Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation al...Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation allows the Burgers-STO(BSTO)decomposition,two types of reducible coupled BSTO decompositions and the BSTO-KdV decomposition.Furthermore,we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions.Using the framework of standard Lie point symmetry theory,these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.展开更多
2N line-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation can be presented by resorting to the Hirota bilinear method. By extending the real parameters into complex parameters, this paper o...2N line-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation can be presented by resorting to the Hirota bilinear method. By extending the real parameters into complex parameters, this paper obtains N periodic-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation from the 2N line-soliton solutions.展开更多
In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an ex...In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.展开更多
In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plas...In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.展开更多
Solutions of a noncommutative nonisospectral Kadomtsev-Petviashvili equation are given in terms of quasiwronskian and quasigrammian respectively. These solutions are verified by direct substitutions. Dynamics of some ...Solutions of a noncommutative nonisospectral Kadomtsev-Petviashvili equation are given in terms of quasiwronskian and quasigrammian respectively. These solutions are verified by direct substitutions. Dynamics of some obtained solutions are illustrated.展开更多
In this paper, we theoretically investigate the four-soliton interaction and their resonance phenomena of the （3＋1）- dimensional Kadomtsev Petviashvili （KP） equation. We find that the maximum amplitude of the res...In this paper, we theoretically investigate the four-soliton interaction and their resonance phenomena of the （3＋1）- dimensional Kadomtsev Petviashvili （KP） equation. We find that the maximum amplitude of the resonantly created soliton can be 16 times that of one of the four equi-amplitude initial interacting solitons. We also find that the maximum amplitude can only be 4 times the initial soliton amplitude when the resonance phenomena does not take place. The case of four solitons with different amplitudes also has been studied analytically. The results indicate that the resonance phenomena still exists in this case. Numerical results confirm the theoretical predictions.展开更多
In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pa...In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.展开更多
By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including t...By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.展开更多
In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to es...In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.展开更多
By using Xu's stable-range method,families of explicit exact solutions with multiple parameter functions for the(2+1)-dimensional breaking soliton and KadomtsevPetviashvili equations.These parameter functions make...By using Xu's stable-range method,families of explicit exact solutions with multiple parameter functions for the(2+1)-dimensional breaking soliton and KadomtsevPetviashvili equations.These parameter functions make our solutions more applicable to related practical models and boundary value problems.展开更多
An extended form of the modified Kadomtsev-Petviashvili （mKP） equation is investigated. The simplified form of the Hirota bilinear method established by Hereman and Nuseir is employed. Multi-front wave solutions are...An extended form of the modified Kadomtsev-Petviashvili （mKP） equation is investigated. The simplified form of the Hirota bilinear method established by Hereman and Nuseir is employed. Multi-front wave solutions are formally derived to the extended mKP equation and the mKP equation. The results show that the extension terms do not kill the integrability of the mKP equation.展开更多
(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equ...(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.展开更多
Using the solution of general Korteweg-de Vries （KdV） equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili （KP） equation are constructed, and then its new solitary wave-like solut...Using the solution of general Korteweg-de Vries （KdV） equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili （KP） equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.展开更多
In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing ...In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.展开更多
This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it...This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation （VCKP）, the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.展开更多
The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the...The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the bilinear Backlund transformation from its Lax pairs and find solutions with the help of the obtained bilinear Bgcklund transformation.展开更多
The spatial-temporal bifurcation for Kadomtsev-Petviashvili （KP） equations is considered. Exact two-soliton solution and doubly periodic solution to the KP-I equation, and two classes of periodic soliton solutions i...The spatial-temporal bifurcation for Kadomtsev-Petviashvili （KP） equations is considered. Exact two-soliton solution and doubly periodic solution to the KP-I equation, and two classes of periodic soliton solutions in different directions to KP-Ⅱ are obtained using the bilinear form, homoclinic test technique and temporal and 1 spatial transformation method, respectively. The equilibrium solution uo =-1/6, a unique spatial-temporal bifurcation which is periodic bifurcation for KP-I and deflexion of soliton for KP-Ⅱ, is investigated.展开更多
Using the standard truncated Painlevé analysis approach, we have obtained some new special types of multisoliton solutions of a (2+ 1)-dimensionM integrable model, the modified Kadomtsev-Petviashvili (mKP) equation.
Solutions in the Crammian form for a non-isospectral Kadomtsev-Petviashvili equation are derived by means of Pfaffian derivative formulae. Explicit entries of the Crammian are given. Non-isospectral dynamics of the so...Solutions in the Crammian form for a non-isospectral Kadomtsev-Petviashvili equation are derived by means of Pfaffian derivative formulae. Explicit entries of the Crammian are given. Non-isospectral dynamics of the solutions generated from the Crammian are investigated in an analytic way. The solutions obtained can describe line solitons in non-uniform media travelling with time-dependent amplitude and time-dependent direction. In addition, some other solutions have singularities.展开更多
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 12235007, 11975131, and 12275144)the K. C. Wong Magna Fund in Ningbo Universitythe Natural Science Foundation of Zhejiang Province of China (Grant No. LQ20A010009)
文摘Starting with a decomposition conjecture,we carefully explain the basic decompositions for the Kadomtsev-Petviashvili(KP)equation as well as the necessary calculation procedures,and it is shown that the KP equation allows the Burgers-STO(BSTO)decomposition,two types of reducible coupled BSTO decompositions and the BSTO-KdV decomposition.Furthermore,we concentrate ourselves on pointing out the main idea and result of Bäcklund transformation of the KP equation based on a special superposition principle in the particular context of the BSTO decompositions.Using the framework of standard Lie point symmetry theory,these decompositions are studied and the problem of computing the corresponding symmetry constraints is treated.
基金supported by the National Key Basic Research Project of China (Grant No 2004CB318000)the National Natural Science Foundation of China (Grant No 10771072)+1 种基金the PhD Program Scholarship Fund of ECNU2008 (Grant No 20080052)the Youth Foundation of Inner Mongolia Normal University, China (Grant No QN07035)
文摘2N line-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation can be presented by resorting to the Hirota bilinear method. By extending the real parameters into complex parameters, this paper obtains N periodic-soliton solutions of the （2＋1）-dimensional Kadomtsev-Petviashvili equation from the 2N line-soliton solutions.
基金The project supported by the Key Project of the Ministry of Education under Grant No.106033the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024+2 种基金National Natural Science Foundation of China under Grant Nos.60372095 and 60772023the Open Fund of the State Key Laboratory of Software Development Environment under Grant No.SKLSDE07-001Beijing University of Aeronautics and Astronautics,and the National Basic Research Program of China(973 Program)under Grant No.2005CB321901
文摘In this paper, we derive the bilinear form for a variable-coefficient Kadomtsev Petviashvili-typed equation. Based on the bilinear form, we obtain the Wronskian determinant solution, which is proved to be indeed an exact solution of this equation through the Wronskian technique. In addition, we testify that this equation can be reduced to a Jacobi identity by considering its solution as a Grammian determinant by means of Pfaffian derivative formulae.
基金Supported by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant Nos. 20060006024 and 20080013006Chinese Ministry of Education, by the National Natural Science Foundation of China under Grant No. 60772023+2 种基金by the Open Fund of the State Key Laboratory of Software Development Environment under Grant No. SKLSDE-07-001Beijing University of Aeronautics and Astronauticsby the National Basic Research Program of China (973 Program) under Grant No. 2005CB321901
文摘In this paper, the investigation is focused on a （3＋1）-dimensional variable-coefficient Kadomtsev- Petviashvili （vcKP） equation, which can describe the realistic nonlinear phenomena in the fluid dynamics and plasma in three spatial dimensions. In order to study the integrability property of such an equation, the Painlevé analysis is performed on it. And then, based on the truncated Painlevé expansion, the bilinear form of the （3＋1）-dimensionaJ vcKP equation is obtained under certain coefficients constraint, and its solution in the Wronskian determinant form is constructed and verified by virtue of the Wronskian technique. Besides the Wronskian determinant solution, it is shown that the （3＋1）-dimensional vcKP equation also possesses a solution in the form of the Grammian determinant.
基金Supported by the National Natural Science Foundation of China under Grant No.11071157Shanghai Leading Academic Discipline Project under Grant No.J50101Beijing Natural Science Foundation under Grant No.1101024 and PHR(IHLB)
文摘Solutions of a noncommutative nonisospectral Kadomtsev-Petviashvili equation are given in terms of quasiwronskian and quasigrammian respectively. These solutions are verified by direct substitutions. Dynamics of some obtained solutions are illustrated.
基金Project supported by the National Natural Science Foundation of China(Grant No.10575082)the Key Project Foundation of the Education Ministry of China(Grant No.209128)the Natural Science Foundation of Northwest Normal University(Grant No.NWNU-KJCXGC-03-53)
文摘In this paper, we theoretically investigate the four-soliton interaction and their resonance phenomena of the （3＋1）- dimensional Kadomtsev Petviashvili （KP） equation. We find that the maximum amplitude of the resonantly created soliton can be 16 times that of one of the four equi-amplitude initial interacting solitons. We also find that the maximum amplitude can only be 4 times the initial soliton amplitude when the resonance phenomena does not take place. The case of four solitons with different amplitudes also has been studied analytically. The results indicate that the resonance phenomena still exists in this case. Numerical results confirm the theoretical predictions.
基金supported by 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 Astronautics,the National Basic Research Program of China (973 Program) under Grant No.2005CB321901by the Specialized Research Fund for the Doctoral Program of Higher Education under Grant No.20060006024the Ministry of Education
文摘In the present paper, under investigation is a nonisospectral modified Kadomtsev-Petviashvili equation, which is shown to have two Painleve branches through the Painleve analysis. With symbolic computation, two Lax pairs for such an equation are derived by applying the generalized singular manifold method. Furthermore, based on the two obtained Lax pairs, the binary Darboux transformation is constructed and then the N-th-iterated potential transformation formula in the form of Grammian is also presented.
基金supported by National Natural Science Foundation of China (Grant Nos.10735030 and 40775042)Ningbo Natural Science Foundation (Grant No. 2008A610017)+1 种基金National Basic Research Program of China (973 Program) (Grant Nos. 2005CB422301 and 2007CB814800)K.C. Wong Magna Fund in Ningbo University
文摘By means of the reductive perturbation method, three types of generalized （2＋l）-dimensional Kadomtsev- Petviashvili （KP） equations are derived from the baroclinic potential vorticity （BPV） equation, including the modified KP （mKP） equation, standard KP equation and cylindrical KP （cKP） equation. Then some solutions of generalized cKP and KP equations with certain conditions are given directly and a relationship between the generalized mKP equation and the mKP equation is established by the symmetry group direct method proposed by Lou et al. From the relationship and the solutions of the mKP equation, some solutions of the generalized mKP equation can be obtained. Furthermore, some approximate solutions of the baroclinic potential vorticity equation are derived from three types of generalized KP equations.
基金The NSF(10971046 and 11371117) of Chinathe Shandong Provincial Natural Science Foundation(ZR2013AM009)+2 种基金GIIFSDU(yzc12063)IIFSDU(2012TS020)the Project of Shandong Province Higher Educational Science and Technology Program(J09LA55)
文摘In this work, we are mainly concerned with the existence of stationary solutions for a generalized Kadomtsev-Petviashvili equation in bounded domain of Rn. We utilize variational method and critical point theory to establish our main results.
基金Foundation item： Supported by the Program of Shannxi Provincial Department of Education（11JK0482） Supported by the NSF of China（11101332） Supported by the Natural Science Foundation of Henan Province （2007140020）
文摘By using Xu's stable-range method,families of explicit exact solutions with multiple parameter functions for the(2+1)-dimensional breaking soliton and KadomtsevPetviashvili equations.These parameter functions make our solutions more applicable to related practical models and boundary value problems.
文摘An extended form of the modified Kadomtsev-Petviashvili （mKP） equation is investigated. The simplified form of the Hirota bilinear method established by Hereman and Nuseir is employed. Multi-front wave solutions are formally derived to the extended mKP equation and the mKP equation. The results show that the extension terms do not kill the integrability of the mKP equation.
文摘(2 + 1) dimensional Boussinesq and Kadomtsev-Petviashvili equation are investigated by employing Jacobi elliptic function expansion method in this paper. As a result, some new forms traveling wave solutions of the equation are reported. Numerical simulation results are shown. These new solutions may be important for the explanation of some practical physical problems. The results of this paper show that Jacobi elliptic function method can be a useful tool in obtaining evolution solutions of nonlinear system.
文摘Using the solution of general Korteweg-de Vries （KdV） equation, the solutions of the generalized variable coefficient Kadomtsev-Petviashvili （KP） equation are constructed, and then its new solitary wave-like solution and Jacobi elliptic function solution are obtained.
文摘In the paper, we will discuss the Kadomtsev-Petviashvili Equation which is used to model shallow-water waves with weakly non-linear restoring forces and is also used to model waves in ferromagnetic media by employing the method of variable separation. Abundant exact solutions including global smooth solutions and local blow up solutions are obtained. These solutions would contribute to studying the behavior and blow up properties of the solution of the Kadomtsev-Petviashvili Equation.
基金Project supported by the National Key Basic Research Project of China （2004CB318000）, the National Science Foundation of China （Grant No 10371023） and Shanghai Shuguang Project of China （Grant No 02SG02）.
文摘This paper constructs more general exact solutions than N-soliton solution and Wronskian solution for variable- coefficient Kadomtsev-Petviashvili （KP） equation. By using the Hirota method and Pfaffian technique, it finds the Grammian determinant-type solution for the variable-coefficient KP equation （VCKP）, the Wronski-type Pfaffian solution and the Gram-type Pfaffian solutions for the Pfaffianized VCKP equation.
基金Supported by the National Natural Science Foundation of China under Grant No 10371070, and the Postdoctoral Science Foundation of China.
文摘The bilinear form for a nonisospectral and variable-coefficient Kadomtsev-Petviashvili equation is obtained and some exact soliton solutions are derived by the Hirota method and Wronskian technique. We also derive the bilinear Backlund transformation from its Lax pairs and find solutions with the help of the obtained bilinear Bgcklund transformation.
基金Supported by the National Natural Science Foundation of China under Grant Nos 10361007 and 10661002, the Yunnan Natural Science Foundation （No 2004A0001M）, and The IMS, CUHK.
文摘The spatial-temporal bifurcation for Kadomtsev-Petviashvili （KP） equations is considered. Exact two-soliton solution and doubly periodic solution to the KP-I equation, and two classes of periodic soliton solutions in different directions to KP-Ⅱ are obtained using the bilinear form, homoclinic test technique and temporal and 1 spatial transformation method, respectively. The equilibrium solution uo =-1/6, a unique spatial-temporal bifurcation which is periodic bifurcation for KP-I and deflexion of soliton for KP-Ⅱ, is investigated.
文摘Using the standard truncated Painlevé analysis approach, we have obtained some new special types of multisoliton solutions of a (2+ 1)-dimensionM integrable model, the modified Kadomtsev-Petviashvili (mKP) equation.
基金Supported by the National Natural Science Foundation of China under Grant No10371070, and the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers.
文摘Solutions in the Crammian form for a non-isospectral Kadomtsev-Petviashvili equation are derived by means of Pfaffian derivative formulae. Explicit entries of the Crammian are given. Non-isospectral dynamics of the solutions generated from the Crammian are investigated in an analytic way. The solutions obtained can describe line solitons in non-uniform media travelling with time-dependent amplitude and time-dependent direction. In addition, some other solutions have singularities.
