A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hi...A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.展开更多
A type of new loop algebra GM is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential f...A type of new loop algebra GM is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.展开更多
By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a ge...By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a generalized nonisospectral integrable hierarchy(GNIH).The GNIH further reduces to the standard nonlinear Schr¨odinger equation and the Kd V equation which have important applications in physics science. Based on this, we discuss the K symmetries and the τ symmetries of the generalized AKNS hierarchy u_t = K_m(u) with isospectral condition coming from the GNIH. Furthermore, we also consider the K symmetries and the τ symmetries of the nonisospectral AKNS hierarchy ut = τ_N~l +1. Finally, we obtain the symmetry Lie algebras for the both integrable hierarchies, and present some applications for the symmetries and the Lie algebras, which means that some Lie groups of transformations and the infinitesimal operators of reduced equations are generated.展开更多
A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature...A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.展开更多
Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identi...Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.展开更多
Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identit...Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.展开更多
In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy...In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.展开更多
Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system...Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system of the above hierarchy is presented. Finally, A multi-component integrable hierarchy is obtained by employing a multi-component loop algebra ↑-GM.展开更多
Based on a kind of Lie a/gebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using th...Based on a kind of Lie a/gebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using the methods proposed by Ma and Gao. With the help of variational identity, we get the Hamiltonian structure of the hierarchy.展开更多
We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and the...We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.展开更多
This paper aims to construct six-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation.Their Hamiltonian formulations are furnished by the trace identity,which ...This paper aims to construct six-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation.Their Hamiltonian formulations are furnished by the trace identity,which guarantee the commuting property of infinitely many symmetries and conserved Hamiltonian functionals.Illustrative examples of the resulting integrable equations of second and third orders are explicitly computed.展开更多
A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obt...A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.展开更多
Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation Ut-Vx ＋[U, V] = 0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian st...Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation Ut-Vx ＋[U, V] = 0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator ^~J is presented by constructing a subalgebra ^~G of the loop algebra -^~A2. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their （2＋1）-dimensional forms are derived from a （2＋1）-dimensional zero-curvature equation.展开更多
A new higher-dimensional Lie algebra is constructed,which is used to generate multiple integrable couplingssimultaneously.From this,we come to a general approach for seeking multi-integrable couplings of the known int...A new higher-dimensional Lie algebra is constructed,which is used to generate multiple integrable couplingssimultaneously.From this,we come to a general approach for seeking multi-integrable couplings of the known integrablesoliton equations.展开更多
It is shown that the Kaup-Newell hierarchy can be derived from the so-called generating equations which are Lax integrable. Positive and negative flows in the hierarchy are derived simultaneously. The generating equat...It is shown that the Kaup-Newell hierarchy can be derived from the so-called generating equations which are Lax integrable. Positive and negative flows in the hierarchy are derived simultaneously. The generating equations and mutual commutativity of these flows enable us to construct new Lax integrable equations.展开更多
In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, ...In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.展开更多
The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system an...The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.展开更多
The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokho...The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.展开更多
Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand...Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand-join equation.Furthermore,H G·(t;p +,p) is a product of several copies of tau functions of the 2-Toda hierarchy,in independent variables.These generalize the corresponding results for ordinary Hurwitz numbers.展开更多
In the paper,we want to derive a few of nonlinear Schrodinger equations with various formats and investigate their properties,such as symmetries,single soliton solutions,multi-soliton solutions,and so on.First of all,...In the paper,we want to derive a few of nonlinear Schrodinger equations with various formats and investigate their properties,such as symmetries,single soliton solutions,multi-soliton solutions,and so on.First of all,we propose an efficient and straightforward scheme for generating nonisospectral integrable hierarchies of evolution equations for which a generalized nonisospectral integrable Schrodinger hierarchy(briefly GNISH)singles out,from which we get a derivative nonlinear Schrodinger equation,a generalized nonlocal Schrodinger integrable system and furthermore we investigate the symmetries and conserved qualities of the GNISH.Next,we apply the dbar method to obtain a generalized nonlinear Schr?dinger-Maxwell-Bloch(GNLS-MB)equation and its hierarchy by introducing a generalized Zakhrov-Shabat spectral problem,whose soliton solutions and gauge transformations are obtained.展开更多
基金supported by China Postdoctoral Science Foundation and National Natural Science Foundation of China under Grant No.10471139
文摘A 3 × 3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra A^-2. Furthermore, high-order binary symmetry constraints of the corresponding hierarchy are obtained by using the binary nonlinearization method. Finally, according to another new subalgebra of loop algebra A^-2, its integrable couplings are established.
基金The project supported by National Natural Science Foundation of China under.Grant No. 10371070
文摘A type of new loop algebra GM is constructed by making use of the concept of cycled numbers. As its application, an isospectral problem is designed and a new multi-component integrable hierarchy with multi-potential functions is worked out, which can be reduced to the famous KN hierarchy.
基金Supported by the National Natural Science Foundation of China (Grant Nos. 1197147512371256)。
文摘By applying a subgroup of the Lie group Mn C we introduce a linear nonisospectral problem whose compatibility condition gives rise to a nonisospectral integrable hierarchy of evolution equations, which reduces to a generalized nonisospectral integrable hierarchy(GNIH).The GNIH further reduces to the standard nonlinear Schr¨odinger equation and the Kd V equation which have important applications in physics science. Based on this, we discuss the K symmetries and the τ symmetries of the generalized AKNS hierarchy u_t = K_m(u) with isospectral condition coming from the GNIH. Furthermore, we also consider the K symmetries and the τ symmetries of the nonisospectral AKNS hierarchy ut = τ_N~l +1. Finally, we obtain the symmetry Lie algebras for the both integrable hierarchies, and present some applications for the symmetries and the Lie algebras, which means that some Lie groups of transformations and the infinitesimal operators of reduced equations are generated.
文摘A new multi-component Lie algebra is constructed, and a type of new loop algebra is presented. A （2＋1）-dimensional multi-component DLW integrable hierarchy is obtained by using a （2＋1）-dimensional zero curvature equation. Furthermore, the loop algebra is expanded into a larger one and a type of integrable coupling system and its corresponding Hamiltonian structure are worked out.
基金Supported by the Natural Science Foundation of Henan Province（162300410075） the Science and Technology Key Research Foundation of the Education Department of Henan Province（14A110010） the Youth Backbone Teacher Foundationof Shangqiu Normal University（2013GGJS02）
文摘Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. Then its super Hamiltonian structures were established by using super trace identity, and the conserved functionals were proved to be in involution in pairs under the defined Poisson bracket. As its reduction,special cases of this nonlinear super integrable couplings were obtained.
文摘Nonlinear super integrable couplings of a super integrable hierarchy based upon an enlarged matrix Lie super algebra were constructed. And its super Hamiltonian structures were established by using super trace identity. As its reduction, special cases of this nonlinear super integrable coupling were obtained.
文摘In general, Liouville integrable hierarchies of evolution equations were obtained by choosing proper U in zero curvature frame Ut - Vx + [U, V] = 0 first. But in the present paper, a new Liouville integrable hierarchy possessing bi-Hamiltonian structure is obtained by choosing V with derivatives in x and spectral potentials. Then integrable coupling, i.e. expanding Lax integrable model of the hierarchy obtained is presented by constructing a subalgebra of loop algebra A2.
文摘Under the frame of the （2＋1）-dimensional zero curvature equation and Tu model, （2＋1）-dimensional Tu hierarchy is obtained. Again by employing a subalgebra of the loop algebra ↑-A2 the integrable coupling system of the above hierarchy is presented. Finally, A multi-component integrable hierarchy is obtained by employing a multi-component loop algebra ↑-GM.
基金Supported by the Natural Science Foundation of China under Grant Nos.11271008,61072147,11071159the Shanghai Leading Academic Discipline Project under Grant No.J50101the Shanghai Univ.Leading Academic Discipline Project(A.13-0101-12-004)
文摘Based on a kind of Lie a/gebra G proposed by Zhang, one isospectral problem is designed. Under the framework of zero curvature equation, a new kind of integrable coupling of an equation hierarchy is generated using the methods proposed by Ma and Gao. With the help of variational identity, we get the Hamiltonian structure of the hierarchy.
文摘We present an eight component integrable Hamiltonian hierarchy, based on a reduced seventh order matrix spectral problem, with the aim of aiding the study and classification of multicomponent integrable models and their underlying mathematical structures. The zero-curvature formulation is the tool to construct a recursion operator from the spatial matrix problem. The second and third set of integrable equations present integrable nonlinear Schrödinger and modified Korteweg-de Vries type equations, respectively. The trace identity is used to construct Hamiltonian structures, and the first three Hamiltonian functionals so generated are computed.
基金supported in part by the NSFC(12271488,11975145,11972291)the Ministry of Science and Technology of China(G2021016032L,G2023016011L)the Natural Science Foundation for Colleges and Universities in Jiangsu Province(17 KJB 110020)。
文摘This paper aims to construct six-component integrable hierarchies from a kind of matrix spectral problems within the zero curvature formulation.Their Hamiltonian formulations are furnished by the trace identity,which guarantee the commuting property of infinitely many symmetries and conserved Hamiltonian functionals.Illustrative examples of the resulting integrable equations of second and third orders are explicitly computed.
基金Supported by the Natural Science Foundation of Shanghai under Grant No.09ZR1410800the Science Foundation of Key Laboratory of Mathematics Mechanization under Grant No.KLMM0806
文摘A new Lie algebra G and its two types of loop algebras G1 and G2 are constructed. Basing on G1 and G2, two different isospectral problems are designed, furthermore, two Liouville integrable soliton hierarchies are obtained respectively under the framework of zero curvature equation, which is derived from the compatibility of the isospectral problems expressed by Hirota operators. At the same time, we obtain the Hamiltonian structure of the first hierarchy and the bi-Hamiltonian structure of the second one with the help of the quadratic-form identity.
文摘Though various integrable hierarchies of evolution equations were obtained by choosing proper U in zero-curvature equation Ut-Vx ＋[U, V] = 0, but in this paper, a new integrable hierarchy possessing bi-Hamiltonian structure is worked out by selecting V with spectral potentials. Then its expanding Lax integrable model of the hierarchy possessing a simple Hamiltonian operator ^~J is presented by constructing a subalgebra ^~G of the loop algebra -^~A2. As linear expansions of the above-mentioned integrable hierarchy and its expanding Lax integrable model with respect to their dimensional numbers, their （2＋1）-dimensional forms are derived from a （2＋1）-dimensional zero-curvature equation.
基金The project supported by National Natural Science Foundation of China under Grant No.10471139
文摘A new higher-dimensional Lie algebra is constructed,which is used to generate multiple integrable couplingssimultaneously.From this,we come to a general approach for seeking multi-integrable couplings of the known integrablesoliton equations.
基金Supported by the Chinese Basic Research Project"Nonlinear Science"
文摘It is shown that the Kaup-Newell hierarchy can be derived from the so-called generating equations which are Lax integrable. Positive and negative flows in the hierarchy are derived simultaneously. The generating equations and mutual commutativity of these flows enable us to construct new Lax integrable equations.
基金Supported by the Research Grant Council of the Hong Kong Special Administrative Region(Grant No.City U101211)the National Natural Science Foundation of China(Grant No.11371361)the Natural Science Foundation of Shandong Province(Grant No.ZR2013AL016)
文摘In this paper we first present a 3-dimensional Lie algebra H and enlarge it into a 6-dimensional Lie algebra T with corresponding loop algebras？H and？T, respectively. By using the loop algebra？H and the Tu scheme, we obtain an integrable hierarchy from which we derive a new Darboux transformation to produce a set of exact periodic solutions. With the loop algebra？T, a new integrable-coupling hierarchy is obtained and reduced to some variable-coefficient nonlinear equations, whose Hamiltonian structure is derived by using the variational identity. Furthermore, we construct a higher-dimensional loop algebraˉH of the Lie algebra H from which a new Liouville-integrable hierarchy with 5-potential functions is produced and reduced to a complex m Kd V equation, whose 3-Hamiltonian structure can be obtained by using the trace identity. A new approach is then given for deriving multiHamiltonian structures of integrable hierarchies. Finally, we extend the loop algebra？H to obtain an integrable hierarchy with variable coefficients.
文摘The completely integrable Hamiltonian systems generated by the general confocal involutive system are proposed. It is proved that the nonlinearized eigenvalue problem for AKNS hierarchy is such an integrable system and showed that the time evolution equations for n≤3 obtained by nonlinearizing the time parts of Lax systems for AKNS hierarchy are Liouville integrable under the constraint of the spatial part.
基金supported in part by NSFC(11975145 and 11972291)the Natural Science Foundation for Colleges and Universities in Jiangsu Province(17 KJB 110020)。
文摘The paper aims at establishing Riemann-Hilbert problems and presenting soliton solutions for nonlocal reverse-time nonlinear Schrodinger(NLS) hierarchies associated with higher-order matrix spectral problems.The Sokhotski-Plemelj formula is used to transform the Riemann-Hilbert problems into Gelfand-Levitan-Marchenko type integral equations.A new formulation of solutions to special Riemann-Hilbert problems with the identity jump matrix,corresponding to the reflectionless inverse scattering transforms,is proposed and applied to construction of soliton solutions to each system in the considered nonlocal reversetime NLS hierarchies.
基金supported by National Natural Science Foundation of China(Grant Nos.10425101,10631050)National Basic Research Program of China(973Project)(Grant No.2006cB805905)
文摘Let G be arbitrary finite group,define H G· (t;p +,p) to be the generating function of G-wreath double Hurwitz numbers.We prove that H G· (t;p +,p) satisfies a differential equation called the colored cutand-join equation.Furthermore,H G·(t;p +,p) is a product of several copies of tau functions of the 2-Toda hierarchy,in independent variables.These generalize the corresponding results for ordinary Hurwitz numbers.
基金supported by the National Natural Science Foundation of China(No.11971475)。
文摘In the paper,we want to derive a few of nonlinear Schrodinger equations with various formats and investigate their properties,such as symmetries,single soliton solutions,multi-soliton solutions,and so on.First of all,we propose an efficient and straightforward scheme for generating nonisospectral integrable hierarchies of evolution equations for which a generalized nonisospectral integrable Schrodinger hierarchy(briefly GNISH)singles out,from which we get a derivative nonlinear Schrodinger equation,a generalized nonlocal Schrodinger integrable system and furthermore we investigate the symmetries and conserved qualities of the GNISH.Next,we apply the dbar method to obtain a generalized nonlinear Schr?dinger-Maxwell-Bloch(GNLS-MB)equation and its hierarchy by introducing a generalized Zakhrov-Shabat spectral problem,whose soliton solutions and gauge transformations are obtained.
