Bifurcation of travelling wave solutions for (2+1)-dimension nonlinear dispersive long wave equation

In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.

Bifurcations, Analytical and Non-Analytical Traveling Wave Solutions of (2 + 1)-Dimensional Nonlinear Dispersive Boussinesq Equation

For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.

Exact solutions of (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq equations

The symmetries and the exact solutions of the （3＋l）-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq （INHB） equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the （3＋ 1）-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the （3＋1）-dimensional INHB system.

Truncated series solutions to the(2+1)-dimensional perturbed Boussinesq equation by using the approximate symmetry method

In this paper, the（2＋1）-dimensional perturbed Boussinesq equation is transformed into a series of two-dimensional（2 D） similarity reduction equations by using the approximate symmetry method. A step-by-step procedure is used to acquire Jacobi elliptic function solutions to these similarity equations, which generate the truncated series solutions to the original perturbed Boussinesq equation. Aside from some singular area, the series solutions are convergent when the perturbation parameter is diminished.

(2+1)-dimensional dissipation nonlinear Schrdinger equatio for envelope Rossby solitary waves and chirp effect

In the past few decades, the （1 ＋ 1）-dimensional nonlinear Schr6dinger （NLS） equation had been derived for envelope Rossby solitary waves in a line by employing the perturbation expansion method. But, with the development of theory, we note that the （1＋1）-dimensional model cannot reflect the evolution of envelope Rossby solitary waves in a plane. In this paper, by constructing a new （2＋1）-dimensional multiscale transform, we derive the （2＋1）-dimensional dissipation nonlinear Schrodinger equation （DNLS） to describe envelope Rossby solitary waves under the influence of dissipation which propagate in a plane. Especially, the previous researches about envelope Rossby solitary waves were established in the zonal area and could not be applied directly to the spherical earth, while we adopt the plane polar coordinate and overcome the problem. By theoretical analyses, the conservation laws of （2＋ 1）-dimensional envelope Rossby solitary waves as well as their variation under the influence of dissipation are studied. Finally, the one-soliton and two-soliton solutions of the （2＋ 1）-dimensional NLS equation are obtained with the Hirota method. Based on these solutions, by virtue of the chirp concept from fiber soliton communication, the chirp effect of envelope Rossby solitary waves is discussed, and the related impact factors of the chirp effect are given.

Symbolic Computation and New Exact Travelling Solutions for the (2+1)-Dimensional Zoomeron Equation

In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbolic computation. The study confirms the power of the two schemes.

Periodic Wave Solutions and Solitary Wave Solutions of the (2+1)-Dimensional Korteweg-de-Vries Equatio

In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different coefficients p, q and r in the elliptic equation. Then these solutions are coupled into an auxiliary equation and substituted into the (2+1)-dimensional KDV equation. As a result, a large number of complex Jacobi elliptic function solutions are obtained, and many of them have not been found in other documents. As , some complex solitary solutions are also obtained correspondingly. These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.

Searching For(2+1)-dimensional nonlinear Boussinesq equation from(1+1)-dimensional nonlinear Boussinesq equation

A novel(2+1)-dimensional nonlinear Boussinesq equation is derived from a(1+1)-dimensional Boussinesq equation in nonlinear Schr?dinger type based on a deformation algorithm.The integrability of the obtained(2+1)-dimensional Boussinesq equation is guaranteed by its Lax pair obtained directly from the Lax pair of the(1+1)-dimensional Boussinesq equation.Because of the effects of the deformation,the(2+1)-dimensional Boussinesq equation admits a special travelling wave solution with a shape that can be deformed to be asymmetric and/or multivalued.

广义(2+1)维Boussinesq方程的新的椭圆函数有理形式解

基于符号计算软件Maple和椭圆方程,提出构造非线性发展方程有理形式解的改进的椭圆方程展开法,该方法可有效地构造出更多新的椭圆函数形式解.利用该方法研究广义(2+1)维Boussinesq方程并获得该方程的一系列新的精确解.

Nonclassical Symmetries for Nonlinear Partial Differential Equations via Compatibility

为有任意的顺序的非线性的部分微分方程的 nonclassical 对称减小的决定方程能被要求在原来的方程和不变的表面条件之间的相容性获得。(2+1 ) 维的浅水波浪方程， Boussinesq 方程，和浅水里的散波浪方程用作说明相容性怎么为他们的 nonclassical 对称快速并且容易带到决定方程的例子。

Multiple exp-function method for soliton solutions of nonlinear evolution equations

We applied the multiple exp-function scheme to the（2＋1）-dimensional Sawada-Kotera（SK） equation and（3＋1）-dimensional nonlinear evolution equation and analytic particular solutions have been deduced. The analytic particular solutions contain one-soliton, two-soliton, and three-soliton type solutions. With the assistance of Maple, we demonstrated the efficiency and advantages of the procedure that generalizes Hirota＇s perturbation scheme. The obtained solutions can be used as a benchmark for numerical solutions and describe the physical phenomena behind the model.

Symmetry Analysis and Conservation Laws to the(2+1)-Dimensional Coupled Nonlinear Extension of the Reaction-Diffusion Equation

In this paper, a detailed Lie symmetry analysis of the(2+1)-dimensional coupled nonlinear extension of the reaction-diffusion equation is presented. The general finite transformation group is derived via a simple direct method,which is equivalent to Lie point symmetry group actually. Similarity reduction and some exact solutions of the original equation are obtained based on the optimal system of one-dimensional subalgebras. In addition, conservation laws are constructed by employing the new conservation theorem.

(2+1)-dimensional coupled Boussinesq equations for Rossby waves in two-layer cylindrical fluid

In this paper,the existence and propagation characteristics of Rossby waves in a two-layer cylindrical fluid are studied.Firstly,based on the dimensionless baroclinic quasi-geostrophic vortex equations including exogenous and dissipative,we derive new(2+1)-dimensional coupled Boussinesq equations describing wave propagation in polar coordinates by employing a multiscale analysis and perturbation method.Then,the Lie symmetries and conservation laws of the coupled Boussinesq equations are analyzed.Subsequently,by using the(G’/G)-expansion method,the exact solutions of the(2+1)-dimensional coupled Boussinesq equations are obtained.Finally,the effects of coupling term coefficients on the propagation characteristics of Rossby waves are analyzed.

Soliton and other solutions to the (1+2)-dimensional chiral nonlinear Schrodinger equation

The(1+2)-dimensional chiral nonlinear Schr?dinger equation(2D-CNLSE)as a nonlinear evolution equation is considered and studied in a detailed manner.To this end,a complex transform is firstly adopted to arrive at the real and imaginary parts of the model,and then,the modified Jacobi elliptic expansion method is formally utilized to derive soliton and other solutions of the 2D-CNLSE.The exact solutions presented in this paper can be classified as topological and nontopological solitons as well as Jacobi elliptic function solutions.

Multiple-order rogue wave solutions to a(2+1)-dimensional Boussinesq type equation

In this paper,based on the Hirota bilinear method and symbolic computation approach,multipleorder rogue waves of(2+1)-dimensional Boussinesq type equation are constructed.The reduced bilinear form of the equation is deduced by the transformation of variables.Three kinds of rogue wave solutions are derived by means of bilinear equation.The maximum and minimum values of the first-order rogue wave solution are given at a specific moment.Furthermore,the second-order and third-order rogue waves are explicitly derived.The dynamic characteristics of three kinds of rogue wave solutions are shown by three-dimensional plot.

SITEM for the conformable space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations

In the present paper,new analytical solutions for the space-time fractional Boussinesq and(2+1)-dimensional breaking soliton equations are obtained by using the simplified tan(φ(ξ)2)-expansion method.Here,fractional derivatives are defined in the conformable sense.To show the correctness of the obtained traveling wave solutions,residual error function is defined.It is observed that the new solutions are very close to the exact solutions.The solutions obtained by the presented method have not been reported in former literature.

Modulation instability analysis of Rossby waves based on(2+1)-dimensional high-order Schrodinger equation

Modulational instability is an important area of research with important practical and theoretical significance in fluid mechanics,optics,plasma physics,and military and communication engineering.In this paper,using multiscale analysis and a perturbation expansion method,starting from the quasi-geostrophic potential vortex equation,a new(2+1)-dimensional highorder nonlinear Schrodinger equation describing Rossby waves in stratified fluids is obtained.Based on this equation,conditions for the occurrence of modulational instability of Rossby waves are analyzed.Moreover,the effects of factors such as the dimension and order of the equation and the latitude at which Rossby waves occur on modulational instability are discussed.It is found that the(2+1)-dimensional equation provides a good description of the modulational instability of Rossby waves on a plane.The high-order terms affect the modulational instability,and it is found that instability is more likely to occur at high latitudes.

Dynamics of optical rogue waves in inhomogeneous nonlinear waveguides

We propose a unified theory to construct exact rogue wave solutions of the （2＋1）-dimensional nonlinear Schr6dinger equation with varying coefficients. And then the dynamics of the first- and the second-order optical rogues are investigated. Finally, the controllability of the optical rogue propagating in inhomogeneous nonlinear waveguides is discussed. By properly choosing the distributed coefficients, we demonstrate analytically that rogue waves can be restrained or even be annihilated, or emerge periodically and sustain forever. We also figure out the center-of-mass motion of the rogue waves.

Lie symmetry analysis and invariant solutions for(2+1) dimensional Bogoyavlensky-Konopelchenko equation with variable-coefficient in wave propagation

This work aims to present nonlinear models that arise in ocean engineering.There are many models of ocean waves that are present in nature.In shallow water,the linearization of the equations requires critical conditions on wave capacity than it make in deep water,and the strong nonlinear belongings are spotted.We use Lie symmetry analysis to obtain different types of soliton solutions like one,two,and three-soliton solutions in a(2+1)dimensional variable-coefficient Bogoyavlensky Konopelchenko(VCBK)equation that describes the interaction of a Riemann wave reproducing along the y-axis and a long wave reproducing along the x-axis in engineering and science.We use the Lie symmetry analysis then the integrating factor method to obtain new solutions of the VCBK equation.To demonstrate the physical meaning of the solutions obtained by the presented techniques,the graphical performance has been demonstrated with some values.The presented equation has fewer dimensions and is reduced to ordinary differential equations using the Lie symmetry technique.

二维色散长波方程组的Backlund变换及其精确解

利用齐次平衡法推导出二维色散长波方程组的Backlund变换,并根据Backlund变换构造出4种精确解,其中包括多孤子解、双曲函数和指数函数混合型解、三角函数和指数函数混合型解等.