Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by u...Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.展开更多
By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other ...By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.展开更多
The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konop...The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.展开更多
A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lo...A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.展开更多
Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the gr...Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou＇s direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.展开更多
Based on the general direct method developed by Lou et al. [J. Phys. A： Math. Gen. 38 （2005） L129], the symmetry group theorem is obtained, from that both the Lie point groups and the non-Lie symmetry groups of the...Based on the general direct method developed by Lou et al. [J. Phys. A： Math. Gen. 38 （2005） L129], the symmetry group theorem is obtained, from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenk-Dubrovsky （KD） equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution.展开更多
In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractiona...In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.展开更多
In this paper, we investigate a(2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. The lump waves, lumpoff waves, and rogue waves are presented based on the Hirota bilinear form of this eq...In this paper, we investigate a(2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. The lump waves, lumpoff waves, and rogue waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the moving path as well as the appearance time and place of the lump waves are given. Moreover, the special rogue waves are considered when lump solution is swallowed by double solitons. Finally,the corresponding characteristics of the dynamical behavior are displayed.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
Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kau...Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.展开更多
By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-br...By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.展开更多
基金The project partially supported by the State Key Basic Research Program of China under Grant No. 2004CB318000
文摘Taking the Konopelchenko-Dubrovsky system as a simple example, some familles of rational formal hyperbolic function solutions, rational formal triangular periodic solutions, and rational solutions are constructed by using the extended Riccati equation rational expansion method presented by us. The method can also be applied to solve more nonlinear partial differential equation or equations.
基金Supported by the Natural Science Foundation of Education Committee of Henan Province(2003110003)Supported by the Natural Science Foundation of Henan Province(0111050200)
文摘By using F-expansion method proposed recently, we derive the periodic wave solution expressed by Jacobi elliptic functions for Konopelchenko-Dubrovsky equation. In the limit case, the solitary wave solution and other type of the traveling wave solutions are derived.
基金supported by the National Natural Science Foundation of China under Grant No.10672053the Scientific Research Fund of the Education Department of Hunan Province under Grant No.07D064
文摘The sinh-Gordon equation expansion method is further extended by generMizing the sinh-Gordon equation and constructing new ansatz solution of the considered equation. As its application, the （2＋1）-dimensional Konopelchenko-Dubrovsky equation is investigated and abundant exact travelling wave solutions are explicitly obtained including solitary wave solutions, trigonometric function solutions and Jacobi elliptic doubly periodic function solutions, some of which are new exact solutions that we have never seen before within our knowledge. The method can be applied to other nonlinear evolution equations in mathematical physics.
基金浙江省自然科学基金，浙江省宁波市博士基金，the State Key Laboratory of Oil/Gas Reservoir Geology and Exploitation，Scientific Research Fund of Education Department of Zhejiang Province under
文摘A set of generalized symmetries with arbitrary functions of t for the Konopelchenko-Dubrovsky (KD)equation in 2+1 space dimensions is given by using a direct method called formal function series method presented by Lou. These symmetries constitute an infinite-dimensional generalized w∞ algebra.
基金the State Key Basic Research Program of China under Grant No.2004CB318000
文摘Based on the symbolic computation system Maple, the infinite-dimensional symmetry group of the （2＋1）- dimensional Sawada-Kotera equation is found by the classical Lie group method and the characterization of the group properties is given. The symmetry groups are used to perform the symmetry reduction. Moreover, with Lou＇s direct method that is based on Lax pairs, we obtain the symmetry transformations of the Sawada-Kotera and Konopelchenko Dubrovsky equations, respectively.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10747141 and 10735030Zhejiang Provincial Natural Science Foundations of China under Grant No.605408+3 种基金Ningbo Natural Science Foundation under Grant Nos.2007A610049 and 2008A610017National Basic Research Program of China (973 Program 2007CB814800)Shanghai Leading Academic Discipline Project under Grant No.B412K.C.Wong Magna Fund in Ningbo University
文摘Based on the general direct method developed by Lou et al. [J. Phys. A： Math. Gen. 38 （2005） L129], the symmetry group theorem is obtained, from that both the Lie point groups and the non-Lie symmetry groups of the Konopelchenk-Dubrovsky （KD） equation are obtained. From the theorem, some exact solutions of KD equation are derived from a simple travelling wave solution and a multi-soliton solution.
基金Supported by the National Natural Science Foundation of China（11462019） Supported by the Scientific Research Foundation of Inner Mongolia University for Nationalities（NMDYB1750, NMDGP1713）
文摘In the current paper, based on fractional complex transformation, the GG2-expansion method which is used to solve differential equations of integer order is developed for finding exact solutions of nonlinear fractional differential equations with Jumarie's modified Riemann-Liouville derivative. And then, time-fractional Burgers equation and space-fractional coupled Konopelchenko-Dubrovsky equations are provided to show that this method is effective in solving nonlinear fractional differential equations.
基金Supported by the Postgraduate Research&Practice Innovation Program of Jiansu Province under Grant No.SJKY19 1877the Fundamental Research Funds for the Central University under Grant No.2017XKZD11
文摘In this paper, we investigate a(2+1)-dimensional generalized Konopelchenko-Dubrovsky-Kaup-Kupershmidt equation. The lump waves, lumpoff waves, and rogue waves are presented based on the Hirota bilinear form of this equation. It is worth noting that the moving path as well as the appearance time and place of the lump waves are given. Moreover, the special rogue waves are considered when lump solution is swallowed by double solitons. Finally,the corresponding characteristics of the dynamical behavior are displayed.
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supported by the National Natural Science Foundation of China (project Nos. 11371086,11671258,11975145)the Fund of Science and Technology Commission of Shanghai Municipality (project No. 13ZR1400100)the Fund of Donghua University,Institute for Nonlinear Sciences and the Fundamental Research Funds for the Central Universities。
文摘Soliton molecules have become one of the hot topics in recent years. In this article, we investigate soliton molecules and some novel hybrid solutions for the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(gKDKK) equation by using the velocity resonance, module resonance, and long wave limits methods. By selecting some specific parameters, we can obtain soliton molecules and asymmetric soliton molecules of the gKDKK equation. And the interactions among N-soliton molecules are elastic. Furthermore, some novel hybrid solutions of the gKDKK equation can be obtained, which are composed of lumps,breathers, soliton molecules and asymmetric soliton molecules. Finally, the images of soliton molecules and some novel hybrid solutions are given, and their dynamic behavior is analyzed.
基金Jiangsu Provincial Natural Science Foundation of China(Grant Nos.BK20221508,11775116,BK20210380,and JSSCBS20210277)SRT(Grant No.202210307165Y)Jiangsu Qinglan High-level Talent Project.
文摘By employing the complexification method and velocity resonant principle to N-solitons of the(2+1)-dimensional generalized Konopelchenko–Dubrovsky–Kaup–Kupershmidt(KDKK)equation,we obtain the soliton molecules,T-breather molecules,T-breather–L-soliton molecules and some interaction solutions when N≤6.Dynamical behaviors of these solutions are discussed analytically and graphically.The method adopted can be effectively used to construct soliton molecules and T-breather molecules of other nonlinear evolution equations.The results obtained may be helpful for experts to study the related phenomenon in oceanography and atmospheric science.
