Lump,lumpoff and predictable rogue wave solutions to the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation

This paper mainly uses Hirota bilinear form to investigate the (2+1)-dimensional asymmetrical Nizhnik-Novikov-Veselov equation. We obtain the general lump solutions and discuss its positiveness, the propagation path, amplitude and position at any time. Based on the general lump solutions, lumpoff solutions which a combination of lump solitons and stripe solitons, are also triumphantly acquired. Similarly, according to the general lump solutions, we are also consider a particular rogue wave by introducing a pair of stripe solitons, and research its predictability which include the time of the rogue wave appearance, position at time, propagation path and the maximum value of wave height. Finally, some figures are given to explain the movement mechanism of these solutions.     

Lump and interaction solutions to the (3+1)-dimensional Burgers equation

The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.     

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.     

Multiple rogue wave solutions for (2+1)-dimensional Boussinesq equation

In this paper, a modified symbolic computation approach is proposed. The multiple rogue wave solutions of a generalized (2+1)-dimensional Boussinesq equation are obtained by using the modified symbolic computation approach. Dynamics features of these obtained multiple rogue wave solutions are displayed in 3D and contour plots. Compared with the original symbolic computation approach, our method does not need to find Hirota bilinear form of nonlinear system.     

扩展的(2+1)维浅水波方程的尖峰孤子解及其相互作用

利用改进的 Riccati方程映射法和变量分离法, 得到了扩展的(2+1)维浅水波方程的变量分离解(包括孤波解, 周期波解和有理函数解). 根据得到的孤波解, 构造出了方程的几种不同形状的尖峰孤子结构, 研究了孤子的相互作用.     

Multiple rogue wave solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation

In this paper, a (3+1)-dimensional generalized Kadomtsev-Petviashvili Benjamin-Bona-Mahony equation are investigated. The first-order, second-order and third-order rogue wave solutions of this equation are derived based on a symbolic computation approach. Their dynamics features are shown in some 3D and contour plots. Compared with the previous literatures, our work does not require the Hirota bilinear form of the equation.     

Novel interaction phenomena of the(3+1)-dimensional Jimbo–Miwa equation

In this paper,we give the general interaction solution to the(3+1)-dimensional Jimbo–Miwa equation.The general interaction solution contains the classical interaction solution.As an example,by using the generalized bilinear method and symbolic computation by using Maple software,novel interaction solutions under certain constraints of the(3+1)-dimensional Jimbo–Miwa equation are obtained.Via three-dimensional plots,contour plots and density plots with the help of Maple,the physical characteristics and structures of these waves are described very well.These solutions greatly enrich the exact solutions to the(3+1)-dimensional Jimbo–Miwa equation found in the existing literature.     

Exact solutions of a(2+1)-dimensional extended shallow water wave equation

We give the bilinear form and n-soliton solutions of a(2+1)-dimensional [(2+1)-D] extended shallow water wave(eSWW) equation associated with two functions v and r by using Hirota bilinear method. We provide solitons, breathers,and hybrid solutions of them. Four cases of a crucial φ(y), which is an arbitrary real continuous function appeared in f of bilinear form, are selected by using Jacobi elliptic functions, which yield a periodic solution and three kinds of doubly localized dormion-type solution. The first order Jacobi-type solution travels parallelly along the x axis with the velocity(3k_1~2+ α, 0) on(x, y)-plane. If φ(y) = sn(y, 3/10), it is a periodic solution. If φ(y) = cn(y, 1), it is a dormion-type-Ⅰ solutions which has a maximum(3/4)k_1p_1 and a minimum-(3/4)k_1p_1. The width of the contour line is ln■. If φ(y) = sn(y, 1), we get a dormion-type-Ⅱ solution(26) which has only one extreme value-(3/2)k_1p_1. The width of the contour line is ln■. If φ(y) = sn(y, 1/2)/(1 + y~2), we get a dormion-type-Ⅲ solution(21) which shows very strong doubly localized feature on(x, y) plane. Moreover, several interesting patterns of the mixture of periodic and localized solutions are also given in graphic way.     

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, multiple-order 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.     

Lump waves,solitary waves and interaction phenomena to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation

In this paper, we investigated the $(2+1)$-dimensional Konopelchenko–Dubrovsky equation. The lump waves, solitary waves as well as interaction between lump waves and solitary waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the rational solutions are obtained by taking a long wave limit, and we also discussed the lump solutions and rogue wave solutions. Moreover, all these solutions are presented via 3-dimensional plots and density plots with choosing some special parameters to show the dynamic graphs.     

Lump,lumpoff and predictable rogue wave solutions to a dimensionally reduced Hirota bilinear equation

We study a simplified(3+1)-dimensional model equation and construct a lump solution for the special case of z=y using the Hirota bilinear method.Then,a more general form of lump solution is constructed,which contains more arbitrary autocephalous parameters.In addition,a lumpoff solution is also derived based on the general lump solutions and a stripe soliton.Furthermore,we figure out instanton/rogue wave solutions via introducing two stripe solitons.Finally,one can better illustrate these propagation phenomena of these solutions by analyzing images.     

Analytical solutions and rogue waves in (3+1)-dimensional nonlinear Schrödinger equation

Analytical solutions in terms of rational-like functions are presented for a (3+1)-dimensional nonlinear Schrödinger equation with time-varying coefficients and a harmonica potential using the similarity transformation and a direct ansatz. Several free functions of time t are involved to generate abundant wave structures. Three types of elementary functions are chosen to exhibit the corresponding nonlinear rogue wave propagations.     

Multiwave interaction solutions for a (3+1)-dimensional generalized Kadomtsev-Petviashvili equation

To construct a class of new multiwave interaction solutions for the (3+1)-dimensional generalized Kadomtsev-Petviashvili equation, we calculate different types of interaction solutions among solitons, periodic waves and rational waves using the direct algebraic method together with the inheritance solving skill. Moreover, a new algorithm is proposed with the aid of the simplified Hirota method, the conjugated parameters assignment and long wave limit strategies, from which multiwave interaction solutions among solitons, breathers and lump waves are generated.     

Exact travelling wave solutions for （1＋ 1）-dimensional dispersive long wave equation

A complete discrimination system for the fourth order polynomial is given. As an application, we have reduced a （1＋1）-dimensional dispersive long wave equation with general coefficients to an elementary integral form and obtained its all possible exact travelling wave solutions including rational function type solutions, solitary wave solutions, triangle function type periodic solutions and Jacobian elliptic functions double periodic solutions. This method can be also applied to many other similar problems.     

Novel methods for finding general forms of new multi-soliton solutions to (1+1)-dimensional KdV equation and (2+1)-dimensional Kadomtsev–Petviashvili (KP) equation

In this paper, two novel methods used to solve (1+1) and (2+1)-dimensional completely integrable equations are proposed. The methods are applied to handle the KdV and Kadomtsev–Petviashvili (KP) equations with variable coefficients, and the general forms of new multi-soliton solutions are formally obtained, respectively. In addition, the new multi-soliton solution is suitable to two different type KP equations. Comparing with the Hirota’s method, the results show that new methods are straightforward handling the KdV and KP equations without conjecturing the transformation and good in dealing the equations with variable coefficients.     

Quasi-periodic wave solutions,soliton solutions,and integrability to a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation

In this paper, a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko (gBK) equation is investigated, which can be used to describe the interaction of a Riemann wave propagating along y-axis and a long wave propagating along x-axis. The complete integrability of the gBK equation is systematically presented. By employing Bell’s polynomials, a lucid and systematic approach is proposed to systematically study its bilinear formalism, bilinear Bäcklund transformations, Lax pairs, respectively. Furthermore, based on multidimensional Riemann theta functions, the periodic wave solutions and soliton solutions of the gBK equation are derived. Finally, an asymptotic relation between the periodic wave solutions and soliton solutions are strictly established under a certain limit condition.     

(2+1)维耗散长波方程的变量分离解

借助Mathematica符号计算软件,利用拓展的F/G展开法和变量分离法,得到(2+1)维耗散长波方程的精确解.通过选择适当的函数,获得(2+1)维耗散长波方程的亮暗dromion解和周期孤波解.     

New lump,lump-kink,breather waves and other interaction solutions to the (3+1)-dimensional soliton equation

This study investigates the (3+1)-dimensional soliton equation via the Hirota bilinear approach and symbolic computations. We successfully construct some new lump, lump-kink, breather wave, lump periodic, and some other new interaction solutions. All the reported solutions are verified by inserting them into the original equation with the help of the Wolfram Mathematica package. The solution's visual characteristics are graphically represented in order to shed more light on the results obtained. The findings obtained are useful in understanding the basic nonlinear fluid dynamic scenarios as well as the dynamics of computational physics and engineering sciences in the related nonlinear higher dimensional wave fields.