to the modified nonlinear Schrödinger equation

This paper studies the new families of exact traveling wave solutions with the modified nonlinear Schrödinger equation, which models the propagation of rogue waves in ocean engineering. The extended Fan sub-equation method with five parameters is used to find exact traveling wave solutions. It has been observed that the equation exhibits a collection of traveling wave solutions for limiting values of parameters. This method is beneficial for solving nonlinear partial differential equations, because it is not only useful for finding the new exact traveling wave solutions, but also gives us the solutions obtained previously by the usage of other techniques (Riccati equation, or first-kind elliptic equation, or the generalized Riccati equation as mapping equation, or auxiliary ordinary differential equation method) in a combined approach. Moreover, by means of the concept of linear stability, we prove that the governing model is stable. 3D figures are plotted for showing the physical behavior of the obtained solutions for the different values of unknown parameters with constraint conditions.     

无阻尼单摆运动微分方程的反正切分式变换精确解

无阻尼单摆运动微分方程是一种具有物理背景的非线性常微分方程,研究其精确解和解法是非线性科学中的一个重要内容.在F展开法的基础上,应用反正切分式变换正弦函数方法,并引入Riccati辅助方程,得到了4种无阻尼单摆方程精确解的结果.达到了丰富此类方程求解技巧和精确解的目的.总结得出此类方程应用反正切分式变换方法具有一定普适性的结论.     

A new auxiliary equation and exact travelling wave solutions of nonlinear equations

A new auxiliary ordinary differential equation and its solutions are used for constructing exact travelling wave solutions of nonlinear partial differential equations in a unified way. The main idea of this method is to take full advantage of the auxiliary equation which has more new exact solutions. More new exact travelling wave solutions are obtained for the quadratic nonlinear Klein–Gordon equation, the combined KdV and mKdV equation, the sine-Gordon equation and the Whitham–Broer–Kaup equations.     

Exact solutions to the generalized lienard equation and its applications

Some new exact solutions of the generalized Lienard equation are obtained, and the solutions of the equation are applied to solve nonlinear wave equations with nonlinear terms of any order directly. The generalized one-dimensional Klein-Gordon equation, the generalized Ablowitz (A) equation and the generalized Gerdjikov-Ivanov (GI) equation are investigated and abundant new exact travelling wave solutions are obtained that include solitary wave solutions and triangular periodic wave solutions.      

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.     

New exact solutions of the Schrödinger equation for a one-dimensional multiparameter potential

A one-dimensional multiparameter Schrödinger equation is constructed on the basis of the theory of Lie group representations and its exact solutions are found. It is shown that the potential of this equation has a singularity which can lead to a change in the eigenvalue spectrum. The asymptotic behavior of the eigenvalues as a function of the parameters defining the potential are investigated.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 102–108, February, 1988.     

构造非线性发展方程无穷序列类孤子精确解的一种方法

辅助方程法已构造了非线性发展方程的有限多个新精确解. 本文为了构造非线性发展方程的无穷序列类孤子精确解, 分析总结了辅助方程法的构造性和机械化性特点. 在此基础上,给出了一种辅助方程的新解与Riccati方程之间的拟Bäcklund变换. 选择了非线性发展方程的两种形式解,借助符号计算系统 Mathematica,用改进的(2+1) 维色散水波系统为应用实例,构造了该方程的无穷序列类孤子新精确解. 这些解包括无穷序列光滑类孤子解, 紧孤立子解和尖峰类孤立子解.     

Motion of neutral fermions with anomalous magnetic and electric moments in external fields

A covariant spin operator is found for fermions with anomalous magnetic and electric dipole moments in constant external fields. The spin behavior of a neutral fermion in constant magnetic and electric fields is investigated using exact solutions obtained for the Dirac equation.     

Determination of the exact solutions to the inhomogeneous burgers equation with the use of the darboux transformation

A method of determining the exact solutions to the Burgers equation on the basis of the Darboux transformation is described. It is shown that a single application of the Darboux transformation to the homogeneous Burgers equation transforms the latter into the inhomogeneous equation describing acoustic wave propagation against transonic flow in the de Laval nozzle. In this case, the contraction ratio of the nozzle is fixed and determined by the viscosity coefficient of the medium. Based on the exact solution of the homogeneous Burgers equation, for the aforementioned problem of the flow in the nozzle, all the possible regular steady-state solutions are presented and the evolution of nonstationary solutions is investigated. The algorithm of a multiple Darboux transformation, which allows an increase in the strength of inhomogeneity, i.e., in the contraction ratio of the nozzle, is determined. This approach leads to a discrete set of possible contraction ratios at which exact solutions can be obtained. The Crum’s theorem is used to derive a formula that allows determination of the exact solutions to the inhomogeneous Burgers equation from the solutions to the homogeneous heat transfer equation. It is noted that, in fact, the proposed algorithm of the multiple Darboux transformation makes it possible to decrease the viscosity coefficient of the medium in a discrete way.     

New exact travelling wave solutions of the generalized zakharov equations

A direct algebraic method is introduced for constructing exact travelling wave solutions of nonlinear partial differential equations with complex phases. The scheme is implemented for obtaining multiple soliton solutions of the generalized Zakharov equations, and then new exact travelling wave solutions with complex phases are obtained. In addition, by using new exact solutions of an auxiliary ordinary differential equation, new exact travelling wave solutions for the generalized Zakharov equations are obtained.     

Dynamics of combined soliton solutions of unstable nonlinear Schrodinger equation with new version of the trial equation method

In this article, a new version of the trial equation method is suggested. With this method, it is possible to find the new exact solutions of the nonlinear partial differential equations. The developed method is applied to unstable nonlinear Schrödinger equation. New exact solutions are expressed with Jacobi elliptic function solutions, 1-soliton solutions and rational function solutions. When the obtained results are examined, we can say the unstable nonlinear Schrödinger equation shows different dynamic behaviors. In addition, the physical behaviors of these new exact solution are given with two and three dimensional graphs.     

Maximum entropy approach to the collisional Vlasov equation: Exact solutions

We show that the maximum entropy approach proposed by El-Wakil, Elhanbaly, and Abdou (from now on EEA) in [S.A. El-Wakil, A. Elhanbaly, M.A. Abdou, Physica A 323 (2003) 213] for solving approximately the collisional Vlasov equation actually can provide exact solutions if properly implemented. We consider here two alternative procedures for obtaining exact maximum entropy solutions of the aforementioned equation. On the one hand, after identifying an appropriate set of relevant mean values (moments), we show that there are exact maximum entropy solutions associated with that set of moments. These solutions can be studied focusing either on the equations of motion of the moments themselves, or on the equations of motion of the corresponding Lagrange multipliers. On the other hand, it is possible to find exact solutions of the reduced equation considered by EEA, if one takes explicitly into account the zeroth-order moment of the solutions.     

New analytical solutions for nonlinear physical models of the coupled Higgs equation and the Maccari system via rational exp <Emphasis Type="BoldItalic">(−φ(η))</Emphasis>-expansion method

In this article, a variety of solitary wave solutions are found for some nonlinear equations. In mathematical physics, we studied two complex systems, the Maccari system and the coupled Higgs field equation. We construct sufficient exact solutions for nonlinear evolution equations. To study travelling wave solutions, we used a fractional complex transform to convert the particular partial differential equation of fractional order into the corresponding partial differential equation and the rational exp (?φ(η))-expansion method is implemented to find exact solutions of nonlinear equation. We find hyperbolic, trigonometric, rational and exponential function solutions using the above equation. The results of various studies show that the suggested method is very effective and can be used as an alternative for finding exact solutions of nonlinear equations in mathematical physics. A comparative study with the other methods gives validity to the technique and shows that the method provides additional solutions. Graphical representations along with the numerical data reinforce the efficacy of the procedure used. The specified idea is very effective, pragmatic for partial differential equations of fractional order and could be protracted to other physical phenomena.     

Exact solutions of the generalized Bretherton equation

The generalized Bretherton equation is studied. The Bäcklund transformations between traveling wave solutions of the generalized Bretherton equation and solutions of polynomial ordinary differential equation are constructed. The classification problem for meromorphic solutions of the latter equation is discussed. Several new families of exact solutions for the generalized Brethenton equation are given.     

A new auxiliary equation method and its application to the Sharma-Tasso-Olver model

A new auxiliary equation method, constructed by a first order nonlinear ordinary differential equation with at most an eighth-degree nonlinear term, is first proposed for exploring more exact solutions to nonlinear evolution equations. Being concise and straightforward, the method, with the aid of symbolic computation, is applied to the Sharma-Tasso-Olver model, and some new exact solitary wave solutions are obtained. The approach is also applicable to searches for exact solutions of other nonlinear evolution equations.     

第二类变系数KdV方程的新类型无穷序列精确解

为了构造变系数非线性发展方程的无穷序列新精确解, 发掘第一种椭圆辅助方程的构造性和机械化性特点, 获得了该方程的 新类型解和相应的 Bäcklund 变换. 在符号计算系统 Mathematica 的帮助下, 以第二类变系数 KdV 方程为应用实例, 构造了三种类型的无穷序列新精确解. 这里包括无穷序列光滑类孤子解、无穷序列尖峰孤立子解和无穷序列紧孤立子解. 这种方法也可以获得其他变系数非线性发展方程的无穷序列新精确解.