A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique.Exactsolutions of the new equation are studied by means of the singular manifold method.Bcklund transformation in termsof th...A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique.Exactsolutions of the new equation are studied by means of the singular manifold method.Bcklund transformation in termsof the singular manifold is obtained.And localized structures are also investigated.展开更多
We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations...We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations of the periodic solutions to the (2+1)-dimensional KdV equations obtained by means of the Jacobian elliptic function method, but they possess different periods and velocities.展开更多
The real physics models are usually quite complex with some arbitrary paramcters which will lead to thenonintegrability of the model. To find some cxact solutions of a nonintegrable modcl with some arbitrary parameter...The real physics models are usually quite complex with some arbitrary paramcters which will lead to thenonintegrability of the model. To find some cxact solutions of a nonintegrable modcl with some arbitrary parametersis much more difficult than to find the solutions of a model with some special parameters. In this paper, we make amodification for the usual direct method to find some conditional similarity solutions of a (2+1)-dimensional generalnonintegrable KdV equation.展开更多
Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear...Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.展开更多
A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painl...A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.展开更多
The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we invest...The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.展开更多
An algebraic method is proposed to solve a new (2+1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, lin...An algebraic method is proposed to solve a new (2+1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.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 Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé...The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.展开更多
This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensiona...This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding density plots.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
In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tio...In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1:o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized (2+1)- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.展开更多
A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary...A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.展开更多
New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solu...New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.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,by using the symmetry method,the relationships between new explicit solutions and oldones of the(2+1)-dimensional Kaup-Kupershmidt(KK)equation are presented.We successfully obtain more generalexact trave...In this paper,by using the symmetry method,the relationships between new explicit solutions and oldones of the(2+1)-dimensional Kaup-Kupershmidt(KK)equation are presented.We successfully obtain more generalexact travelhng wave solutions for(2+1)-dimensional KK equation by the symmetry method and the(G'/G)-expansionmethod.Consequently,we find some new solutions of(2+1)-dimensional KK equation,including similarity solutions,solitary wave solutions,and periodic solutions.展开更多
The singular manifold method is used to obtain two general solutions to a (2+1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of t...The singular manifold method is used to obtain two general solutions to a (2+1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.展开更多
基金Supported by Research Fund for the Doctoral Program of Higher Education of China under Grant No.20070486094
文摘A new (2+1)-dimensional KdV equation is constructed by using Lax pair generating technique.Exactsolutions of the new equation are studied by means of the singular manifold method.Bcklund transformation in termsof the singular manifold is obtained.And localized structures are also investigated.
基金国家自然科学基金,Research Foundation for Young Skeleton Teacher in College of Zhejiang Province,the Science Research Foundation of Huzhou University
文摘We investigate a new class of periodic solutions to (2+1)-dimensional KdV equations, by both the linear superposition approach and the mapping deformation method. These new periodic solutions are suitable combinations of the periodic solutions to the (2+1)-dimensional KdV equations obtained by means of the Jacobian elliptic function method, but they possess different periods and velocities.
文摘The real physics models are usually quite complex with some arbitrary paramcters which will lead to thenonintegrability of the model. To find some cxact solutions of a nonintegrable modcl with some arbitrary parametersis much more difficult than to find the solutions of a model with some special parameters. In this paper, we make amodification for the usual direct method to find some conditional similarity solutions of a (2+1)-dimensional generalnonintegrable KdV equation.
文摘Using the extension homogeneous balance method,we have obtained some new special types of soliton solutions of the (2+1)-dimensional KdV equation.Starting from the homogeneous balance method,one can obtain a nonlinear transformation to simple (2+1)-dimensional KdV equation into a linear partial differential equation and two bilinear partial differential equations.Usually,one can obtain only a kind of soliton-like solutions.In this letter,we find further some special types of the multisoliton solutions from the linear and bilinear partial differential equations.
基金supported by Chinese National Social Science Foundation(Grant Number:CNSSF:13CJY037)Research on the indemnificatory Apartment Construction Based on Residential Integration.
文摘A (2 + 1) dimensional KdV-mKdV equation is proposed and integrability in the sense of Painlevé and some exact solutions are discussed. The B?cklund transformation and bilinear equations are obtained through Painlevé analysis. Some exact solutions are deduced by Hirota method and generalized Wronskian method.
文摘The (2 + 1)-dimensional fifth-order KdV equation is an important higher-dimensional and higher-order extension of the famous KdV equation in fluid dynamics. In this paper, by constructing new test functions, we investigate the periodic solitary wave solutions for the (2 + 1)-dimensional fifth-order KdV equation by virtue of the Hirota bilinear form. Several novel analytic solutions for such a model are obtained and verified with the help of symbolic computation.
文摘An algebraic method is proposed to solve a new (2+1)-dimensional Calogero KdV equation and explicitly construct a series of exact solutions including rational solutions, triangular solutions, exponential solution, line soliton solutions, and doubly periodic wave solutions.
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Project supported by the National Natural Science Foundation of China(Grant Nos.11975131 and 11435005)the K C Wong Magna Fund in Ningbo University。
文摘The Painlevé property for a(2+1)-dimensional Korteweg–de Vries(KdV) extension, the combined KP3(Kadomtsev–Petviashvili) and KP4(cKP3-4), is proved by using Kruskal’s simplification. The truncated Painlevé expansion is used to find the Schwartz form, the Bäcklund/Levi transformations, and the residual nonlocal symmetry. The residual symmetry is localized to find its finite Bäcklund transformation. The local point symmetries of the model constitute a centerless Kac–Moody–Virasoro algebra. The local point symmetries are used to find the related group-invariant reductions including a new Lax integrable model with a fourth-order spectral problem. The finite transformation theorem or the Lie point symmetry group is obtained by using a direct method.
文摘This paper constructs exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation with the help of symbolic computation. By means of the truncated Painlev expansion, the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation can be written as a trilinear equation, through the trilinear-linear equation, we can obtain the explicit representation of exact solutions for the (2 + 1)-dimensional KdV-Calogero-Bogoyavlenkskii-Schiff equation. We have depicted the profiles of the exact solutions by presenting their three-dimensional plots and the corresponding 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.
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 (11002115,10972182,11172239)the Science Foundation of Aviation of China (2010ZB53021)+5 种基金the China Postdoctoral Science Special Foundation (201003682)111 project(B07050) to the Northwestern Polytechnical Universitythe NPU Foundation for Fundamental Research (JC200938,JC20110259)the Doctoral Program Foundation of Education Ministry of China(20106102110019)the Open Foundation of State Key Laboratory of Mechanical System & Vibration (MSV-2011-21)the Open Foundation of State Key Laboratory of Structural Analysis of Industrial Equipment (GZ0802)
文摘In the present paper, a general solution involv- ing three arbitrary functions for the generalized (2+1)- dimensional KdV-mKdV equation, which is derived from the generalized (1+1)-dimensional KdV-mKdV equa- tion, is first introduced by means of the Wiess, Tabor, Carnevale (WTC) truncation method. And then multi- symplectic formulations with several conservation laws taken into account are presented for the generalized (2+1)- dimensional KdV-mKdV equation based on the multi- symplectic theory of Bridges. Subsequently, in order to simulate the periodic wave solutions in terms of rational functions of the Jacobi elliptic functions derived from thegeneral solution, a semi-implicit multi-symplectic scheme is constructed that is equivalent 1:o the Preissmann scheme. From the results of the numerical experiments, we can con- clude that the multi-symplectic schemes can accurately sim- ulate the periodic wave solutions of the generalized (2+1)- dimensional KdV-mKdV equation while preserve approxi- mately the conservation laws.
文摘A set of symmetries of a generalized (2+l)-dimensional bilinear equation is given by a formal serins formula. There exist four truncated symmetries for the KdV-Ito model. These trun-cated symmetries with four arourary functions of time t constitute an infinnite-dirmensional Lin algebra which contains two types of the Virasoro subalgebra.
文摘New exact solutions in terms of the Jacobi elliptic functions are obtained to the (2+1)-dimensional breakingsoliton equation by means of the modified mapping method. Limit cases are studied, and new solitary wave solutionsand triangular periodic wave solutions are obtained.
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Supported by the Natural Science Foundation of Shandong Province in China under Grant No.Q2005A01
文摘In this paper,by using the symmetry method,the relationships between new explicit solutions and oldones of the(2+1)-dimensional Kaup-Kupershmidt(KK)equation are presented.We successfully obtain more generalexact travelhng wave solutions for(2+1)-dimensional KK equation by the symmetry method and the(G'/G)-expansionmethod.Consequently,we find some new solutions of(2+1)-dimensional KK equation,including similarity solutions,solitary wave solutions,and periodic solutions.
文摘The singular manifold method is used to obtain two general solutions to a (2+1)-dimensional breaking soliton equation, each of which contains two arbitrary functions. Then the new periodic wave solutions in terms of the Jacobi elliptic functions are generated from the general solutions. The long wave limit yields the new types of dromion and solitary structures.