This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional ...This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.展开更多
Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamic...Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.展开更多
Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic so...Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.展开更多
Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechan...Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.展开更多
Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By fur...Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.展开更多
In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are...In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.展开更多
Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a cl...Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.展开更多
In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the...In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.展开更多
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 soli...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（3k12+ 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）k1p1 and a minimum-（3/4）k1p1. 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）k1p1. The width of the contour line is ln■. If φ（y） = sn（y, 1/2）/（1 + y2）, 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.展开更多
For the （2＋1）-dimensional Broer–Kaup–Kupershmidt（BKK） system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the ...For the （2＋1）-dimensional Broer–Kaup–Kupershmidt（BKK） system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion（CRE） is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.展开更多
Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmet...Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmetric solitons upon assigning appropriate values to some parameters.Furthermore,a double-peaked lump solution can be constructed with breather degeneration approach.By applying a mixed technique of a resonance ansatz and conjugate complexes of partial parameters to multisoliton solutions,various kinds of interactional structures are constructed;There include the soliton molecule(SM),the breather molecule(BM)and the soliton-breather molecule(SBM).Graphical investigation and theoretical analysis show that the interactions composed of SM,BM and SBM are inelastic.展开更多
The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is exp...The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.展开更多
The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and thei...The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and their interactions in(2+1)-dimensional potential Boiti–Leon-Manna–Pempinelli equation.Dromion molecules,ring molecules,lump molecules,multi-instantaneous molecules,and their interactions are obtained.Then we draw corresponding images with maple software to study their dynamic behavior.展开更多
By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equa...By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the （2 ＋ 1）-dimensional sine-Poisson equation are derived in a simple manner by this technique.展开更多
The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions ...The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.展开更多
This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé...This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.展开更多
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 proce...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.展开更多
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 de...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.展开更多
We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)eq...We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.展开更多
A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contain...A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos.12275172 and 11905124)。
文摘This paper studies the(2+1)-dimensional Hirota-Satsuma-Ito equation.Based on an associated Hirota bilinear form,lump-type solution,two types of interaction solutions,and breather wave solution of the(2+1)-dimensional Hirota-Satsuma-Ito equation are obtained,which are all related to the seed solution of the equation.It is interesting that the rogue wave is aroused by the interaction between one-lump soliton and a pair of resonance stripe solitons,and the fusion and fission phenomena are also found in the interaction between lump solitons and one-stripe soliton.Furthermore,the breather wave solution is also obtained by reducing the two-soliton solutions.The trajectory and period of the one-order breather wave are analyzed.The corresponding dynamical characteristics are demonstrated by the graphs.
基金Supported by the National Natural Science Foundation of China(12275172)。
文摘Based on the Hirota bilinear method,this study derived N-soliton solutions,breather solutions,lump solutions and interaction solutions for the(2+1)-dimensional extended Boiti-Leon-Manna-Pempinelli equation.The dynamical characteristics of these solutions were displayed through graphical,particularly revealing fusion and ssion phenomena in the interaction of lump and the one-stripe soliton.
文摘Based on the travelling wave method, a(2 + 1)-dimensional AKNS equation is considered. Elliptic solution and soliton solution are presented and it is shown that the soliton solution can be reduced from the elliptic solution. It also proves that the result is consistent with the soliton solution of simplify Hirota bilinear method by Wazwaz and illustrate the solution are right travelling wave solution.
基金Project supported by the Natural Science Foundation of Guangdong Province of China (Grant No.10452840301004616)the National Natural Science Foundation of China (Grant No.61001018)the Scientific Research Foundation for the Doctors of University of Electronic Science and Technology of China Zhongshan Institute (Grant No.408YKQ09)
文摘Recently, a new （2＋1）-dimensional shallow water wave system, the （2＋1）-dlmenslonal displacement shallow water wave system （2DDSWWS）, was constructed by applying the variational principle of the analytic mechanics in the Lagrange coordinates. The disadvantage is that fluid viscidity is not considered in the 2DDSWWS, which is the same as the famous Kadomtsev-Petviashvili equation and Korteweg-de Vries equation. Applying dimensional analysis, we modify the 2DDSWWS and add the term related to the fluid viscidity to the 2DDSWWS. The approximate similarity solutions of the modified 2DDSWWS （M2DDSWWS） is studied and four similarity solutions are obtained. For the perfect fluids, the coefficient of kinematic viscosity is zero, then the M2DDSWWS will degenerate to the 2DDSWWS.
基金Project supported by the National Natural Science Foundation of China （Grant No 10672147） and Natural Science Foundation of Zhejiang Forestry University, China （Grant No 2006FR035）. Acknowledgments The authors are indebted to Professor Zhang J F for his helpful suggestions and fruitful discussions, and also express their sincere thanks to the editors and the anonymous referees for their constructive suggestions and kind help.
文摘Starting from the extended tanh-function method （ETM） based on the mapping method, the variable separation solutions of the （2＋1）-dimensional asymmetric Nizhnik Novikov Veselov （ANNV） system are derived. By further study, we find that these variable separation solutions are seemingly independent of but actually dependent on each other. Based on the variable separation solution and by choosing appropriate functions, some novel and interesting interactions between special solitons, such as bell-like compacton, peakon-like compacton and compacton-like semifoldon, are investigated.
基金Project supported by the Anhui Key Laboratory of Information Materials and Devices (Anhui University),China
文摘In this paper, we make use of the auxiliary equation and the expanded mapping methods to find the new exact periodic solutions for （2＋1）-dimensional dispersive long wave equations in mathematical physics, which are expressed by Jacobi elliptic functions, and obtain some new solitary wave solutions （m → 1）. This method can also be used to explore new periodic wave solutions for other nonlinear evolution equations.
基金Project supported by the National Natural Science Foundation of China (Grant No 10647112)the Foundation of Donghua University
文摘Based on the B/icklund method and the multilinear variable separation approach （MLVSA）, this paper finds a general solution including two arbitrary functions for the （2＋1）-dimensional Burgers equations. Then a class of new doubly periodic wave solutions for （2＋l）-dimensional Burgers equations is obtained by introducing appropriate Jacobi elliptic functions, Weierstrass elliptic functions and their combination in the general solutions （which contains two arbitrary functions）. Two types of limit cases are considered. Firstly, taking one of the moduli to be unity and the other zero, it obtains particular wave （called semi-localized） patterns, which is periodic in one direction, but localized in the other direction. Secondly, if both moduli are tending to 1 as a limit, it derives some novel localized excitations （two-dromion solution）.
基金Project supported by the Global Change Research Program of China(Grant No.2015CB953904)the National Natural Science Foundation of China(Grant Nos.11275072 and 11435005)+2 种基金the Doctoral Program of Higher Education of China(Grant No.20120076110024)the Network Information Physics Calculation of Basic Research Innovation Research Group of China(Grant No.61321064)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)
文摘In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the（2＋1）-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the（2＋1）-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion（CRE） solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11671219 and 11871446)
文摘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（3k12+ 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）k1p1 and a minimum-（3/4）k1p1. 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）k1p1. The width of the contour line is ln■. If φ（y） = sn（y, 1/2）/（1 + y2）, 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.
基金Project supported by the Zhejiang Provincial Natural Science Foundation of China(Grant No.LQ13A010014)the National Natural Science Foundation of China(Grant Nos.11326164,11401528,11435005,and 11375090)
文摘For the （2＋1）-dimensional Broer–Kaup–Kupershmidt（BKK） system, the nonlocal symmetries related to the Schwarzian variable and the corresponding transformation group are found. Moreover, the integrability of the BKK system in the sense of having a consistent Riccati expansion（CRE） is investigated. The interaction solutions between soliton and cnoidal periodic wave are explicitly studied.
基金Supported by the National Natural Science Foundation of China(12001424)the Natural Science Basic Research Program of Shaanxi Province(2021JZ-21)the Fundamental Research Funds for the Central Universities(2020CBLY013)。
文摘Soliton molecules(SMs)of the(2+1)-dimensional generalized KonopelchenkoDubrovsky-Kaup-Kupershmidt(gKDKK)equation are found by utilizing a velocity resonance ansatz to N-soliton solutions,which can transform to asymmetric solitons upon assigning appropriate values to some parameters.Furthermore,a double-peaked lump solution can be constructed with breather degeneration approach.By applying a mixed technique of a resonance ansatz and conjugate complexes of partial parameters to multisoliton solutions,various kinds of interactional structures are constructed;There include the soliton molecule(SM),the breather molecule(BM)and the soliton-breather molecule(SBM).Graphical investigation and theoretical analysis show that the interactions composed of SM,BM and SBM are inelastic.
基金the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)and K.C.Wong Magna Fund in Ningbo University.
文摘The celebrated(1+1)-dimensional Korteweg de-Vries(KdV)equation and its(2+1)-dimensional extension,the Kadomtsev-Petviashvili(KP)equation,are two of the most important models in physical science.The KP hierarchy is explicitly written out by means of the linearized operator of the KP equation.A novel(2+1)-dimensional KdV extension,the cKP3-4 equation,is obtained by combining the third member(KP3,the usual KP equation)and the fourth member(KP4)of the KP hierarchy.The integrability of the cKP3-4 equation is guaranteed by the existence of the Lax pair and dual Lax pair.The cKP3-4 system can be bilinearized by using Hirota's bilinear operators after introducing an additional auxiliary variable.Exact solutions of the cKP3-4 equation possess some peculiar and interesting properties which are not valid for the KP3 and KP4 equations.For instance,the soliton molecules and the missing D'Alembert type solutions(the arbitrary travelling waves moving in one direction with a fixed model dependent velocity)including periodic kink molecules,periodic kink-antikink molecules,few-cycle solitons,and envelope solitons exist for the cKP3-4 equation but not for the separated KP3 equation and the KP4 equation.
基金the National Natural Science Foundation of China(Grant Nos.11371086,11671258,and 11975145)the Fund of Science and Technology Commission of Shanghai Municipality(Grant No.13ZR1400100)。
文摘The method of variable separation has always been regarded as a crucial method for solving nonlinear evolution equations.In this paper,we use a new form of variable separation to study novel soliton molecules and their interactions in(2+1)-dimensional potential Boiti–Leon-Manna–Pempinelli equation.Dromion molecules,ring molecules,lump molecules,multi-instantaneous molecules,and their interactions are obtained.Then we draw corresponding images with maple software to study their dynamic behavior.
基金Project supported by the National Natural Science Foundation of China (Grant No. 10672053)
文摘By introducing a more general auxiliary ordinary differential equation （ODE）, a modified variable separated ordinary differential equation method is presented for solving the （2 ＋ 1）-dimensional sine-Poisson equation. As a result, many explicit and exact solutions of the （2 ＋ 1）-dimensional sine-Poisson equation are derived in a simple manner by this technique.
基金the National Natural Science Foundation of China(Grant Nos.12061051 and 11965014)。
文摘The(2+1)-dimensional elliptic Toda equation is a high-dimensional generalization of the Toda lattice and a semidiscrete Kadomtsev–Petviashvili I equation.This paper focuses on investigating the resonant interactions between two breathers,a breather/lump and line solitons as well as lump molecules for the(2+1)-dimensional elliptic Toda equation.Based on the N-soliton solution,we obtain the hybrid solutions consisting of line solitons,breathers and lumps.Through the asymptotic analysis of these hybrid solutions,we derive the phase shifts of the breather,lump and line solitons before and after the interaction between a breather/lump and line solitons.By making the phase shifts infinite,we obtain the resonant solution of two breathers and the resonant solutions of a breather/lump and line solitons.Through the asymptotic analysis of these resonant solutions,we demonstrate that the resonant interactions exhibit the fusion,fission,time-localized breather and rogue lump phenomena.Utilizing the velocity resonance method,we obtain lump–soliton,lump–breather,lump–soliton–breather and lump–breather–breather molecules.The above works have not been reported in the(2+1)-dimensional discrete nonlinear wave equations.
基金This work was supported by the National Natural Science Foundation of China(Grant No.11505090)Research Award Foundation for Outstanding Young Scientists of Shandong Province(Grant No.BS2015SF009)+2 种基金the Doctoral Foundation of Liaocheng University(Grant No.318051413)Liaocheng University Level Science and Technology Research Fund(Grant No.318012018)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology(Grant No.319462208).
文摘This article investigates the Hirota-Satsuma-Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method.The equation is proved to be Painlevé integrable by Painlevé analysis.On the basis of the bilinear form,the forms of two-soliton solutions,three-soliton solutions,and four-soliton solutions are studied specifically.The appropriate parameter values are chosen and the corresponding figures are presented.The breather waves solutions,lump solutions,periodic solutions and the interaction of breather waves solutions and soliton solutions,etc.are given.In addition,we also analyze the different effects of the parameters on the figures.The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions.These are important for describing water waves in nature.
基金Project supported by the National Natural Science Foundation of China(Grant No.11505094)the Natural Science Foundation of Jiangsu Province,China(Grant No.BK20150984)
文摘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.
基金supported by the National Natural Science Foundation of China(Grant No.41406018)
文摘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.
文摘We present new lemmas,theorem and corollaries to construct interactions among higher-order rogue waves,n-periodic waves and n-solitons solutions(n→∞)to the(2+1)-dimensional asymmetric Nizhnik-Novikov-Veselov(ANNV)equation.Several examples for theories are given by choosing definite interactions of the wave solutions for the model.In particular,we exhibit dynamical interactions between a rogue and a cross bright-dark bell wave,a rogue and a cross-bright bell wave,a rogue and a one-,two-,three-,four-periodic wave.In addition,we also present multi-types interactions between a rogue and a periodic cross-bright bell wave,a rogue and a periodic cross-bright-bark bell wave.Finally,we physically explain such interaction solutions of the model in the 3D and density plots.
基金Foundation item： Supported by the National Natural Science Foundation of China（10647112, 10871040） Acknowledgement The authors are in debt to thank the helpful discussions with Prof Qin and Dr A P Deng.
文摘A class of new doubly periodic wave solutions for （2＋1）-dimensional KdV equation are obtained by introducing appropriate Jacobi elliptic functions and Weierstrass elliptic functions in the general solution（contains two arbitrary functions） got by means of multilinear variable separation approach for （2＋1）-dimensional KdV equation. Limiting cases are considered and some localized excitations are derived, such as dromion, multidromions, dromion-antidromion, multidromions-antidromions, and so on. Some solutions of the dromion-antidromion and multidromions-antidromions are periodic in one direction but localized in the other direction. The interaction properties of these solutions, which are numerically studied, reveal that some of them are nonelastic and some are completely elastic. Furthermore, these results are visualized.
