The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensivel...The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.展开更多
By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solu...By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.展开更多
By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of ...By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.展开更多
Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equation...Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.展开更多
To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are pr...To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.展开更多
The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. ...The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.展开更多
In this paper, exact solutions are derived for four coupled complex nonlinear different equations by using simplified transformation method and algebraic equations.
Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular s...Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations.展开更多
The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid i...The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.展开更多
By using the (G'/G)-expansion method and the variable separation method, a new family of exact solutions of the (3+1)-dimensional Jimbo-Miwa system is obtained. Based on the derived solitary wave solutions, we o...By using the (G'/G)-expansion method and the variable separation method, a new family of exact solutions of the (3+1)-dimensional Jimbo-Miwa system is obtained. Based on the derived solitary wave solutions, we obtain some special localized excitations and study the interactions between two solitary waves of the system.展开更多
The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-P...The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.展开更多
The exact solutions of a chain of type Ⅱ are investigated. The chain of type Ⅱ is first transformed to an integrable differential-difference equation, which has the Kaup Newell spectral problem as its continuous spa...The exact solutions of a chain of type Ⅱ are investigated. The chain of type Ⅱ is first transformed to an integrable differential-difference equation, which has the Kaup Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup-Newell equation as its discrete temporal spectral problem. Then, with these spectral problems, a Darboux transformation of the transformed equation is constructed. Finally, as an application of the Darboux transformation, an exact solution of the transformed equation and thus the chain of type Ⅱ are presented.展开更多
The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The cal...The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+ 1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.展开更多
We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the gover...We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations (PDEs) and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.展开更多
The non-isospectral sine-Gordon equation with self-consistent sources is derived.Its solutions are obtainedby means of Hirota method and Wronskian technique,respectively.Non-isospectral dynamics including one-solitonc...The non-isospectral sine-Gordon equation with self-consistent sources is derived.Its solutions are obtainedby means of Hirota method and Wronskian technique,respectively.Non-isospectral dynamics including one-solitoncharacteristics,two-soliton scattering,and ghost solitons,are investigated.展开更多
The symmetry reduction method based on the Frechet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields, which is a coupled system ...The symmetry reduction method based on the Frechet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields, which is a coupled system of nonlinear partial differential equations of the second order. The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied to obtain the exact solutions.展开更多
In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions ...In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.展开更多
We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-...We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.展开更多
基金Project supported by the Natural Science Foundation of Shandong Province (Grant No.ZR2021MA084)the Natural Science Foundation of Liaocheng University (Grant No.318012025)Discipline with Strong Characteristics of Liaocheng University–Intelligent Science and Technology (Grant No.319462208)。
文摘The time-fractional modified Korteweg-de Vries(KdV)equation is committed to establish exact solutions by employing the bifurcation method.Firstly,the phase portraits and related qualitative analysis are comprehensively provided.Then,we give parametric expressions of different types of solutions matching with the corresponding orbits.Finally,solution profiles,3D and density plots of some solutions are presented with proper parametric choices.
文摘By using an improved projective Riccati equation method, this paper obtains several types of exact travelling wave solutions to the Benjamin Ono equation which include multiple soliton solutions, periodic soliton solutions and Weierstrass function solutions. Some of them are found for the first time. The method can be applied to other nonlinear evolution equations in mathematical physics.
基金Project supported by the National Natural Science Foundation of China(Grant No 10461006), the High Education Science Research Program(Grant No NJ02035) of Inner Mongolia Autonomous Region, Natural Science Foundation of Inner Mongolia Autonomous Region(Grant No 2004080201103) and the Youth Research Program of Inner Mongolia Normal University(Grant No QN005023).
文摘By use of an auxiliary equation and through a function transformation, the Jacobi elliptic function wave-like solutions, the degenerated soliton-like solutions and the triangle function wave solutions to two kinds of Korteweg de Vries (KdV) equations with variable coefficients and a KdV equation with a forcible term are constructed with the help of symbolic computation system Mathematica, where the new solutions are also constructed.
基金Supported by the Natural Science Foundation of Zhejiang Province(1 0 2 0 3 7)
文摘Nonlinear partial differetial equation(NLPDE) is converted into ordinary differential equation(ODE) via a new ansatz.Using undetermined function method,the ODE obtained above is replaced by a set of algebraic equations which are solved out with the aid of Mathematica.The exact solutions and solitary solutions of NLPDE are obtained.
基金supported by the National Natural Science Foundation of China(Grant No.10862003)the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China(Grant No.NJZZ07031)the Natural Science Foundation of Inner Mongolia Autonomous Region,China(Grant No.2010MS0111)
文摘To construct the infinite sequence new exact solutions of nonlinear evolution equations and study the first kind of elliptic function, new solutions and the corresponding B^cklund transformation of the equation are presented. Based on this, the generalized pentavalent KdV equation and the breaking soliton equation are chosen as applicable examples and infinite sequence smooth soliton solutions, infinite sequence peak solitary wave solutions and infinite sequence compact soliton solutions are obtained with the help of symbolic computation system Mathematica. The method is of significance to search for infinite sequence new exact solutions to other nonlinear evolution equations.
基金Project supported by the National Nature Science Foundation of China (Grant No 49894190) of the Chinese Academy of Science (Grant No KZCXI-sw-18), and Knowledge Innovation Program.
文摘The extended F-expansion method or mapping method is used to construct exact solutions for the coupled KleinGordon Schr/Sdinger equations (K-G-S equations) by the aid of the symbolic computation system Mathematica. More solutions in the Jacobi elliptic function form are obtained, including the single Jacobi elliptic function solutions, combined Jacobi elliptic function solutions, rational solutions, triangular solutions, soliton solutions and combined soliton solutions.
文摘In this paper, exact solutions are derived for four coupled complex nonlinear different equations by using simplified transformation method and algebraic equations.
文摘Some new exact solutions of the Burgers-Fisher equation and generalized Burgers-Fisher equation have been obtained by using the first integral method. These solutions include exponential function solutions, singular solitary wave solutions and some more complex solutions whose figures are given in the article. The result shows that the first integral method is one of the most effective approaches to obtain the solutions of the nonlinear partial differential equations.
文摘The exact solutions are obtained for unsteady unidirectional flows of a generalized second-order fluid through a rectangular conduit. The fractional calculus in the constitutive relationship of a non-Newtonian fluid is introduced. We construct the solutions by means of Fourier transform and the discrete Laplace transform of the sequential derivatives and the double finite Fourier transform. The solutions for Newtonian fluid between two infinite parallel plates appear as limiting cases of our solutions.
基金Project supported by the Scientific Research Foundation of Lishui University, China (Grant No. KZ201110)
文摘By using the (G'/G)-expansion method and the variable separation method, a new family of exact solutions of the (3+1)-dimensional Jimbo-Miwa system is obtained. Based on the derived solitary wave solutions, we obtain some special localized excitations and study the interactions between two solitary waves of the system.
基金Project Supported by The National Natural Science Foundation of China
文摘The exact solutions for stationary responses of one class of the second order and three classes of higher order nonlinear systems to parametric and/or external while noise excitations are constructed by using Fokkcr-Planck-Kolmogorov et/ualion approach. The conditions for the existence and uniqueness and the behavior of the solutions are discussed. All the systems under consideration are characterized by the dependence ofnonconservative fqrces on the first integrals of the corresponding conservative systems and arc catted generalized-energy-dependent f G.E.D.) systems. It is shown taht for each of the four classes of G.E.D. nonlinear stochastic systems there is a family of non-G.E.D. systems which are equivalent to the G.E.D. system in the sense of having identical stationary solution. The way to find the equivalent stochastic systems for a given G.E.D. system is indicated and. as an example, the equivalent stochastic systems for the second order G.E. D. nonlinear stochastic system are given. It is pointed out and illustrated with example that the exact stationary solutions for many non-G.E.D. nonlinear stochastic systems may he found by searching the equivalent G.E.D. systems.
基金Supported by the National Natural Science Foundation of China under Grant Nos 11271168 and 11671177the Priority Academic Program Development of Jiangsu Higher Education Institutionsthe Innovation Project of the Graduate Students in Jiangsu Normal University
文摘The exact solutions of a chain of type Ⅱ are investigated. The chain of type Ⅱ is first transformed to an integrable differential-difference equation, which has the Kaup Newell spectral problem as its continuous spatial spectral problem and a Darboux transformation of the Kaup-Newell equation as its discrete temporal spectral problem. Then, with these spectral problems, a Darboux transformation of the transformed equation is constructed. Finally, as an application of the Darboux transformation, an exact solution of the transformed equation and thus the chain of type Ⅱ are presented.
基金Project supported by the Natural Science Foundation of Guangdong Province, China (Grant Nos. 10452840301004616 and S2011040000403)the National Natural Science Foundation of China (Grant No. 41176005)the Science and Technology Project Foundation of Zhongshan, China (Grnat No. 20123A326)
文摘The symmetries and the exact solutions of the (3+l)-dimensional nonlinear incompressible non-hydrostatic Boussi- nesq (INHB) equations, which describe atmospheric gravity waves, are studied in this paper. The calculation on symmetry shows that the equations are invariant under the Galilean transformations, the scaling transformations, and the space-time translations. Three types of symmetry reduction equations and similar solutions for the (3+ 1)-dimensional INHB equations are proposed. Traveling and non-traveling wave solutions of the INHB equations are demonstrated. The evolutions of the wind velocities in latitudinal, longitudinal, and vertical directions with space-time are demonstrated. The periodicity and the atmosphere viscosity are displayed in the (3+1)-dimensional INHB system.
文摘We investigate the Einstein field equations corresponding to the Weyl-Lewis-Papapetrou form for an axisymmetric rotating field by using the classical symmetry method. Using the invafiance group properties of the governing system of partial differential equations (PDEs) and admitting a Lie group of point transformations with commuting infinitesimal generators, we obtain exact solutions to the system of PDEs describing the Einstein field equations. Some appropriate canonical variables are characterized that transform the equations at hand to an equivalent system of ordinary differential equations and some physically important analytic solutions of field equations are constructed. Also, the class of axially symmetric solutions of Einstein field equations including the Papapetrou solution as a particular case has been found.
基金The project supported by National Natural Science Foundation of China under Grant No.10371070 the Foundation of Shanghai Education Committee for Shanghai Prospective Excellent Young Teachers
文摘The non-isospectral sine-Gordon equation with self-consistent sources is derived.Its solutions are obtainedby means of Hirota method and Wronskian technique,respectively.Non-isospectral dynamics including one-solitoncharacteristics,two-soliton scattering,and ghost solitons,are investigated.
基金Project supported by the Human Resource Development Group Council of Scientific Industrial Research (CSIR),India (Grant No. 09/677(0014)2009-EMR-1)
文摘The symmetry reduction method based on the Frechet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields, which is a coupled system of nonlinear partial differential equations of the second order. The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied to obtain the exact solutions.
文摘In this paper, the nonlinear Schrodinger (NLS) equation was analytically solved. Firstly, the stationary solutions of NLS equation were explicitly given by the elliptic functions. Then a family of exact solutions of NLS equation were obtained from these stationary solutions by a method for finding new exact solutions from the stationary solutions of integrable evolution equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11275072,11075055,and 11271211)the Research Fund for the Doctoral Program of Higher Education of China(Grant No.20120076110024)+3 种基金the Innovative Research Team Program of the National Natural Science Foundation of China(Grant No.61021004)the Shanghai Leading Academic Discipline Project,China(Grant No.B412)the National High Technology Research and Development Program of China(Grant No.2011AA010101)the Shanghai Knowledge Service Platform for Trustworthy Internet of Things,China(Grant No.ZF1213)
文摘We construct various novel exact solutions of two coupled dynamical nonlinear Schrōdinger equations. Based on the similarity transformation, we reduce the coupled nonlinear Schrōdinger equations with time-and space-dependent potentials, nonlinearities, and gain or loss to the coupled dynamical nonlinear Schrrdinger equations. Some special types of non-travelling wave solutions, such as periodic, resonant, and quasiperiodically oscillating solitons, are used to exhibit the wave propagations by choosing some arbitrary functions. Our results show that the number of the localized wave of one component is always twice that of the other one. In addition, the stability analysis of the solutions is discussed numerically.