Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable coup...Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.展开更多
This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of f...This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.展开更多
Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We ap...Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure. The method can be generalized to the other fractional soliton hierarchy.展开更多
In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. Prom the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquis...In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. Prom the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.展开更多
A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetrie...A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetries and solutions to Riemann-Hilbert problems in soltion theory.展开更多
A Bargmann symmetry constraint is proposed for the Lax pairg and the adjoint Lax pairs of the Dirac systems.It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensiona...A Bargmann symmetry constraint is proposed for the Lax pairg and the adjoint Lax pairs of the Dirac systems.It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville integrable Hamiltonian system and that nnder the control of the spatial part,the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are interpreted as a hierarchy of commntative,finite dimensional Lionville integrable Hamiltonian systems whose Hamiltonian functions consist of a series of integrals of motion for the spatial part.Moreover an involutive representation of solutions of the Dirac systema exhibits their integrability by quadratures.This kind of symmetry constraint procedure involving the spectral problem and the adjoint spectral problem is referred to as a binary nonlinearization technique like a binary Darboux transformation.展开更多
基金supported by the National Natural Science Foundation of China(1127100861072147+1 种基金11071159)the First-Class Discipline of Universities in Shanghai and the Shanghai University Leading Academic Discipline Project(A13-0101-12-004)
文摘Based on fractional isospectral problems and general bilinear forms, the gener-alized fractional trace identity is presented. Then, a new explicit Lie algebra is introduced for which the new fractional integrable couplings of a fractional soliton hierarchy are derived from a fractional zero-curvature equation. Finally, we obtain the fractional Hamiltonian structures of the fractional integrable couplings of the soliton hierarchy.
基金Project supported by the National Natural Science Foundation of China （Grant No 10471132）, and the National Key Basic Research Special Foundation of China （Grant No 113000531034）.Acknowledgments The authors are obliged to the anonymous referee for his valuable remarks and suggestions.
文摘This paper is devoted to the study of the underlying linearities of the coupled Harry-Dym （cHD） soliton hierarchy, including the well-known cHD equation. Resorting to the nonlinearization of Lax pairs, a family of finite-dimensional Hamiltonian systems associated with soliton equations are presented, constituting the decomposition of the cHD soliton hierarchy. After suitably introducing the Abel-Jacobi coordinates on a Riemann surface, the cHD soliton hierarchy can be ultimately reduced to linear superpositions, expressed by the Abel-Jacobi variables.
基金Project supported by the National Natural Science Foundation of China (Grant Nos. 11271008,61072147,and 11071159 )the Shanghai Leading Academic Discipline Project,China (Grant No. J50101)
文摘Based on the differential forms and exterior derivatives of fractional orders, Wu first presented the generalized Tu formula to construct the generalized Hamiltonian structure of the fractional soliton equation. We apply the generalized Tu formula to calculate the fractional Dirac soliton equation hierarchy and its Hamiltonian structure. The method can be generalized to the other fractional soliton hierarchy.
基金Project supported by the Scientific Research Fundation of the Education Department of Liaoning Province,China(GrantNo.L2010513)the China Postdoctoral Science Foundation(Grant No.2011M500404)
文摘In this paper, a new nonlinear integrable coupling system of the soliton hierarchy is presented. Prom the Lax pairs, the coupled KdV equations are constructed successfully. Based on the prolongation method of Wahlquist and Estabrook, we study the prolongation structure of the nonlinear integrable couplings of the KdV equation.
基金Supported by the National Natural Science Foundation of China(11975145,11972291)The National Science Foundation(DMS-1664561)。
文摘A generalized Liouville’s formula is established for linear matrix differential equations involving left and right multiplications.Its special cases are used to determine the localness of characteristics of symmetries and solutions to Riemann-Hilbert problems in soltion theory.
文摘A Bargmann symmetry constraint is proposed for the Lax pairg and the adjoint Lax pairs of the Dirac systems.It is shown that the spatial part of the nonlinearized Lax pairs and adjoint Lax pairs is a finite dimensional Liouville integrable Hamiltonian system and that nnder the control of the spatial part,the time parts of the nonlinearized Lax pairs and adjoint Lax pairs are interpreted as a hierarchy of commntative,finite dimensional Lionville integrable Hamiltonian systems whose Hamiltonian functions consist of a series of integrals of motion for the spatial part.Moreover an involutive representation of solutions of the Dirac systema exhibits their integrability by quadratures.This kind of symmetry constraint procedure involving the spectral problem and the adjoint spectral problem is referred to as a binary nonlinearization technique like a binary Darboux transformation.
