For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits ...For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.展开更多
A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the （2＋1）-dimensio...A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the （2＋1）-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.展开更多
In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurca...In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.展开更多
With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transfor...With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.展开更多
In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transf...In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to （2＋1）-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.展开更多
Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）-...Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.展开更多
The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtaine...The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtained detalledly in this paper. The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain （loss）.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 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
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
Recently some (1+1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of ...Recently some (1+1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of (2+1)-dimensional generalization of mKd V equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this (2+ 1)-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.展开更多
By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation th...By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.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
In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evo...In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evolution equations. To illustrate the effectiveness of our method, we take the （2＋ 1）-dimensional stochastic dispersive long wave system as an example. We not only have obtained some known solutions, but also have constructed some new rational formal stochastic Jacobi elliptic function solutions.展开更多
In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbol...In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbolic computation. The study confirms the power of the two schemes.展开更多
In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:"">...In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.展开更多
In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and othe...In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on （2＋1）-dimensional dispersive long-wave equation.展开更多
The(2 + 1)-dimensional generalized fifth-order Kd V(2GKd V) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero backgr...The(2 + 1)-dimensional generalized fifth-order Kd V(2GKd V) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKd V equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped(XS/YS),asymmetric M-Shaped(MS), W-Shaped(WS), Space-Curved(SC) and Oscillation M-Shaped(OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.展开更多
The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the...The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.展开更多
文摘For the (2 + 1)-dimensional nonlinear dispersive Boussinesq equation, by using the bifurcation theory of planar dynamical systems to study its corresponding traveling wave system, the bifurcations and phase portraits of the regular system are obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of analytical and non-analytical solutions of the singular system are given by using singular traveling wave theory. For certain special cases, some explicit and exact parametric representations of traveling wave solutions are derived such as analytical periodic waves and non-analytical periodic cusp waves. Further, two-dimensional wave plots of analytical periodic solutions and non-analytical periodic cusp wave solutions are drawn to visualize the dynamics of the equation.
基金The project supported in part by National Natural Science Foundation of China under Grant No. 10272071 and the Science Research Foundation of Huzhou University under Grant No. KX21025
文摘A new generalized extended F-expansion method is presented for finding periodic wave solutions of nonlinear evolution equations in mathematical physics. As an application of this method, we study the （2＋1）-dimensional dispersive long wave equation. With the aid of computerized symbolic computation, a number of doubly periodic wave solutions expressed by various Jacobi elliptic functions are obtained. In the limit cases, the solitary wave solutions are derived as well.
基金Supported by the National Natural Science Foundation of China (10871206)Program for Excellent Talents in Guangxi Higher Education Institutions
文摘In this paper,the bifurcation of solitary,kink,anti-kink,and periodic waves for （2＋1）-dimension nonlinear dispersive long wave equation is studied by using the bifurcation theory of planar dynamical systems.Bifurcation parameter sets are shown,and under various parameter conditions,all exact explicit formulas of solitary travelling wave solutions and kink travelling wave solutions and periodic travelling wave solutions are listed.
基金supported by the Scientific Research Foundation of Beijing Information Science and Technology UniversityScientific Creative Platform Foundation of Beijing Municipal Commission of Education
文摘With the aid of symbolic computation system Maple, some families of new rational variable separation solutions of the （2＋1）-dimensional dispersive long wave equations are constructed by means of a function transformation, improved mapping approach, and variable separation approach, among which there are rational solitary wave solutions, periodic wave solutions and rational wave solutions.
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004 CB 318000
文摘In this paper, a further extended Jacobi elliptic function rationM expansion method is proposed for constructing new forms of exact solutions to nonlinear partial differential equations by making a more general transformation. For illustration, we apply the method to （2＋1）-dimensionM dispersive long wave equation and successfully obtain many new doubly periodic solutions. When the modulus m→1, these sohitions degenerate as soliton solutions. The method can be also applied to other nonlinear partial differential equations.
基金supported by the Natural Science Foundation of Shandong Province of China under Grant Nos.Q2005A01
文摘Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.
基金Supported by the National Natural Science Foundation of China under Grant No.11072219the Zhejiang Provincial Natural Science Foundation under Grant No.Y1100099
文摘The exact chirped soliton-like and quasi-periodic wave solutions of （2 ＋ 1）-dimensional generalized nonlinear Schr6dinger equation including linear and nonlinear gain （loss） with variable coefficients are obtained detalledly in this paper. The form and the behavior of solutions are strongly affected by the modulation of both the dispersion coefficient and the nonlinearity coefficient. In addition, self-similar soliton-like waves precisely piloted from our obtained solutions by tailoring the dispersion and linear gain （loss）.
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.
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.
基金浙江省自然科学基金，中国博士后科学基金，中国科学院资助项目，教育部留学回国人员科研启动基金，Scientific Research Foundation for Returned Overseas Chinese Scholars of Ministry of Education of China
文摘Recently some (1+1)-dimensional nonlinear wave equations with linearly dispersive terms were shown to possess compacton-like and solitary pattern-like solutions. In this paper, with the aid of Maple, new solutions of (2+1)-dimensional generalization of mKd V equation, which is of only linearly dispersive terms, are investigated using three new transformations. As a consequence, it is shown that this (2+ 1)-dimensional equation also possesses new compacton-like solutions and solitary pattern-like solutions.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10735030, 90718041, and 40975038Shanghai Leading Academic Discipline Project under Grant No. B412Program for Changjiang Scholars and Innovative Research Team in University (IRT0734)
文摘By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.
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.
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000
文摘In this work, by means of a generalized method and symbolic computation, we extend the Jacobi elliptic function rational expansion method to uniformly construct a series of stochastic wave solutions for stochastic evolution equations. To illustrate the effectiveness of our method, we take the （2＋ 1）-dimensional stochastic dispersive long wave system as an example. We not only have obtained some known solutions, but also have constructed some new rational formal stochastic Jacobi elliptic function solutions.
文摘In this paper, we present Yan’s sine-cosine method and Wazwaz’s sine-cosine method to solve the (2+1)-dimensional Zoomeron equation. New exact travelling wave solutions are explicitly obtained with the aid of symbolic computation. The study confirms the power of the two schemes.
文摘In this paper, we investigate the periodic wave solutions and solitary wave solutions of a (2+1)-dimensional Korteweg-de Vries (KDV) equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">by applying Jacobi elliptic function expansion method. Abundant types of Jacobi elliptic function solutions are obtained by choosing different </span><span style="font-size:10.0pt;font-family:"">coefficient</span><span style="font-size:10.0pt;font-family:"">s</span><span style="font-size:10pt;font-family:""> <i>p</i>, <i>q</i> and <i>r</i> in the</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">elliptic equation. Then these solutions are</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">coupled into an auxiliary equation</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">and substituted into the (2+1)-dimensional KDV equation. As <span>a result,</span></span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">a large number of complex Jacobi elliptic function solutions are ob</span><span style="font-size:10pt;font-family:"">tained, and many of them have not been found in other documents. As</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10.0pt;font-family:""><span></span></span><span style="font-size:10pt;font-family:"">, some complex solitary solutions are also obtained correspondingly.</span><span style="font-size:10pt;font-family:""> </span><span style="font-size:10pt;font-family:"">These solutions that we obtained in this paper will be helpful to understand the physics of the (2+1)-dimensional KDV equation.
基金The project supported by National Natural Science Foundation of China, the Natural Science Foundation of Shandong Province of China, and the Natural Science Foundation of Liaocheng University .
文摘In this paper, we present a new rational algebraic approach to uniformly construct a series of exact analytical solutions for nonlinear partial differential equations. Compared with most existing tanh methods and other sophisticated methods, the proposed method not only recovers some known solutions, but also finds some new and general solutions. The solutions obtained in this paper include rational form triangular periodic wave solutions, solitary wave solutions, and elliptic doubly periodic wave solutions. The efficiency of the method can be demonstrated on （2＋1）-dimensional dispersive long-wave equation.
基金provided by the National Natural Science Foundation of China (Grant No. 12271324)the Natural Science Basic Research Program of Shaanxi Province (Grant No. 2024JC-YBQN-0069)+2 种基金the China Postdoctoral Science Foundation (Grant No. 2024M751921)the 2023 Shaanxi Province Postdoctoral Research Project (Grant No.2023BSHEDZZ186)the Fundamental Research Funds for the Central Universities (Grant No. 1301032598)。
文摘The(2 + 1)-dimensional generalized fifth-order Kd V(2GKd V) equation is revisited via combined physical and mathematical methods. By using the Hirota perturbation expansion technique and via setting the nonzero background wave on the multiple soliton solution of the 2GKd V equation, breather waves are constructed, for which some transformed wave conditions are considered that yield abundant novel nonlinear waves including X/Y-Shaped(XS/YS),asymmetric M-Shaped(MS), W-Shaped(WS), Space-Curved(SC) and Oscillation M-Shaped(OMS) solitons. Furthermore, distinct nonlinear wave molecules and interactional structures involving the asymmetric MS, WS, XS/YS, SC solitons, and breathers, lumps are constructed after considering the corresponding existence conditions. The dynamical properties of the nonlinear molecular waves and interactional structures are revealed via analyzing the trajectory equations along with the change of the phase shifts.
文摘The determining equations for the nonclassical symmetry reductions of nonlinear partial differential equations with arbitrary order can be obtained by requiring the compatibility between the original equations and the invariant surface conditions. The （2＋1）-dimensional shallow water wave equation, Boussinesq equation, and the dispersive wave equations in shallow water serve as examples i11ustrating how compatibility leads quickly and easily to the determining equations for their nonclassical symmetries.
