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.     

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

The symmetries and the exact solutions of the (3+1)-dimensional nonlinear incompressible non-hydrostatic Boussinesq (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.     

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.     

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     

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.     

Bosonized Supersymmetric Sawada–Kotera Equations: Symmetries and Exact Solutions

The Bosonized Supersymmetric Sawada–Kotera(BSSK) system is constructed by applying bosonization method to a Supersymmetric Sawada–Kotera system in this paper. The symmetries on the BSSK equations are researched and the calculation shows that the BSSK equations are invariant under the scaling transformations, the space-time translations and Galilean boosts. The one-parameter invariant subgroups and the corresponding invariant solutions are researched for the BSSK equations. Four types of reduction equations and similarity solutions are proposed. Period Cnoidal wave solutions, dark solitary wave solutions and bright solitary wave solutions of the BSSK equations are demonstrated and some evolution curves of the exact solutions are figured out.     

Approximate Similarity Reduction for Perturbed Kaup-Kupershmidt Equation via Lie Symmetry Method and Direct Method

The perturbed Kaup-Kupershmidt equation is investigated in terms of the approximate symmetry perturbation method and the approximate direct method. The similarity reduction solutions of different orders are obtained for both methods, series reduction solutions are consequently derived. Higher order similarity reduction equations are linear variable coefficients ordinary differential equations. By comparison, it is find that the results generated from the approximate direct method are more general than the results generated from the approximate symmetry perturbation method.     

Similarity structure of wave-collapse

Similarity transformations of the cubic Schrödinger equation (CSE) are investigated. The transformations are used to remove the explicit time variation in the CSE and reduce it to differential equations in the spatial variables only. Two different methods for similarity reduction are employed and the significance of similarity in the evolution of a collapsing wave packet is investigated. Numerical solutions in radial symmetry demonstrate that the similarity behaviour is local in space and time, and that some similarity solutions must be classified as improper solutions. The nature of the collapsing singularity is reexamined.     

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.     

Zakharov方程的显式行波解

借助Mathematica软件，采用双函数法和吴文俊消元法，获得了等离子体物理中的重要方程组Zakharov方程的十组行波解，其中包括包络孤波解，孤子解. 关键词： Zakharov方程 孤子解     

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     

Symmetry violating trial wave functions

A variational procedure using trial wave functions projected on to symmetry eigenstates is investigated. In the individual particle model it leads to generalized self-consistent single-particle equations being dependent on the symmetry eigenvalue. A perturbation expansion appropriate for the case of strong symmetry violations is described. Model independent expressions for the moment of inertia and the decoupling parameter are derived by applying the method to rotational symmetry.     

Lie symmetry analysis and reduction of a new integrable coupled KdV system

In this paper, Lie symmetry is investigated for a new integrable coupled Korteweg--de Vries (KdV) equation system. Using some symmetry subalgebra of the equation system, we obtain five types of the significant similarity reductions. Abundant solutions of the coupled KdV equation system, such as the solitary wave solution, exponential solution, rational solution and polynomial solution, etc. are obtained from the reduced equations. Especially, one type of group-invariant solution of reduced equations can be acquired by means of the Painlev\'e I transcendent function.     

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.     

Dynamical Analysis and Exact Solutions of a New (2+1)-Dimensional Generalized Boussinesq Model Equation for Nonlinear Rossby Waves*

In this paper, we study the higher dimensional nonlinear Rossby waves under the generalized beta effect. Using methods of the multiple scales and weak nonlinear perturbation expansions [Q. S. Liu, et al., Phys. Lett. A 383 (2019) 514], we derive a new $(2+1)$-dimensional generalized Boussinesq equation from the barotropic potential vorticity equation. Based on bifurcation theory of planar dynamical systems and the qualitative theory of ordinary differential equations, the dynamical analysis and exact traveling wave solutions of the new generalized Boussinesq equation are obtained. Moreover, we provide the numerical simulations of these exact solutions under some conditions of all parameters. The numerical results show that these traveling wave solutions are all the Rossby solitary waves.     

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.     

变系数非线性方程的Jacobi椭圆函数展开解

把Jacobi椭圆函数展开法扩展并应用到求解变系数的非线性演化方程,比较方便地得到新的解析解 关键词： Jacobi椭圆函数 变系数非线性方程 类椭圆余弦波解 类孤子解     

Extended Complex tanh-Function Method and Exact Solutions to (2+1)-Dimensional Hirota Equation

In this paper,a new extended complex tanh-function method is presented for constructing traveling wave,non-traveling wave,and coefficient functions' soliton-like solutions of nonlinear equations.This method is nore powerful than the complex tanh-function method [Chaos,Solitons and Fractals 20 (2004) 1037].Abundant new solutions of (2 1)-dimensional Hirota equation are obtained by using this method and symbolic computation system Maple.     

Linear perturbation amplification in self-similar ablation flows of inertial confinement fusion

Exact similarity solutions for inviscid compressible ablative flows in slab symmetry with nonlinear heat conduction are proposed for studying unsteadiness and compressibility effects on the hydrodynamic stability of ablation fronts relevant to inertial confinement fusion. Both the similarity solutions and their linear perturbations are numerically computed with a dynamical multidomain Chebyshev pseudospectral method. Herewith the first analysis of laser-imprinting based on a dynamic solution is presented, showing that maximum perturbation amplification occurs for a laser-intensity modulation of zero transverse wave number, with growth dominated by the mean flow stretching.