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
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(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
For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by...For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.展开更多
The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryf...The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.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
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
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
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, ...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(3k1^2+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)k1p1 and a minimum-(3/4)k1p1. The width of the contour line is ln[(2+√6+√2+√3)/(2+√6-√2-√3)]. 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[(√2+1)/(√2-1)]. 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.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
N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constru...N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.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
In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the co...In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.展开更多
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 cal...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.展开更多
By using the extended homogeneous balance method, the localized coherent structures are studied. A nonlinear transformation was first established, and then the linearization form was obtained based on the extended hom...By using the extended homogeneous balance method, the localized coherent structures are studied. A nonlinear transformation was first established, and then the linearization form was obtained based on the extended homogeneous balance method for the higher order (2 + 1)-dimensional Broer-Kaup equations. Starting from this linearization form equation, a variable separation solution with the entrance of some arbitrary functions and some arbitrary parameters was constructed. The quite rich localized coherent structures were revealed. This method, which can be generalized to other (2 + I) -dimensional nonlinear evolution equation, is simple and powerful.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
The (2+1)-dimensional BKP equation in the Hirota bilinear form is studied during this work. Wronskian and Grammian techniques are applied to the construction of Wronskian and Grammian solutions of this equation, re...The (2+1)-dimensional BKP equation in the Hirota bilinear form is studied during this work. Wronskian and Grammian techniques are applied to the construction of Wronskian and Grammian solutions of this equation, respectively. It is shown that these solutions can be expressed as not only Pfaffians but also Wronskians and Grammians.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(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.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 under Grant Nos.60772023 and 60372095the Key Project of the Ministry of Education under Grant No.106033+3 种基金the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE-07-001Beijing University of Aeronautics and Astronauticsthe National Basic Research Program of China(973 Program)under Grant No.2005CB321901the Specialized Research Fund for the Doctoral Program of Higher Education of the Ministry of Education under Grant No.20060006024
文摘For describing various complex nonlinear phenomena in the realistic world,the higher-dimensional nonlinearevolution equations appear more attractive in many fields of physical and engineering sciences.In this paper,by virtueof the Hirota bilinear method and Riemann theta functions,the periodic wave solutions for the(2+1)-dimensionalBoussinesq equation and(3+1)-dimensional Kadomtsev-Petviashvili(KP)equation are obtained.Furthermore,it isshown that the known soliton solutions for the two equations can be reduced from the periodic wave solutions.
基金The project supported by National Natural Science Foundation of China
文摘The variable separation approach is used to obtain localized coherent structures of the new (2+1)-dimensional nonlinear partialdifferential equation. Applying the Backlund transformation and introducing the arbitraryfunctions of the seed solutions, the abundance of the localized structures of this model are derived. Some special types ofsolutions solitoff, dromions, dromion lattice, breathers and instantons are discussed by selecting the arbitrary functionsappropriately. The breathers may breath in their amplititudes, shapes, distances among the peaks and even the numberof the peaks.
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 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 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 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(3k1^2+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)k1p1 and a minimum-(3/4)k1p1. The width of the contour line is ln[(2+√6+√2+√3)/(2+√6-√2-√3)]. 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[(√2+1)/(√2-1)]. 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.
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.
文摘N-soliton solutions and the bilinear form of the (2 + 1)-dimensional AKNS equation are obtained by using the Hirota method. Moreover, the double Wronskian solution and generalized double Wronskian solution are constructed through the Wronskian technique. Furthermore, rational solutions, Matveev solutions and complexitons of the (2 + 1)-dimensional AKNS equation are given through a matrix method for constructing double Wronskian entries. The three solutions are new.
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.
文摘In this paper, the new mapping approach and the new extended auxiliary equation approach were used to investigate the exact traveling wave solutions of (2 + 1)-dimensional time-fractional Zoomeron equation with the conformable fractional derivative. As a result, the singular soliton solutions, kink and anti-kink soliton solutions, periodic function soliton solutions, Jacobi elliptic function solutions and hyperbolic function solutions of (2 + 1)-dimensional time-fractional Zoomeron equation were obtained. Finally, the 3D and 2D graphs of some solutions were drawn by setting the suitable values of parameters with Maple, and analyze the dynamic behaviors of the solutions.
基金Project supported by the Natural Science Foundation of Guangdong Province, China (Grant Nos. 10452840301004616 and S2011040000403)the National Natural Science Foundation of China (Grant No. 41176005)the Science and Technology Project Foundation of Zhongshan, China (Grnat No. 20123A326)
文摘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.
文摘By using the extended homogeneous balance method, the localized coherent structures are studied. A nonlinear transformation was first established, and then the linearization form was obtained based on the extended homogeneous balance method for the higher order (2 + 1)-dimensional Broer-Kaup equations. Starting from this linearization form equation, a variable separation solution with the entrance of some arbitrary functions and some arbitrary parameters was constructed. The quite rich localized coherent structures were revealed. This method, which can be generalized to other (2 + I) -dimensional nonlinear evolution equation, is simple and powerful.
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supported by the National Natural Science Foundation of China(11202161 and 11172233)the Basic Research Fund of the Northwestern Polytechnical University(GBKY1034)
文摘The (2+1)-dimensional BKP equation in the Hirota bilinear form is studied during this work. Wronskian and Grammian techniques are applied to the construction of Wronskian and Grammian solutions of this equation, respectively. It is shown that these solutions can be expressed as not only Pfaffians but also Wronskians and Grammians.