Analytical studies on the Benney–Luke equation in mathematical physics

The enhanced (G′/G)-expansion method presents wide applicability to handling nonlinear wave equations. In this article, we find the new exact traveling wave solutions of the Benney–Luke equation by using the enhanced (G′/G)-expansion method. This method is a useful, reliable, and concise method to easily solve the nonlinear evaluation equations (NLEEs). The traveling wave solutions have expressed in term of the hyperbolic and trigonometric functions. We also have plotted the 2D and 3D graphics of some analytical solutions obtained in this paper.     

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.     

Linear superposition solutions to nonlinear wave equations

The solutions to a linear wave equation can satisfy the principle of superposition,i.e.,the linear superposition of two or more known solutions is still a solution of the linear wave equation.We show in this article that many nonlinear wave equations possess exact traveling wave solutions involving hyperbolic,triangle,and exponential functions,and the suitable linear combinations of these known solutions can also constitute linear superposition solutions to some nonlinear wave equations with special structural characteristics.The linear superposition solutions to the generalized KdV equation K(2,2,1),the Oliver water wave equation,and the k(n,n) equation are given.The structure characteristic of the nonlinear wave equations having linear superposition solutions is analyzed,and the reason why the solutions with the forms of hyperbolic,triangle,and exponential functions can form the linear superposition solutions is also discussed.     

Special solutions for nonlinear wave-particle interaction

A class of exact solutions for a coupled set of nonlinear equations describing the interaction between two propagating waves and a system of particles is found. These solutions include traveling wave solutions of the non-linear coupled equations.     

Wave Resonance in Media with Modular,Quadratic and Quadratically-Cubic Nonlinearities Described by Inhomogeneous Burgers-Type Equations

The phenomenon of “wave resonance” which occurs at excitation of traveling waves in dissipative media possessing modular, quadratic and quadratically-cubic nonlinearities is studied. The mathematical model of this phenomenon is the inhomogeneous (or “forced”) equation of Burgers type. Such nonlinearities are of interest because the corresponding equations admit exact linearization and describe real physical objects. The presence of “accompanying sources” (traveling with the wave) on the right-hand side of the inhomogeneous equations ensures the inflow of energy into the wave, which thereafter spreads throughout the wave profile, flows to emerging shock fronts, and then dissipates due to linear and nonlinear losses. As an introduction, the phenomenon of wave resonance in ideal and dissipative media is described and physical examples are given. Exact expressions for nonlinear steady-state wave profiles are derived. Non-stationary processes of wave generation, spatial “beating” of amplitudes with different relationship between the speed of motion of the sources and the natural wave velocity in the medium are studied. Resonance curves are constructed that contain a nonlinear shift of the absolute maxima to the “supersonic” region. The features of the resonance in each of the three types of nonlinearity are discussed.     

柱面非线性麦克斯韦方程组的行波解

柱面电磁波在各种非均匀非线性介质中的传播问题具有非常重要的研究价值.对描述该问题的柱面非线性麦克斯韦方程组进行精确求解,则是最近几年新兴的研究热点.但由于非线性偏微分方程组的极端复杂性,针对任意初边值条件的精确求解在客观上具有极高的难度,已有工作仅解决了柱面电磁波在指数非线性因子的非色散介质中的传播情况.因此,针对更为确定的物理场景,寻求能够精确描述其中更为广泛的物理性质的解,是一种更为有效的处理方法.本文讨论了具有任意非线性因子与幂律非均匀因子的非色散介质中柱面麦克斯韦方程组的行波精确解,理论分析表明这种情况下柱面电磁波的电场分量E已不存在通常形如E=g(r-kt)的平面行波解;继而通过适当的变量替换与求解相应的非线性常微分方程,给出电场分量E=g(lnr-kt)形式的广义行波解,并以例子展示所得到的解中蕴含的类似于自陡效应的物理现象.     

Analytic study of sixth-order thin-film equation by tan(<Emphasis Type="BoldItalic">ϕ</Emphasis>/2)-expansion method

A improvement of the expansion methods, namely, the improved \(\tan (\phi (\xi )/2)\)-expansion method for solving the sixth-order thin-film equation is proposed. As a result, many new and more general exact traveling wave solutions are obtained including singular kink-type solutions. We obtained the further solutions comparing with other methods as Flitton and King (Eur J Appl Math 15:713–754, 2004) and Taha et al. (J King Saud Univ Sci 26:75–78, 2014). Recently this method is developed for searching exact traveling wave solutions of nonlinear partial differential equations. Abundant exact traveling wave solutions including kink and rational solutions have been found. These solutions might play important role in engineering and physics fields. Also the results demonstrate that the introduced method is powerful tools for solving the nonlinear partial differential equations.     

A Class of Traveling Wave Solutions to Some Nonlinear Partial Differential Equations

For the Noyes-Fields equations, two-dimensional hyperbolic equations of conversation laws, and theBurgers-KdV equation, a class of traveling wave solutions has been obtained by constructing appropriate functiontransformations. The main idea of solving the equations is that nonlinear partial differential equations are changed intosolving algebraic equations. This method has a wide-rangingpracticability.     

Exact Solutions of Atmospheric (2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric (2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

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.     

试探方程法及其在非线性发展方程中的应用

提出了一种比较系统的求解非线性发展方程精确解的新方法, 即试探方程法. 以一个带5阶 导数项的非线性发展方程为例, 利用试探方程法化成初等积分形式，再利用三阶多项式的完 全判别系统求解，由此求得的精确解包括有理函数型解, 孤波解, 三角函数型周期解, 多项 式型Jacobi椭圆函数周期解和分式型Jacobi椭圆函数周期解 关键词： 试探方程法 非线性发展方程 孤波解 Jacobi椭圆函数 周期解     

A new approach for seeking coefficient function solutions of conformable fractional partial differential equations based on the Jacobi elliptic equation

In this paper, we present an approach for seeking exact solutions with coefficient function forms of conformable fractional partial differential equations. By a combination of an under-determined fractional transformation and the Jacobi elliptic equation, exact solutions with coefficient function forms can be obtained for fractional partial differential equations. The innovation point of the present approach lies in two aspects. One is the fractional transformation, which involve the traveling wave transformations used by many articles as special cases. The other is that more general exact solutions with coefficient function forms can be found, and traveling wave solutions with constants coefficients are only special cases of our results. As of applications, we apply this method to the space-time fractional (2+1)-dimensional dispersive long wave equations and the time fractional Bogoyavlenskii equations. As a result, some exact solutions with coefficient function forms for the two equations are successfully found.     

Exact Solutions of Atmospheric(2+1)-Dimensional Nonlinear Incompressible Non-hydrostatic Boussinesq Equations

Exact solutions of the atmospheric(2+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq(INHB) equations are researched by Combining function expansion and symmetry method. By function expansion, several expansion coefficient equations are derived. Symmetries and similarity solutions are researched in order to obtain exact solutions of the INHB equations. Three types of symmetry reduction equations and similarity solutions for the expansion coefficient equations are proposed. Non-traveling wave solutions for the INHB equations are obtained by symmetries of the expansion coefficient equations. Making traveling wave transformations on expansion coefficient equations, we demonstrate some traveling wave solutions of the INHB equations. The evolutions on the wind velocities, temperature perturbation and pressure perturbation are demonstrated by figures, which demonstrate the periodic evolutions with time and space.     

非线性Schroedinger方程组的精确解组

运用Maple语言程序，在没有假设的条件下，得到了具有耦合特性的非线性Schrfidinger方程组的行波精确解组及其约束条件方程，它们的表达式涵盖了所有的耦合解组与非耦合解组，具有任意性。耦合解组的算例函数及其特性分析，解释了α螺旋蛋白质螺旋链运动模型的行波孤立子解的耦合效应，揭示了增加、稳定和控制蛋白质活性和功能的方向。文章的研究方法，为求解耦合的非线性微分方程组的行波精确解组探索了蹊径。     

A note on the homogeneous balance method

Based on the idea of the homogeneous balance method, a simple and efficient method is proposed for obtaining exact solutions of nonlinear partial differential equations. Some equations are investigated by this means and new solitary wave solutions or singular traveling wave solutions are found.     

一般变换下的Jacobi椭圆函数展开法及应用

将在行波变换下的Jacobi椭圆函数展开法推广到范围非常广泛的一般函数变换下进行，利用这一方法求得了一些非线性发展方程的精确周期解，这些解包括了在行波变换下所求得的周期解. 证明了一些非线性发展方程的周期解一定是行波解. 关键词： 非线性发展方程 周期解 行波解 Jacobi椭圆函数     

Existence and Dynamics of Bounded Traveling Wave Solutions to Getmanou Equation

In this paper, we study the existence and dynamics of bounded traveling wave solutions to Getmanou equations by using the qualitative theory of differential equations and the bifurcation method of dynamical systems. We show that the corresponding traveling wave system is a singular planar dynamical system with two singular straight lines, and obtain the bifurcations of phase portraits of the system under different parameters conditions. Through phase portraits, we show the existence and dynamics of several types of bounded traveling wave solutions including solitary wave solutions, periodic wave solutions, compactons, kink-like and antikink-like wave solutions. Moreover, the expressions of solitary wave solutions are given. Additionally, we confirm abundant dynamical behaviors of the traveling wave s olutions to the equation, which are summarized as follows: i) We confirm that two types of orbits give rise to solitary wave solutions, that is, the homoclinic orbit passing the singular point, and the composed homoclinic orbit which is comprised of two heteroclinic orbits and tangent to the singular line at the singular point of associated system. ii) We confirm that two types of orbits correspond to periodic wave solutions, that is, the periodic orbit surrounding a center, and the homoclinic orbit of associated system, which is tangent to the singular line at the singular point of associated system.     

Basin boundaries in asymmetric vibrations of a circular plate

In order to investigate further nonlinear asymmetric vibrations of a clamped circular plate under a harmonic excitation, we reexamine a primary resonance, studied by Yeo and Lee [Corrected solvability conditions for non-linear asymmetric vibrations of a circular plate, Journal of Sound and Vibration 257 (2002) 653-665] in which at most three stable steady-state responses (one standing wave and two traveling waves) are observed to exist. Further examination, however, tells that there exist at most five stable steady-state responses: one standing wave and four traveling waves. Two of the traveling waves lose their stability by Hopf bifurcation and have a sequence of period-doubling bifurcations leading to chaos. When the system has five attractors: three equilibrium solutions (one standing wave and two traveling waves) and two chaotic attractors (two modulated traveling waves), the basin boundaries of the attractors on the principal plane are obtained. Also examined is how basin boundaries of the modulated motions (quasi-periodic and chaotic motions) evolve as a system parameter varies. The basin boundaries of the modulated motions turn out to have the fractal nature.     

Exact traveling wave solutions of an autonomous system via the enhanced (G′/G)-expansion method

Mathematical modeling of many autonomous physical systems leads to nonlinear evolution equations because most physical systems are inherently nonlinear in nature. The investigation of traveling wave solutions of nonlinear evolution equations plays a significant role in the study of nonlinear physical phenomena. In this article, the enhanced (G′/G)-expansion method has been applied for finding the exact traveling wave solutions of longitudinal wave motion equation in a nonlinear magneto-electro-elastic circular rod. Each of the obtained solutions contains an explicit function of the variables in the considered equations. It has been shown that the applied method provides a powerful mathematical tool for solving nonlinear wave equations in mathematical physics and engineering fields.     

Nonlinear trans-resonant waves,vortices and patterns: From microresonators to the early Universe

Perturbed wave equations are considered. Approximate general solutions of these equations are constructed, which describe wave phenomena in different physical and chemical systems. Analogies between surface waves, nonlinear and atom optics, field theories and acoustics of the early Universe can be seen in the similarities between the general solutions that govern each system. With the help of the general solutions and boundary conditions and/or resonant conditions we have derived the basic highly nonlinear ordinary differential equation or the basic algebraic equation for traveling waves. Then, approximate analytic resonant solutions are constructed, which describe the trans-resonant transformation of harmonic waves into traveling shock-, jet-, or mushroom-like waves. The mushroom-like waves can evolve into cloud-like and vortex-like structures. The motion and oscillations of these waves and structures can be very complex. Under parametric excitation these waves can vary their velocity, stop, and change the direction of their motion. Different dynamic patterns are yielded by these resonant traveling waves in the x-t and x-y planes. They simulate many patterns observed in liquid layers, optical systems, superconductors, Bose-Einstein condensates, micro- and electron resonators. The harmonic excitation may be compressed and transformed inside the resonant band into traveling or standing particle-like waves. The area of application of these solutions and results may possibly vary from the generation of nuclear particles, acoustical turbulence, and catastrophic seismic waves to the formation of galaxies and the Universe. In particular, the formation of galaxies and galaxy clusters may be connected with nonlinear and resonant phenomena in the early Universe. (c) 2001 American Institute of Physics.