The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two ki...The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.展开更多
A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transfor...A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transformation(DT)which is derived by Taylor expansion and determinants.We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions,singular breather and periodic wave interaction solution,singular breather and traveling wave interaction solution,bimodal breather and periodic wave interaction solution by two spectral parameters.We found a general formula for these solutions in the form of determinants.We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the(2+1)-dimensional reverse space–time nonlocal NLS equation.展开更多
The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis...The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.展开更多
This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave i...This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.展开更多
The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method.The nonlocal symmetries a...The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method.The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables.The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly.Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.展开更多
With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation ...With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the(3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.展开更多
Symmetry reduction method is one of the best ways to find exact solutions. In this paper, we study the possibility of symmetry reductions of the well known Burgers equation including the nonlocal symmetry. The related...Symmetry reduction method is one of the best ways to find exact solutions. In this paper, we study the possibility of symmetry reductions of the well known Burgers equation including the nonlocal symmetry. The related new group invariant solutions are obtained. Especially, the interactions among solitons, Airy waves, and Kummer waves are explicitly given.展开更多
We investigate the global structures of the non-selfsimilar solutions for n-dimensional(n-D) nonhomogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a(n-1)-...We investigate the global structures of the non-selfsimilar solutions for n-dimensional(n-D) nonhomogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a(n-1)-dimensional sphere. We first obtain the expressions of n-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves,we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the n-D shock waves. The asymptotic behaviors with geometric structures are also proved.展开更多
The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for...The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.展开更多
Based on the bosonization approach, the supersymmetric Burgers(SB) system is transformed to a coupled bosonic system. By solving the bosonized SB(BSB) equation, the difficulties caused by the anticommutative fermionic...Based on the bosonization approach, the supersymmetric Burgers(SB) system is transformed to a coupled bosonic system. By solving the bosonized SB(BSB) equation, the difficulties caused by the anticommutative fermionic field of the SB equation can be avoided. The nonlocal symmetry for the BSB equation is obtained by the truncated Painlev′e method. By introducing multiple new fields, the finite symmetry transformation for the BSB equation is derived by solving the first Lie's principle of the prolonged systems. Some group invariant solutions are obtained with the similarity reductions related by the nonlocal symmetry.展开更多
In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the ...In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the(2+1)-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion(CRE) solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.展开更多
The nonlocal symmetry of the Sawada–Kotera(SK)equation is constructed with the known Lax pair.By introducing suitable and simple auxiliary variables,the nonlocal symmetry is localized and the finite transformation an...The nonlocal symmetry of the Sawada–Kotera(SK)equation is constructed with the known Lax pair.By introducing suitable and simple auxiliary variables,the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further.On the other hand,the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular,by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.展开更多
A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method.To study various exact solutions of the nonlocal Boussinesq equation,it is converted into two local equations whi...A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method.To study various exact solutions of the nonlocal Boussinesq equation,it is converted into two local equations which contain the local Boussinesq equation.From the N-soliton solutions of the local Boussinesq equation,the N-soliton solutions of the nonlocal Boussinesq equation are obtained,among which the(N=2,3,4)-soliton solutions are analyzed with graphs.Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation.Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.展开更多
In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamen...In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations(PDEs).展开更多
Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to t...Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.展开更多
The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painle...The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.展开更多
In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the clas...In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.展开更多
The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In part...The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular,an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.展开更多
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11501323,11701323,and 11605102)。
文摘The(3+1)-dimensional Burgers equation, which describes nonlinear waves in turbulence and the interface dynamics,is considered. Two types of semi-rational solutions, namely, the lump–kink solution and the lump–two kinks solution, are constructed from the quadratic function ansatz. Some interesting features of interactions between lumps and other solitons are revealed analytically and shown graphically, such as fusion and fission processes.
文摘A chain of novel higher order rational solutions with some parameters and interaction solutions of a(2+1)-dimensional reverse space–time nonlocal Schrodinger(NLS)equation was derived by a generalized Darboux transformation(DT)which is derived by Taylor expansion and determinants.We obtained a series of higher-order rational solutions by one spectral parameter and we could get the periodic wave solution and three kinds of interaction solutions,singular breather and periodic wave interaction solution,singular breather and traveling wave interaction solution,bimodal breather and periodic wave interaction solution by two spectral parameters.We found a general formula for these solutions in the form of determinants.We also analyzed the complex wave structures of the dynamic behaviors and the effects of special parameters and presented exact solutions for the(2+1)-dimensional reverse space–time nonlocal NLS equation.
基金supported by the National Natural Science Foundation of China(Grant Nos.11347183,11405110,11275129,and 11305106)the Natural Science Foundation of Zhejiang Province of China(Grant Nos.Y7080455 and LQ13A050001)
文摘The nonlocal symmetry of the generalized fifth order KdV equation(FOKdV) is first obtained by using the related Lax pair and then localizing it in a new enlarged system by introducing some new variables. On this basis, new Ba¨cklund transformation is obtained through Lie's first theorem. Furthermore, the general form of Lie point symmetry for the enlarged FOKdV system is found and new interaction solutions for the generalized FOKdV equation are explored by using a symmetry reduction method.
基金Supported by National Natural Science Foundation of China under Grant No.11505090Research Award Foundation for Outstanding Young Scientists of Shandong Province under Grant No.BS2015SF009
文摘This paper is concerned with the fifth-order modified Korteweg-de Vries(fmKdV) equation. It is proved that the fmKdV equation is consistent Riccati expansion(CRE) solvable. Three special form of soliton-cnoidal wave interaction solutions are discussed analytically and shown graphically. Furthermore, based on the consistent tanh expansion(CTE) method, the nonlocal symmetry related to the consistent tanh expansion(CTE) is investigated, we also give the relationship between this kind of nonlocal symmetry and the residual symmetry which can be obtained with the truncated Painlev′e method. We further study the spectral function symmetry and derive the Lax pair of the fmKdV equation. The residual symmetry can be localized to the Lie point symmetry of an enlarged system and the corresponding finite transformation group is computed.
基金supported by National Natural Science Foundation of China(No.11471215)Shanghai Natural Science Foundation(No.18ZR142600)。
文摘The nonlocal symmetry is derived for an equation combining the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form from the truncated Painlevéexpansion method.The nonlocal symmetries are localized to the Lie point symmetry by introducing new auxiliary dependent variables.The finite symmetry transformation and the Lie point symmetry for the prolonged system are solved directly.Many new interaction solutions among soliton and other types of interaction solutions for the modified Calogero–Bogoyavlenskii–Schiff equation with its negative-order form can be obtained from the consistent condition of the consistent tanh expansion method by selecting the proper arbitrary constants.
基金supported by the National Natural Science Foundation of China (Grant Nos. 11835011 and 12074343)。
文摘With the aid of the Painlevé analysis, we obtain residual symmetries for a new(3+1)-dimensional generalized Kadomtsev–Petviashvili(gKP) equation. The residual symmetry is localized and the finite transformation is proposed by introducing suitable auxiliary variables. In addition, the interaction solutions of the(3+1)-dimensional gKP equation are constructed via the consistent Riccati expansion method. Particularly, some analytical soliton-cnoidal interaction solutions are discussed in graphical way.
基金Supported by the National Natural Science Foundations of China under Grant Nos. 11175092, 11275123, 11205092Scientific Research Fund of Zhejiang Provincial Education Department under Grant No. Y201017148K.C. Wong Magna Fund in Ningbo University
文摘Symmetry reduction method is one of the best ways to find exact solutions. In this paper, we study the possibility of symmetry reductions of the well known Burgers equation including the nonlocal symmetry. The related new group invariant solutions are obtained. Especially, the interactions among solitons, Airy waves, and Kummer waves are explicitly given.
基金partly supported by the National Natural Science Foundation of China (Grant11701551 and Grant 11971024)partly supported by the National Natural Science Foundation of China (Grant 11471332)。
文摘We investigate the global structures of the non-selfsimilar solutions for n-dimensional(n-D) nonhomogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a(n-1)-dimensional sphere. We first obtain the expressions of n-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves,we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the n-D shock waves. The asymptotic behaviors with geometric structures are also proved.
基金Supported by the Global Change Research Program China under Grant No.2015CB953904the National Natural Science Foundations of China under Grant Nos.11435005,11175092,and 11205092+1 种基金Shanghai Knowledge Service Platform for Trustworthy Internet of Things under Grant No.ZF1213K.C.Wong Magna Fund in Ningbo University
文摘The nonlocal symmetries of the Burgers equation are explicitly given by the truncated Painlevé method.The auto-B?cklund transformation and group invariant solutions are obtained via the localization procedure for the nonlocal residual symmetries. Furthermore, the interaction solutions of the solition-Kummer waves and the solition-Airy waves are obtained.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11675146,11305106,11472177,11275129the Natural Science Foundation of Zhejiang Province of China under Grant No.LZ15A050001
文摘Based on the bosonization approach, the supersymmetric Burgers(SB) system is transformed to a coupled bosonic system. By solving the bosonized SB(BSB) equation, the difficulties caused by the anticommutative fermionic field of the SB equation can be avoided. The nonlocal symmetry for the BSB equation is obtained by the truncated Painlev′e method. By introducing multiple new fields, the finite symmetry transformation for the BSB equation is derived by solving the first Lie's principle of the prolonged systems. Some group invariant solutions are obtained with the similarity reductions related by the nonlocal symmetry.
基金Project supported by the Global Change Research Program of China(Grant No.2015CB953904)the National Natural Science Foundation of China(Grant Nos.11275072 and 11435005)+2 种基金the Doctoral Program of Higher Education of China(Grant No.20120076110024)the Network Information Physics Calculation of Basic Research Innovation Research Group of China(Grant No.61321064)the Fund from Shanghai Collaborative Innovation Center of Trustworthy Software for Internet of Things(Grant No.ZF1213)
文摘In this paper,the truncated Painlev′e analysis,nonlocal symmetry,Bcklund transformation of the(2+1)-dimensional modified Bogoyavlenskii–Schiff equation are presented.Then the nonlocal symmetry is localized to the corresponding nonlocal group by the prolonged system.In addition,the(2+1)-dimensional modified Bogoyavlenskii–Schiff is proved consistent Riccati expansion(CRE) solvable.As a result,the soliton–cnoidal wave interaction solutions of the equation are explicitly given,which are difficult to find by other traditional methods.Moreover figures are given out to show the properties of the explicit analytic interaction solutions.
基金supported by the National Natural Science Foundation of China,Grant No.11835011 and Grant No.11675146。
文摘The nonlocal symmetry of the Sawada–Kotera(SK)equation is constructed with the known Lax pair.By introducing suitable and simple auxiliary variables,the nonlocal symmetry is localized and the finite transformation and some new solutions are obtained further.On the other hand,the group invariant solutions of the SK equation are constructed with the classic Lie group method.In particular,by a Galileo transformation some analytical soliton-cnoidal interaction solutions of a asymptotically integrable equation are discussed in graphical ways.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11975156 and 12175148)the Natural Science Foundation of Zhejiang Province of China(Grant No.LY18A050001)。
文摘A nonlocal Boussinesq equation is deduced from the local one by using consistent correlated bang method.To study various exact solutions of the nonlocal Boussinesq equation,it is converted into two local equations which contain the local Boussinesq equation.From the N-soliton solutions of the local Boussinesq equation,the N-soliton solutions of the nonlocal Boussinesq equation are obtained,among which the(N=2,3,4)-soliton solutions are analyzed with graphs.Some periodic and traveling solutions of the nonlocal Boussinesq equation are derived directly from the known solutions of the local Boussinesq equation.Symmetry reduction solutions of the nonlocal Boussinesq equation are also obtained by using the classical Lie symmetry method.
基金Project supported by the National Natural Science Foundation of China(No.11571008)
文摘In this paper, a symmetry analysis of the modified 2D Burgers vortex equation with a flow parameter is presented. A general form of classical and non-classical symmetries of the equation is derived. These are fundamental tools for obtaining exact solutions to the equation. In several physical cases of the parameter, the specific classical and non-classical symmetries of the equation are then obtained. In addition to rediscovering the existing solutions given by different methods, some new exact solutions are obtained with the symmetry method, showing that the symmetry method is powerful and more general for solving partial differential equations(PDEs).
文摘Using the machinery of Lie group analysis,the nonlinear system of coupled Burgers-type equations is studied.Using the infinitesimal generators in the optimal system of subalgebra of the said Lie algebras,it leads to two nonequivalent similarity transformations by using it we obtain two reductions in the form of system of nonlinear ordinary differential equations.The search for solutions of these systems by using the G'/G-method has yielded certain exact solutions expressed by rational functions,hyperbolic functions,and trigonometric functions.Some figures are given to show the properties of the solutions.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11775047,11775146,and 11865013)the Science and Technology Project Foundation of Zhongshan City,China(Grant No.2017B1016).
文摘The famous Kadomtsev-Petviashvili(KP)equation is a classical equation in soliton theory.A B?cklund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlevéexpansion in this paper.One-parameter group transformations and one-parameter subgroup-invariant solutions for the extended KP equation are obtained.The consistent Riccati expansion(CRE)solvability of the KP equation is proved.Some interaction structures between soliton-cnoidal waves are obtained by CRE and several evolution graphs and density graphs are plotted.
文摘In this paper, we discuss one-dimensional optimal system and the invariant solutions of Coupled Burgers’ equations. By using Wu-differential characteristic set algorithm with the aid of Mathematica software, the classical symmetries of the Coupled Burgers’ equations are calculated, and the one-dimensional optimal system of Lie algebra is constructed. And we obtain the invariant solution of the Coupled Burgers’ equations corresponding to one element in one dimensional optimal system by using the invariant method. The results generalize the exact solutions of the Coupled Burgers’ equations.
文摘The generalized fractional Burgers equation is studied in this paper. Using the classical Lie symmetry method, all of the vector fields and symmetry reduction of the equation with nonlinearity are constructed. In particular,an exact solution is provided by using the ansatz method. In addition, other types of exact solution are obtained via the invariant subspace method. Finally, conservation laws for this equation are derived.