Dynamics of solitons of the generalized(3+1)-dimensional nonlinear Schr(o|¨)dinger equation with distributed coefficients

We present three families of soliton solutions to the generalized （3＋l）-dimensional nonlinear Schrodinger equation with distributed coefficients. We investigate the dynamics of these solitons in nonlinear optics with some selected parameters. Different shapes of bright solitons, a train of bright solitons and dark solitons are observed. The obtained results may raise the possibilities of relevant experiments and potential applications.

A more general form of lump solution,lumpoff,and instanton/rogue wave solutions of a reduced (3+1)-dimensional nonlinear evolution equation

In this manuscript,a reduced(3+1)-dimensional nonlinear evolution equation is studied.We first construct the bilinear formalism of the equation by using the binary Bell polynomials theory,then explore a lump solution to the special case for z=x.Furthermore,a more general form of lump solution of the equation is found which possesses seven arbitrary parameters and four constraint conditions.By cutting the lump by the induced soliton(s),lumpoff and instanton/rogue wave solutions are also constructed by the more general form of lump solution.

Extended symmetry transformation of (3+1)-dimensional generalized nonlinear Schrdinger equation with variable coefficients

In this paper, the extended symmetry transformation of （3＋1）-dimensional （3D） generalized nonlinear Schrodinger （NLS） equations with variable coefficients is investigated by using the extended symmetry approach and symbolic computation. Then based on the extended symmetry, some 3D variable coefficient NLS equations are reduced to other variable coefficient NLS equations or the constant coefficient 3D NLS equation. By using these symmetry transformations, abundant exact solutions of some 3D NLS equations with distributed dispersion, nonlinearity, and gain or loss are obtained from the constant coefficient 3D NLS 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.

Symmetry analysis and explicit solutions of the (3+1)-dimensional baroclinic potential vorticity equation

This paper investigates an important high-dimensional model in the atmospheric and oceanic dynamics-（3＋1）- dimensional nonlinear baroclinic potential vorticity equation by the classical Lie group method. Its symmetry algebra, symmetry group and group-invariant solutions are analysed. Otherwise, some exact explicit solutions are obtained from the corresponding （2＋1）-dimensional equation, the inviscid barotropic nondivergent vorticy equation. To show the properties and characters of these solutions, some plots as well as their possible physical meanings of the atmospheric circulation are given out.

Classification of All Single Traveling Wave Solutions to (3 + 1)-Dimensional Breaking Soliton Equation

In order to get the exact traveling wave solutions to nonlinear partial differential equation, the complete discrimination system for polynomial and direct integral method are applied to the considered equation. All single traveling wave solutions to the equation can be obtained. As an example, we give the solutions to (3 + 1)-dimensional breaking soliton equation.

New lump solutions and several interaction solutions and their dynamics of a generalized(3+1)-dimensional nonlinear differential equation

In this paper,we mainly focus on proving the existence of lump solutions to a generalized(3+1)-dimensional nonlinear differential equation.Hirota’s bilinear method and a quadratic function method are employed to derive the lump solutions localized in the whole plane for a(3+1)-dimensional nonlinear differential equation.Three examples of such a nonlinear equation are presented to investigate the exact expressions of the lump solutions.Moreover,the 3d plots and corresponding density plots of the solutions are given to show the space structures of the lump waves.In addition,the breath-wave solutions and several interaction solutions of the(3+1)-dimensional nonlinear differential equation are obtained and their dynamics are analyzed.

Spatiotemporal Similaritons in (3+l)-Dimensional Inhomogeneous Nonlinear Medium with Cubic-Quintic Nonlinearity

我们获得准确空间与时间的 similaritons 到一(3+1 ) 维的不同类的非线性的 Schr ? dinger 方程，它与分布式的分散和获得在 cubic-quintic 非线性媒介描述光脉搏的繁殖。当某个相容性条件满足时，在椭圆形的方程的如此的自我类似的波浪和答案之间的一对一的通讯被发现。基于准确解决方案，我们在二种典型 soliton 控制系统讨论自我类似的 cnoidal 波浪和啁啾的 similaritons 的进化行为。而且，在啁啾的 similaritons 之间的比较和啁啾免费的 solitons 被给。

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.

New Bilinear B?cklund Transformation and Higher Order Rogue Waves with Controllable Center of a Generalized(3+1)-Dimensional Nonlinear Wave Equation

In this paper, we first obtain a bilinear form with small perturbation u_0 for a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles. Based on that, a new bilinear B?cklund transformation which consists of four bilinear equations and involves seven arbitrary parameters is constructed. After that, by applying a new symbolic computation method, we construct the higher order rogue waves with controllable center to the generalized(3+1)-dimensional nonlinear wave equation. The rogue waves present new structure, which contain two free parametersα and β. The dynamic properties of the higher order rogue waves are demonstrated graphically. The graphs tell that the parameters α and β can control the center of the rogue waves.

Lump wave and hybrid solutions of a generalized(3+1)-dimensional nonlinear wave equation in liquid with gas bubbles

We investigate a generalized (3 + 1)-dimensional nonlinear wave equation, which can be used to depict many nonlinear phenomena in liquid containing gas bubbles. By employing the Hirota bilinear method, we derive its bilinear formalism and soliton solutions succinctly. Meanwhile, the first-order lump wave solution and second-order lump wave solution are well presented based on the corresponding two-soliton solution and four-soliton solution. Furthermore, two types of hybrid solutions are systematically established by using the long wave limit method. Finally, the graphical analyses of the obtained solutions are represented in order to better understand their dynamical behaviors.

Lump and new interaction solutions to the (3+1)-dimensional nonlinear evolution equation

In this paper,a new(3+1)-dimensional nonlinear evolution equation is introduced,through the generalized bilinear operators based on prime number p=3.By Maple symbolic calculation,one-,two-lump,and breather-type periodic soliton solutions are obtained,where the condition of positiveness and analyticity of the lump solution are considered.The interaction solutions between the lump and multi-kink soliton,and the interaction between the lump and breather-type periodic soliton are derived,by combining multi-exponential function or trigonometric sine and cosine functions with a quadratic one.In addition,new interaction solutions between a lump,periodic-solitary waves,and one-,two-or even three-kink solitons are constructed by using the ansatz technique.Finally,the characteristics of these various solutions are exhibited and illustrated graphically.

Rogue Waves and Lump Solitons of the(3+1)-Dimensional Generalized B-type Kadomtsev–Petviashvili Equation for Water Waves

In this paper, the(3+1)-dimensional generalized B-type Kadomtsev–Petviashvili equation for water waves is investigated. Through the Hirota method and Kadomtsev–Petviashvili hierarchy reduction, we obtain the first-order,higher-order, multiple rogue waves and lump solitons based on the solutions in terms of the Gramian. The first-order rogue waves are the line rogue waves which arise from the constant background and then disappear into the constant background again, while the first-order lump solitons propagate stably. Interactions among several first-order rogue waves which are described by the multiple rogue waves are presented. Elastic interactions of several first-order lump solitons are also presented. We find that the higher-order rogue waves and lump solitons can be treated as the superpositions of several first-order ones, while the interaction between the second-order lump solitons is inelastic.

Bifurcation Method to Analysis of Traveling Wave Solutions for(3+l)-Dimensional Nonlinear Models Generated by the Jaulent-Miodek Hierarchy

In this paper,the third model of four(3+1)-dimensional nonlinear evolution equations,generated by the Jaulent-Miodek hierarchy,is investigated by the bifurcation method of planar dynamical systems.The 2-parameters different bifurcation regions are obtained.According to the different phase portraits in 2-parameters different bifurcation regions,we obtain kink(anti-kink)wave solutions,solitary wave solutions and periodic wave solutions for the third of these models in the different subsets of 4-parameters space by dynamical system method.Furthermore,the explicit exact expressions of these bounded traveling waves are obtained.All these wave solutions are characterized by distinct physical structures.

(3+1)维非线性发展方程的显式解

本文利用推广的(W/G)展开法,研究(3+1)维非线性发展方程,并得到了很多该方程新的显式解,包括单循环孤立子解、三角周期解、有理函数解等.