Lump and rogue wave solutions for the Broer-Kaup-Kupershmidt system

This paper retrieves lump solution for (2+1)-dimensional Broer-Kaup-Kupershmidt (BKK) system by the aid of Hirota bilinear method (HBM). We also obtain rogue wave solutions formed by the interaction of lump solution and a pair of stripe solitons. The dynamics of these solutions are figured out graphically by selecting suitable values to parameters.     

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.     

(2+1)维破裂孤子方程的新多孤子解

Hirota双线性方法是一种非常有效的直接方法，使得求解非线性演化方程的多孤子解转化为 代数求解.将这一方法进一步拓展，求得了(2+1)维破裂孤子方程的新多孤子解. 关键词： 双线性方法 多孤子解 (2+1)维破裂孤子方程     

High-Order Lump-Type Solutions and Their Interaction Solutions to a (3+1)-Dimensional Nonlinear Evolution Equation *

By means of the Hirota bilinear method and symbolic computation, high-order lump-type solutions and a kind of interaction solutions are presented for a (3+1)-dimensional nonlinear evolution equation. The high-order lump-type solutions of the associated Hirota bilinear equation are presented, which is a kind of positive quartic-quadratic-function solution. At the same time, the interaction solutions can also be obtained, which are linear combination solutions of quartic-quadratic-functions and hyperbolic cosine functions. Physical properties and dynamical structures of two classes of the presented solutions are demonstrated in detail by their graphs.      

Quasi-periodic solutions and asymptotic properties for the nonlocal Boussinesq equation

We construct the Hirota bilinear form of the nonlocal Boussinesq(nlBq) equation with four arbitrary constants for the first time. It is special because one arbitrary constant appears with a bilinear operator together in a product form. A straightforward method is presented to construct quasiperiodic wave solutions of the nl Bq equation in terms of Riemann theta functions. Due to the specific dispersion relation of the nl Bq equation, relations among the characteristic parameters are nonlinear, then the linear method does not work for them. We adopt the perturbation method to solve the nonlinear relations among parameters in the form of series. In fact, the coefficients of the governing equations are also in series form.The quasiperiodic wave solutions and soliton solutions are given. The relations between the periodic wave solutions and the soliton solutions have also been established and the asymptotic behaviors of the quasiperiodic waves are analyzed by a limiting procedure.     

Grammian and Pfaffian solutions as well as Pfaffianization for a (3+1)-dimensional generalized shallow water equation

Based on the Grammian and Pfaffian derivative formulae, Grammian and Pfaffian solutions are obtained for a (3+1)-dimensional generalized shallow water equation in the Hirota bilinear form. Moreover, a Pfaffian extension is made for the equation by means of the Pfaffianization procedure, the Wronski-type and Gramm-type Pfaffian solutions of the resulting coupled system are presented.     

Periodic-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation

2N line-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation can be presented by resorting to the Hirota bilinear method. By extending the real parameters into complex parameters, this paper obtains N periodic-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation from the 2N line-soliton solutions.     

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.     

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.     

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.     

High-order breather,M-kink lump and semi-rational solutions of potential Kadomtsev-Petviashvili equation

N-kink soliton and high-order synchronized breather solutions for potential Kadomtsev-Petviashvili equation are derived by means of the Hirota bilinear method,and the limit process of high-order synchronized breathers are shown.Furthermore,M-lump solutions are also presented by taking the long wave limit.Additionally,a family of semi-rational solutions with elastic collision are generated by taking a long-wave limit of only a part of exponential functions,their interaction behaviors are shown by three-dimensional plots and contour plots.     

Exact Solutions for a Nonisospectral and Variable-Coefficient KdV Equation

The bilinear form for a nonisospectral and variable-coefficient KdV equation is obtained and some exact soliton solutions are derived through Hirota method and Wronskian technique. We also derive the bilinear transformation from its Lax pairs and find solutions with the help of the obtained bilinear transformation.     

Exact Solutions for a Nonisospectral and Variable-Coefficient Kadomtsev-Petviashvili Equation

The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the bilinear Backlund transformation from its Lax pairs and find solutions with the help of the obtained bilinear Bgcklund transformation.     

Bilinear Bcklund transformation and explicit solutions for a nonlinear evolution equation

The bilinear form of two nonlinear evolution equations are derived by using Hirota derivative. The B\"{a}cklund transformation based on the Hirota bilinear method for these two equations are presented, respectively. As an application, the explicit solutions including soliton and stationary rational solutions for these two equations are obtained.     

Bilinearization and soliton solutions of N=1 supersymmetric coupled dispersionless integrable system

An N=1 supersymmetric generalization of coupled dispersionless (SUSY-CD) integrable system has been proposed by writing its superfield Lax representation. It has been shown that under a suitable variable transformation, the SUSY-CD integrable system is equivalent to N=1 supersymmetric sine-Gordon equation. A superfield bilinear form of SUSY-CD integrable system has been proposed by using super Hirota operator. Explicit expressions of superfield soliton solutions of SUSY-CD integrable system have been obtained by using the Hirota method.     

An improved Hirota bilinear method and new application for a nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation

The Hirota bilinear method has been studied in a lot of local equations, but there are few of works to solve nonlocal equations by Hirota bilinear method. In this letter, we show that the nonlocal integrable complex modified Korteweg-de Vries (MKdV) equation admits multiple complex soliton solutions. A variety of exact solutions including the single bright soliton solutions and two bright soliton solutions are derived via constructing an improved Hirota bilinear method for nonlocal complex MKdV equation. From the gauge equivalence, we can see the difference between the solution of nonlocal integrable complex MKdV equation and the solution of local complex MKdV equation.     

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.     

A Generalized Hirota Ansatz to Obtain Soliton-Like Solutions for a (3+1)-Dimensional Nonlinear Evolution Equation

Instead of the usual Hirota ansatz, i.e., the functions in bilinear equations being chosen as exponential types, a generalized Hirota ansatz is proposed for a (3+1)-dimensional nonlinear evolution equation. Based on the resulting generalized Hirota ansatz, a family of new explicit solutions for the equation are derived.     

The multisoliton solutions for the nonisospectral mKPI equation with self-consistent sources

The nonisospectral mKPI equation with self-consistent sources is derived through the linear problem of the nonisospectral mKPI system. The bilinear form of the nonisospectral mKPI equation with self-consistent sources is given and the N-soliton solutions are obtained through Hirota method and Wronskian technique respectively.     

Lump Solutions for Two Mixed Calogero-Bogoyavlenskii-Schiff and Bogoyavlensky-Konopelchenko

Based on the Hirota bilinear operators and their generalized bilinear derivatives, we formulate two new (2+1)-dimensional nonlinear partial differential equations, which possess lumps. One of the new nonlinear differential equations includes the generalized Calogero-Bogoyavlenskii-Schiff equation and the generalized Bogoyavlensky-Konopelchenko equation as particular examples, and the other has the same bilinear form with different $D_p$-operators. A class explicit lump solutions of the new nonlinear differential equation is constructed by using the Hirota bilinear approaches. A specific case of the presented lump solution is plotted to shed light on the charateristics of the lump.