An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loo...An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loop algebra [AKG～], the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.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 scheme for generating nonisospectral integrable hierarchies is introduced.Based on the method,we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem.It follows that the co...A scheme for generating nonisospectral integrable hierarchies is introduced.Based on the method,we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem.It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space ■.By reducing these integrable hierarchies,we obtain the expanded isospectral and nonisospectral derivative nonlinear Schr?dinger equation.By using the trace identity,the biHamiltonian structure of these two hierarchies are also obtained.Moreover,some symmetries and conserved quantities of the resulting hierarchy are discussed.展开更多
<Abstract>A 3×3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra _2.Furthermore,high-order binary symmetry constraints of the corr...<Abstract>A 3×3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra _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 _2, its integrable couplings are established.展开更多
A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and...A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra ~FM of the loop algebra ~X is presented. Based on the ~FM, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.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
A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6...A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613）. Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl（4）. In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.展开更多
By using a Lie algebra, an integrable couplings of the classical-Boussinesq hierarchy is obtained. Then, theHamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-fo...By using a Lie algebra, an integrable couplings of the classical-Boussinesq hierarchy is obtained. Then, theHamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.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
A new 3M-dimensional Lie algebra X is constructed firstly. Then, the corresponding loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1....A new 3M-dimensional Lie algebra X is constructed firstly. Then, the corresponding loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1.It follows that a generalscheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then well-known multi-component TC hierarchy is obtained. Finally,an expanding loop algebra FM of the loop algebra X is presented. Based on the FM, the multi-component integrable coupling system of the generalized multi-component TC hierarchy has been worked out. The method in this paper can be applied to other nonlinear evolution equations hierarchies. It is easy to find that we can construct any finite-dimensional Lie algebra by this approach.展开更多
A simple 3M-dimensional loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multicompo...A simple 3M-dimensional loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multicomponent integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally, an expanding loop algebra FM of the loop algebra X is presented, based on the FM, the multi-component integrable coupling system of the multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolution equation hierarchies.展开更多
A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the i...A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.展开更多
A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational iden...A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.展开更多
文摘An extension of the Lie algebra A_~n-1 has been proposed [Phys. Lett. A, 2003, [STHZ]310:19-24]. In this paper, the new Lie algebra was used to construct a new higher dimensional loop algebra [AKG～]. Based on the loop algebra [AKG～], the integrable couplings system of the NLS-MKdV equations hierarchy was obtained. As its reduction case, generalized nonlinear NLS-MKdV equations were obtained. The method proposed in this letter can be applied to other hierarchies of evolution equations.
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supported by the National Natural Science Foundation of China (No.12371256)。
文摘A scheme for generating nonisospectral integrable hierarchies is introduced.Based on the method,we deduce a nonisospectral hierarchy of soliton equations by considering a linear spectral problem.It follows that the corresponding expanded isospectral and nonisospectral integrable hierarchies are deduced based on a 6 dimensional complex linear space ■.By reducing these integrable hierarchies,we obtain the expanded isospectral and nonisospectral derivative nonlinear Schr?dinger equation.By using the trace identity,the biHamiltonian structure of these two hierarchies are also obtained.Moreover,some symmetries and conserved quantities of the resulting hierarchy are discussed.
基金supported by China Postdoctoral Science Foundation and National Natural Science Foundation of China under Grant No.10471139
文摘<Abstract>A 3×3 matrix spectral problem and a Liouville integrable hierarchy are constructed by designing a new subalgebra of loop algebra _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 _2, its integrable couplings are established.
文摘A general scheme for generating a multi-component integrable equation hierarchy is proposed. A simple 3M- dimensional loop algebra ~X is produced. By taking advantage of ~X a new isospectral problem is established and then by making use of the Tu scheme the multi-component Dirac equation hierarchy is obtained. Finally, an expanding loop algebra ~FM of the loop algebra ~X is presented. Based on the ~FM, the multi-component integrable coupling system of the multi-component Dirac equation hierarchy is investigated. The method in this paper can be applied to other nonlinear evolution equation hierarchies.
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Project supported by the Natural Science Foundation of Shanghai (Grant No. 09ZR1410800)the Science Foundation of Key Laboratory of Mathematics Mechanization (Grant No. KLMM0806)+2 种基金the Shanghai Leading Academic Discipline Project (Grant No. J50101)the Key Disciplines of Shanghai Municipality (Grant No. S30104)the National Natural Science Foundation of China (Grant Nos. 61072147 and 11071159)
文摘A kind of integrable coupling of soliton equations hierarchy with self-consistent sources associated with s/（4） has been presented （Yu F J and Li L 2009 Appl. Math. Comput. 207 171; Yu F J 2008 Phys. Lett. A 372 6613）. Based on this method, we construct two integrable couplings of the soliton hierarchy with self-consistent sources by using the loop algebra sl（4）. In this paper, we also point out that there are some errors in these references and we have corrected these errors and set up new formula. The method can be generalized to other soliton hierarchy with self-consistent sources.
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+1 种基金the Shanghai Leading Academic Discipline Project under Grant No.J50101by Key Disciplines of Shanghai Municipality (S30104)
文摘By using a Lie algebra, an integrable couplings of the classical-Boussinesq hierarchy is obtained. Then, theHamiltonian structure of the integrable couplings of the classical-Boussinesq is obtained by the quadratic-form identity.
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.
基金中国科学院资助项目，the Science Foundation of Liuhui Center of Tianjin University and Nankai University，辽宁省自然科学基金
文摘A new 3M-dimensional Lie algebra X is constructed firstly. Then, the corresponding loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1.It follows that a generalscheme for generating multi-component integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then well-known multi-component TC hierarchy is obtained. Finally,an expanding loop algebra FM of the loop algebra X is presented. Based on the FM, the multi-component integrable coupling system of the generalized multi-component TC hierarchy has been worked out. The method in this paper can be applied to other nonlinear evolution equations hierarchies. It is easy to find that we can construct any finite-dimensional Lie algebra by this approach.
基金the National Natural Science Foundation of China under，南开大学-天津大学合作项目，天津大学校科研和教改项目，the Educational Committee of Liaoning Province of China under Gant
文摘A simple 3M-dimensional loop algebra X is produced, whose commutation operation defined by us is as simple and straightforward as that in the loop algebra A1. It follows that a general scheme for generating multicomponent integrable hierarchy is proposed. By taking advantage of X, a new isospectral problem is established, and then by making use of the Tu scheme the well-known multi-component Levi hierarchy is obtained. Finally, an expanding loop algebra FM of the loop algebra X is presented, based on the FM, the multi-component integrable coupling system of the multi-component Levi hierarchy is worked out. The method in this paper can be applied to other nonlinear evolution equation hierarchies.
文摘A new loop algebra and a new Lax pair are constructed, respectively. It follows that the integrable coupling of the TC hierarchy of equations, which is also an expanding integrable model, is obtained. Specially, the integrable coupling of the famous KdV equation is presented.
基金Project supported by the State Administration of Foreign Experts Affairs of Chinathe National Natural Science Foundation of China (Nos.10971136,10831003,61072147,11071159)+3 种基金the Chunhui Plan of the Ministry of Education of Chinathe Innovation Project of Zhejiang Province (No.T200905)the Natural Science Foundation of Shanghai (No.09ZR1410800)the Shanghai Leading Academic Discipline Project (No.J50101)
文摘A class of non-semisimple matrix loop algebras consisting of triangular block matrices is introduced and used to generate bi-integrable couplings of soliton equations from zero curvature equations.The variational identities under non-degenerate,symmetric and ad-invariant bilinear forms are used to furnish Hamiltonian structures of the resulting bi-integrable couplings.A special case of the suggested loop algebras yields nonlinear bi-integrable Hamiltonian couplings for the AKNS soliton hierarchy.
