In this paper, the extended symmetry of generalized variable-coeFficient Kadomtsev-Petviashvili （vcKP） equation is investigated by the extended symmetry group method with symbolic computation. Then on the basis of t...In this paper, the extended symmetry of generalized variable-coeFficient Kadomtsev-Petviashvili （vcKP） equation is investigated by the extended symmetry group method with symbolic computation. Then on the basis of the extended symmetry, we can establish relation among some different kinds of vcKP equations. Thus the exact solutions of these veKP equations can be constructed via the simple veKP equations or constant-coefficient KP equations.展开更多
A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, th...A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.展开更多
The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-d...The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.展开更多
In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation...In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.展开更多
Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial d...Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.展开更多
New type of variable-coefficient KP equation with self-consistent sources and its Grammian solutions are obtained by using the source generation procedure.
The purpose of this paper is to introduce a class of generaJized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A KdV-type Wronskian form...The purpose of this paper is to introduce a class of generaJized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A KdV-type Wronskian formulation is constructed by employing the Wronskian conditions of the KdV equation. Applications are made for the （3＋1）- dimensional generalized KP, BKP and Jimbo-Miwa equations, thereby presenting their Wronskian sufficient conditions. An N-soliton solution in terms of Wronskian determinant is obtained. Under a dimensional reduction, our results yield Wronskian solutions of the KdV equation.展开更多
The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of ...The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.展开更多
In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated ...In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.展开更多
A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak...A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak value of the 1-breather solution are calculated.Additionally,through the asymptotic analysis of 2-breather solution,we show that two breathers undergo an elastic collision.By applying the generalized long-wave limit method,the fundamental and second-order rogue wave solutions for the complex m Kd V equation are obtained from the 1-breather and 2-breather solutions,respectively.We also construct the hybrid solution of a breather and a fundamental rogue wave for the complex m Kd V equation from the 2-breather solution.Furthermore,the hybrid solution of two breathers and a fundamental rogue wave as well as the hybrid solution of a breather and a second-order rogue wave for the complex m Kd V equation are derived from the 3-breather solution via the generalized long-wave limit method.By controlling the phase parameters of breathers,the diverse phenomena of interaction between the breathers and the rogue waves are demonstrated.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
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
In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for...In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.展开更多
Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm...Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm,pulses in a blood vessel,or features in a circulatory system,this paper symbolically computes out an auto-B?cklund transformation via a noncharacteristic movable singular manifold,certain families of the solitonic solutions,as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation.Aiming,e.g.,at the dynamical radial displacement superimposed on the original static deformation from an arterial wall,our results rely on the axial stretch of the injured artery,blood as an incompressible Newtonian fluid,radius variation along the axial direction or aneurysmal geometry,viscosity of the fluid,thickness of the artery,mass density of the membrane material,mass density of the fluid,strain energy density of the artery,shear modulus,stretch ratio,etc.We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.展开更多
An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.
In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structur...In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.展开更多
该文考虑了Kac-van de Leur版本下超KP可积系列的一类推广系统,即s-次推广的超KP可积系列.首先,利用克利福德超代数与A型无穷维李超代数给出该系统在超费米Fock空间中的定义,其表现形式是一个用τ函数表达的恒等式.其次,利用A型超玻色-...该文考虑了Kac-van de Leur版本下超KP可积系列的一类推广系统,即s-次推广的超KP可积系列.首先,利用克利福德超代数与A型无穷维李超代数给出该系统在超费米Fock空间中的定义,其表现形式是一个用τ函数表达的恒等式.其次,利用A型超玻色-费米对应给出s-次推广的超KP可积系列在超玻色Fock空间中的像,即一个含有超变量的偏微分方程系统.在此基础上,给出该系统的超Hirota双线性形式,并从中导出KP方程和超KP方程.最后,该系统被推广到多分量情形.展开更多
In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obt...In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.展开更多
In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability prope...In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.展开更多
基金Supported by the National Natural Science Foundation of China under Grant No. 0735030Zhejiang Provincial Natural Science Foundations of China under Grant No. Y6090592+1 种基金National Basic Research Program of China (973 Program 2007CB814800)Ningbo Natural Science Foundation under Grant No. 2008A610017 and K.C. Wong Magna Fund in Ningbo University
文摘In this paper, the extended symmetry of generalized variable-coeFficient Kadomtsev-Petviashvili （vcKP） equation is investigated by the extended symmetry group method with symbolic computation. Then on the basis of the extended symmetry, we can establish relation among some different kinds of vcKP equations. Thus the exact solutions of these veKP equations can be constructed via the simple veKP equations or constant-coefficient KP equations.
文摘A generalized variable-coefficient algebraic method is appfied to construct several new families of exact solutions of physical interest for （3＋1）-dimensional Kadomtsev-Petviashvilli （KP） equation. Among them, the Jacobi elliptic periodic solutions exactly degenerate to the soliton solutions at a certain limit condition. Compared with the existing tanh method, the extended tanh method, the Jacobi elliptic function method, and the algebraic method, the proposed method gives new and more general solutions.
基金the National Natural Science Foundation of China(Grant No.11871116)the Fundamental Research Funds for the Central Universities of China(Grant No.2019XD-A11)the BUPT Innovation and Entrepreneurship Support Program,Beijing University of Posts and Telecommunications,and the National Scholarship for Doctoral Students of China.
文摘The atmosphere is an evolutionary agent essential to the shaping of a planet,while in oceanic science and daily life,liquids are commonly seen.In this paper,we investigate a generalized variable-coefficient Korteweg-de Vriesmodified Korteweg-de Vries equation for the atmosphere,oceanic fluids and plasmas.With symbolic computation,beginning with a presumption,we work out certain scaling transformations,bilinear forms through the binary Bell polynomials and our scaling transformations,N solitons(with N being a positive integer)via the aforementioned bilinear forms and bilinear auto-Bäcklund transformations through the Hirota method with some solitons.In addition,Painlevé-type auto-Bäcklund transformations with some solitons are symbolically computed out.Respective dependences and constraints on the variable/constant coefficients are discussed,while those coefficients correspond to the quadratic-nonlinear,cubic-nonlinear,dispersive,dissipative and line-damping effects in the atmosphere,oceanic fluids and plasmas.
基金The project supported by the Natural Science Foundation of Shandong Province of China under Grant Nos. 2004zx16 and Q2005A01
文摘In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.
基金Supported by the National Natural Science Foundation of China under Grant No.11171312
文摘Utilizing the Wronskian technique, a combined Wronskian condition is established for a （3＋1）-dimensional generalized KP equation. The generating functions for matrix entries satisfy a linear system of new partial differential equations. Moreover, as applications, examples of Wronskian determinant solutions, including N-soliton solutions, periodic solutions and rational solutions, are computed.
基金Supported by the NSF of Henan Province(112300410109)Supported by the NSF of the Education Department(2010A110022)
文摘New type of variable-coefficient KP equation with self-consistent sources and its Grammian solutions are obtained by using the source generation procedure.
基金Supported by the National Natural Science Foundation of China under Grant No.11371326
文摘The purpose of this paper is to introduce a class of generaJized nonlinear evolution equations, which can be widely applied to describing a variety of phenomena in nonlinear physical science. A KdV-type Wronskian formulation is constructed by employing the Wronskian conditions of the KdV equation. Applications are made for the （3＋1）- dimensional generalized KP, BKP and Jimbo-Miwa equations, thereby presenting their Wronskian sufficient conditions. An N-soliton solution in terms of Wronskian determinant is obtained. Under a dimensional reduction, our results yield Wronskian solutions of the KdV equation.
文摘The generalized (n + 1)-dimensional KP equation with variable coefficients is investigated in this paper. The bilinear form of the equation has been obtained by the Hirota direct method. In addition, with the help of Wronskian technique and the Pfaffian properties, Wronskian and Grammian solutions have been generated.
文摘In this work, the (G,/G)- --expansion method is proposed for constructing more general exact solutions of the (2 + 1)--dimensional Kadomtsev-Petviashvili (KP) equation and its generalized forms. Our work is motivated by the fact that the (G,/G)---expansion method provides not only more general forms of solutions but also periodic and solitary waves. If we set the parameters in the obtained wider set of solutions as special values, then some previously known solutions can be recovered. The method appears to be easier and faster by means of a symbolic computation system.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.12061051 and 12461048)。
文摘A compact Grammian form for N-breather solution to the complex m Kd V equation is derived using the bilinear Kadomtsev–Petviashvili hierarchy reduction method.The propagation trajectory,period,maximum points,and peak value of the 1-breather solution are calculated.Additionally,through the asymptotic analysis of 2-breather solution,we show that two breathers undergo an elastic collision.By applying the generalized long-wave limit method,the fundamental and second-order rogue wave solutions for the complex m Kd V equation are obtained from the 1-breather and 2-breather solutions,respectively.We also construct the hybrid solution of a breather and a fundamental rogue wave for the complex m Kd V equation from the 2-breather solution.Furthermore,the hybrid solution of two breathers and a fundamental rogue wave as well as the hybrid solution of a breather and a second-order rogue wave for the complex m Kd V equation are derived from the 3-breather solution via the generalized long-wave limit method.By controlling the phase parameters of breathers,the diverse phenomena of interaction between the breathers and the rogue waves are demonstrated.
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.
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.11272023the Open Fund of State Key Laboratory of Information Photonics and Optical Communications(Beijing University of Posts and Telecommunications)under GrantNo.IPOC2013B008the Fundamental Research Funds for the Central Universities of China under Grant No.2011BUPTYB02
文摘In this paper, we investigate a(3+1)-dimensional generalized variable-coefficient Kadomtsev–Petviashvili equation, which can describe the nonlinear phenomena in fluids or plasmas. Painlev′e analysis is performed for us to study the integrability, and we find that the equation is not completely integrable. By virtue of the binary Bell polynomials,bilinear form and soliton solutions are obtained, and B¨acklund transformation in the binary-Bell-polynomial form and bilinear form are derived. Soliton collisions are graphically discussed: the solitons keep their original shapes unchanged after the collision except for the phase shifts. Variable coefficients are seen to affect the motion of solitons: when the variable coefficients are chosen as the constants, solitons keep their directions unchanged during the collision; with the variable coefficients as the functions of the temporal coordinate, the one soliton changes its direction.
基金supported by the National Natural Science Foundation of China under Grant Nos.11871116 and 11772017the Fundamental Research Funds for the Central Universities of China under Grant No.2019XD-A11.
文摘Recent theoretical physics efforts have been focused on the probes for nonlinear pulse waves in,for example,variable-radius arteries.With respect to the nonlinear waves in an artery full of blood with certain aneurysm,pulses in a blood vessel,or features in a circulatory system,this paper symbolically computes out an auto-B?cklund transformation via a noncharacteristic movable singular manifold,certain families of the solitonic solutions,as well as a family of the similarity reductions for a variable-coefficient generalized forced–perturbed Korteweg–de Vries–Burgers equation.Aiming,e.g.,at the dynamical radial displacement superimposed on the original static deformation from an arterial wall,our results rely on the axial stretch of the injured artery,blood as an incompressible Newtonian fluid,radius variation along the axial direction or aneurysmal geometry,viscosity of the fluid,thickness of the artery,mass density of the membrane material,mass density of the fluid,strain energy density of the artery,shear modulus,stretch ratio,etc.We also highlight that the shock-wave structures from our solutions agree well with those dusty-plasma-experimentally reported.
文摘An infinite number of semi-discrete and continuous conservation laws for the differential-difference KP equation were obtained by using a solvable generalized Riccati equation.
基金Project supported by the National Natural Science Foundation of China (Grant Nos 10747141 and 10735030)National Basic Research Program of China (Grant No 2007CB814800)+2 种基金Natural Science Foundations of Zhejiang Province of China (Grant No605408)Ningbo Natural Science Foundation (Grant Nos 2007A610049 and 2008A610017)K. C.Wong Magna Fund in Ningbo University
文摘In this paper, the Lie symmetry algebra of the coupled Kadomtsev-Petviashvili （cKP） equation is obtained by the classical Lie group method and this algebra is shown to have a Kac-Moody-Virasoro loop algebra structure. Then the general symmetry groups of the cKP equation is also obtained by the symmetry group direct method which is proposed by Lou et alo From the general symmetry groups, the Lie symmetry group can be recovered and a group of discrete transformations can be derived simultaneously. Lastly, from a known simple solution of the cKP equation, we can easily obtain two new solutions by the general symmetry groups.
文摘该文考虑了Kac-van de Leur版本下超KP可积系列的一类推广系统,即s-次推广的超KP可积系列.首先,利用克利福德超代数与A型无穷维李超代数给出该系统在超费米Fock空间中的定义,其表现形式是一个用τ函数表达的恒等式.其次,利用A型超玻色-费米对应给出s-次推广的超KP可积系列在超玻色Fock空间中的像,即一个含有超变量的偏微分方程系统.在此基础上,给出该系统的超Hirota双线性形式,并从中导出KP方程和超KP方程.最后,该系统被推广到多分量情形.
文摘In this paper, we first consider exact solutions for Lienard equation with nonlinear terms of any order. Then,explicit exact bell and kink profile solitary-wave solutions for many nonlinear evolution equations are obtained by means of results of the Lienard equation and proper deductions, which transform original partial differential equations into the Lienard one. These nonlinear equations include compound KdV, compound KdV-Burgers, generalized Boussinesq,generalized KP and Ginzburg-Landau equation. Some new solitary-wave solutions are found.
基金The author would like to thank the Deanship of Scientific Re-search,Majmaah University,Saudi Arabia,for funding this work under project No.R-2021-222.
文摘In this study,the generalized modified variable-coefficient KdV equation with external-force term(gvcmKdV)describing atmospheric blocking located in the mid-high latitudes over ocean is studied for integrability property by using consistent Riccati expansion solvability and the necessary integrability conditions between the function coefficients are obtained.Moreover,several new solutions have been constructed for the gvcmKdV.Additionally,the classical direct similarity reduction method is used to re-duce the gvcmKdV to a nonlinear ordinary differential equation.Building on the solutions given in the previous literature for the reduced equation,many novel solitary and periodic wave solutions have been obtained for the gvcmKdV.
