Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation i...Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.展开更多
The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painl...The famous Kadomtsev-Petviashvili(KP)equation 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion 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, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.
The homogeneous balance method, which is simple and straightforward, is extended to seek for Backlund transformation, exact bell shape soliton solutions and similarity reduction of the Boussinesq equation. The method...The homogeneous balance method, which is simple and straightforward, is extended to seek for Backlund transformation, exact bell shape soliton solutions and similarity reduction of the Boussinesq equation. The method can be used in general.展开更多
A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the non...A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the nonlinear φ4 model are given. Using the symmetry theory, the Lie point symmetries and symmetry reductions of the coupled KdV equation are presented. The results show that the coupled KdV equation possesses infinitely many symmetries and may be considered as an integrable system. Also, the Palnleve test shows the coupled KdV equation possesses Palnleve property. The Backlund transformations of the coupled KdV equation related to Palnleve property and residual symmetry are shown.展开更多
In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series...In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.展开更多
By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are use...By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are used to obtain three families of solutions for WBK equations, on of which is the family of solitary wave solutions.展开更多
In this paper, an explicit Bgcklund transformation （BT） of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669]. Ba...In this paper, an explicit Bgcklund transformation （BT） of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669]. Based on the BT and some newly obtained seed solutions, infinite sequences of exact solutions for the Burgers equation are generated. Further more, this BT of the Burgers equation is applied to solve the variant Boussinesq equations and the approximate equations of long water wave.展开更多
Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund ...Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.展开更多
In this paper, we obtain a 1＋1 dimensional integrable differential-difference model for the sine-Gordon equation by Hirota＇s discretization method. A bilinear Backlund transformation and the associated Lax pair are ...In this paper, we obtain a 1＋1 dimensional integrable differential-difference model for the sine-Gordon equation by Hirota＇s discretization method. A bilinear Backlund transformation and the associated Lax pair are also proposed/or this model.展开更多
Based on the method developed by Nucci, the pseudopotentials, Lax pairs and the mngulanty mamtoia equations of the generalized fifth-order KdV equation are derived. By choosing different coefficient, the corresponding...Based on the method developed by Nucci, the pseudopotentials, Lax pairs and the mngulanty mamtoia equations of the generalized fifth-order KdV equation are derived. By choosing different coefficient, the corresponding results and the Backlund transformations can be obtained on three conditioners which include Caudrey-Dodd-Cibbon- Sawada-Kotera equation, the Lax equation and the Kaup-kupershmidt equation.展开更多
This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application...This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.展开更多
By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation th...By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.展开更多
In this paper, with the help of the Lax representation, we show the existence of infinitely many conservation laws for a differential-difference equation,which is one of the Ladic-Ablowitz hierarchy, and the conservat...In this paper, with the help of the Lax representation, we show the existence of infinitely many conservation laws for a differential-difference equation,which is one of the Ladic-Ablowitz hierarchy, and the conservation density and the associated flux are given formularlly. We also demonstrate the relation between a continuous partial differential equation and the differential-difference equation, and give Backlund transformation for the former.展开更多
This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations.By interpreting conservation laws as closed 1-forms,we formalize reciprocal transforma...This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations.By interpreting conservation laws as closed 1-forms,we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle.This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions,where fiber coordinates are generated by potential functions of the conservation laws.This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-Backlund transformations are composite diffeomorphisms.These results are applied to several classical integrable models.展开更多
To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations （NEE（s））, by developing two characteristics of construction and mechanization on auxiliary equation method, the second ki...To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations （NEE（s））, by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptie equation is highly studied and new type solutions and Backlund transformation are obtained. Then （2＋ l ）-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type.展开更多
Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broe...Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broer-Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.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
A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable syst...A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a B?cklund transformation for the resulting integrable lattice equations.展开更多
The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal r...The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.展开更多
基金supported by the National Nature Science Foundation of China under Grant No.11871116Fundamental Research Funds for the Central Universities of China under Grant No. 2019XD-A11。
文摘Water waves are one of the most common phenomena in nature, the studies of which help energy development, marine/offshore engineering, hydraulic engineering, mechanical engineering, etc. Hereby, symbolic computation is performed on the Boussinesq–Burgers system for shallow water waves in a lake or near an ocean beach. For the water-wave horizontal velocity and height of the water surface above the bottom, two sets of the bilinear forms through the binary Bell polynomials and N-soliton solutions are worked out, while two auto-B?cklund transformations are constructed together with the solitonic solutions, where N is a positive integer. Our bilinear forms, N-soliton solutions and B?cklund transformations are different from those in the existing literature. All of our results are dependent on the waterwave dispersive power.
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 1 s a classical equation In soliton tneory.A Backlund transformation between the KP equation and the Schwarzian KP equation is demonstrated by means of the truncated Painlev6 expansion 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, a new approach to Backlund transformations of nonlinear evolution equations is presented. The results obtained by this procedure are completely the same as that by Painleve truncating expansion.
文摘The homogeneous balance method, which is simple and straightforward, is extended to seek for Backlund transformation, exact bell shape soliton solutions and similarity reduction of the Boussinesq equation. The method can be used in general.
基金Supported by the National Natural Science Foundation of China under Grant Nos.11675084 and 11435005Ningbo Natural Science Foundation under Grant No.2015A610159+1 种基金granted by the Opening Project of Zhejiang Provincial Top Key Discipline of Physics Sciences in Ningbo University under Grant No.xkzwl1502sponsored by K.C.Wong Magna Fund in Ningbo University
文摘A coupled KdV equation is studied in this manuscript. The exact solutions, such as the periodic wave solutions and solitary wave solutions by means of the deformation and mapping approach from the solutions of the nonlinear φ4 model are given. Using the symmetry theory, the Lie point symmetries and symmetry reductions of the coupled KdV equation are presented. The results show that the coupled KdV equation possesses infinitely many symmetries and may be considered as an integrable system. Also, the Palnleve test shows the coupled KdV equation possesses Palnleve property. The Backlund transformations of the coupled KdV equation related to Palnleve property and residual symmetry are shown.
文摘In the present paper, an equation of nonlinear chromatography is derived from the physical chemistry A recursion formula of the symmetries of the equation as well as an infinite number of symmetries is found. A series of Backlund transformations of the equation are constructed by means of the symmetries. The exact solutions of two boundary-initial value problems on the half straight line for the equation are given m terms of the solutions of the corresponding linear problems.
文摘By using a new method and Mathematica, the Backlund transformations for Whitham-Broer-Kaup equations (WBK) are derived. The connections between WBK equation, heat equation and Burgers equation are found, which are used to obtain three families of solutions for WBK equations, on of which is the family of solitary wave solutions.
基金中国博士后科学基金，国家重点基础研究发展计划(973计划)，the National Key Basic Research Project of China under
文摘In this paper, an explicit Bgcklund transformation （BT） of the Burgers equation is obtained by using the further extended tanh method [Phys. Lett. A 307 （2003） 269; Chaos, Solitons ＆ Fractals 17 （2003） 669]. Based on the BT and some newly obtained seed solutions, infinite sequences of exact solutions for the Burgers equation are generated. Further more, this BT of the Burgers equation is applied to solve the variant Boussinesq equations and the approximate equations of long water wave.
基金Supported by National Natural Science Foundation of China under Grant No. 10671121Shanghai Leading Academic Discipline Project under Grant No. J50101
文摘Generalized Casoratian condition and Casoratian solutions of the Toda lattice are given in terms of its bilinear Bgcklund transformation. By choosing suitable Casoratian entries and parameter in the bilinear Bgcklund transformation, we can give transformations among many kinds of solutions.
基金The project supported by National Natural Science Foundation of China under Grant No. 90203001, the Fund of the State Key Laboratory of Scientific and Engineering Computing, the Chinese Academy of Sciences, and Hong Kong Research Grant Council under Grant No. HKBU/2016/03P
文摘In this paper, we obtain a 1＋1 dimensional integrable differential-difference model for the sine-Gordon equation by Hirota＇s discretization method. A bilinear Backlund transformation and the associated Lax pair are also proposed/or this model.
基金Supported by the National Natural Science Foundation of China under Grant Nos.10735030,11075055,and 90718041the Shanghai Leading Academic Discipline Project,China under Grant No.B412+1 种基金the Program for Changjiang Scholars,the Innovative Research Team in University of Ministry of Education of China under Grant No.IRT 0734the K.C.Wong Magna Fund in Ningbo University
文摘Based on the method developed by Nucci, the pseudopotentials, Lax pairs and the mngulanty mamtoia equations of the generalized fifth-order KdV equation are derived. By choosing different coefficient, the corresponding results and the Backlund transformations can be obtained on three conditioners which include Caudrey-Dodd-Cibbon- Sawada-Kotera equation, the Lax equation and the Kaup-kupershmidt equation.
基金sponsored by the National Natural Science Foundations of China(Nos.12301315,12235007,11975131)the Natural Science Foundation of Zhejiang Province(No.LQ20A010009).
文摘This paper introduces a modified formal variable separation approach,showcasing a systematic and notably straightforward methodology for analyzing the B-type Kadomtsev-Petviashvili(BKP)equation.Through the application of this approach,we successfully ascertain decomposition solutions,Bäcklund transformations,the Lax pair,and the linear superposition solution associated with the aforementioned equation.Furthermore,we expand the utilization of this technique to the C-type Kadomtsev-Petviashvili(CKP)equation,leading to the derivation of decomposition solutions,Bäcklund transformations,and the Lax pair specific to this equation.The results obtained not only underscore the efficacy of the proposed approach,but also highlight its potential in offering a profound comprehension of integrable behaviors in nonlinear systems.Moreover,this approach demonstrates an efficient framework for establishing interrelations between diverse systems.
基金Supported by the National Natural Science Foundation of China under Grant Nos. 10735030, 90718041, and 40975038Shanghai Leading Academic Discipline Project under Grant No. B412Program for Changjiang Scholars and Innovative Research Team in University (IRT0734)
文摘By the Backlund transformation method, an important （2＋1）-dimensional nonlinear barotropie and quasigeostrophic potential vorticity （BQGPV） equation is investigated. Some simple special Backlund transformation theorems are proposed and used to get explicit solutions of the BQGPV equation. Furthermore, all solutions of a second order linear ordinary differential equation including an arbitrary function can be used to construct explicit solutions of the （2＋1）-dimensional BQGPV equation. Some figures are also given out to describe these solutions.
基金Supported by the NSF of Henan Prevince(062110300)
文摘In this paper, with the help of the Lax representation, we show the existence of infinitely many conservation laws for a differential-difference equation,which is one of the Ladic-Ablowitz hierarchy, and the conservation density and the associated flux are given formularlly. We also demonstrate the relation between a continuous partial differential equation and the differential-difference equation, and give Backlund transformation for the former.
基金sponsored by the National Natural Science Foundation of China(Nos.12235007,11975131)。
文摘This paper presents a geometric perspective that connects reciprocal transformations with multidimensional integrable deformations.By interpreting conservation laws as closed 1-forms,we formalize reciprocal transformations as induced local diffeomorphisms on the jet bundle.This allows us to characterize higher-dimensional deformations as systematic fiber bundle extensions,where fiber coordinates are generated by potential functions of the conservation laws.This perspective provides an interpretation for the covariant lifting of Lax pairs to higher dimensions and reveals that auto-Backlund transformations are composite diffeomorphisms.These results are applied to several classical integrable models.
基金Supported by the Natural Natural Science Foundation of China under Grant No.10461006the Science Research Foundation of Institution of Higher Education of Inner Mongolia Autonomous Region,China under Grant No.NJZZ07031the Natural Science Foundation of Inner Mongolia Autonomous Region,China under Grant No.2010MS0111
文摘To seek new infinite sequence soliton-like exact solutions to nonlinear evolution equations （NEE（s））, by developing two characteristics of construction and mechanization on auxiliary equation method, the second kind of elliptie equation is highly studied and new type solutions and Backlund transformation are obtained. Then （2＋ l ）-dimensional breaking soliton equation is chosen as an example and its infinite sequence soliton-like exact solutions are constructed with the help of symbolic computation system Mathematica, which include infinite sequence smooth soliton-like solutions of Jacobi elliptic type, infinite sequence compact soliton solutions of Jacobi elliptic type and infinite sequence peak soliton solutions of exponential function type and triangular function type.
文摘Starting from a Backlund transformation and taking a special ansatz for the function f, we can obtain a much more generalexpression of solution that includes some variable separated functions for the higher-order Broer-Kaup system. From this expression, we investigate the interactions of localized coherent structures such as the multi-solitonic excitations and find the novel phenomenon that their interactions have non-elastic behavior because the fission/fusion may occur after the interaction of each localized coherent structure.
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.
文摘A discrete spectral problem is discussed, and a hierarchy of integrable nonlinear lattice equations related to this spectral problem is devised. The new integrable symplectic map and finite-dimensional integrable systems are given by nonlinearization method. The binary Bargmann constraint gives rise to a B?cklund transformation for the resulting integrable lattice equations.
基金Project supported by the National Natural Science Foundation of China(Grant Nos.11305031,11365017,and 11305106)the Natural Science Foundation of Guangdong Province,China(Grant No.S2013010011546)+2 种基金the Natural Science Foundation of Zhejiang Province,China(Grant No.LQ13A050001)the Science and Technology Project Foundation of Zhongshan,China(Grant Nos.2013A3FC0264 and 2013A3FC0334)the Training Programme Foundation for Outstanding Young Teachers in Higher Education Institutions of Guangdong Province,China(Grant No.Yq2013205)
文摘The residual symmetries of the Ablowitz-Kaup-Newell-Segur （AKNS） equations are obtained by the truncated Painleve analysis. The residual symmetries for the AKNS equations are proved to be nonlocal and the nonlocal residual symmetries are extended to the local Lie point symmetries of a prolonged AKNS system. The local Lie point symme- tries of the prolonged AKNS equations are composed of the residual symmetries and the standard Lie point symmetries, which suggests that the residual symmetry method is a useful complement to the classical Lie group theory. The calcula- tion on the symmetries shows that the enlarged equations are invariant under the scaling transformations, the space-time translations, and the shift translations. Three types of similarity solutions and the reduction equations are demonstrated. Furthermore, several types of exact solutions for the AKNS equations are obtained with the help of the symmetry method and the Backlund transformations between the AKNS equations and the Schwarzian AKNS equation.
