Within the framework of zero-curvature representation theory, the decompositions of eachequation in a hierarchy of zero-curvature equations associated with loop algebra 81(2) by meansof higher-order constraints on pot...Within the framework of zero-curvature representation theory, the decompositions of eachequation in a hierarchy of zero-curvature equations associated with loop algebra 81(2) by meansof higher-order constraints on potential are given a unified treatment, and the general schemeand uniform formulas for the decompositions are proposed. This provides a method of separationof variables to solve a hierarchy of (1+1)-dimensional integrable systems. TO illustrate the general scheme, new higher-order decompositions of two hierarchies of zero-curvature equations arepresented.展开更多
We present an integrable sl(2)-matrix Camassa-Holm(CH)equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undeter...We present an integrable sl(2)-matrix Camassa-Holm(CH)equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.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
In this paper,the translation of the Lax pairs of the Levi equations is pre- sented.Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the tempo...In this paper,the translation of the Lax pairs of the Levi equations is pre- sented.Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally,the involutive solutions of the Levi equations are presented.展开更多
文摘Within the framework of zero-curvature representation theory, the decompositions of eachequation in a hierarchy of zero-curvature equations associated with loop algebra 81(2) by meansof higher-order constraints on potential are given a unified treatment, and the general schemeand uniform formulas for the decompositions are proposed. This provides a method of separationof variables to solve a hierarchy of (1+1)-dimensional integrable systems. TO illustrate the general scheme, new higher-order decompositions of two hierarchies of zero-curvature equations arepresented.
基金Supported by National Natural Science Foundation of China under Grant Nos.11771186 and 11671177
文摘We present an integrable sl(2)-matrix Camassa-Holm(CH)equation.The integrability means that the equation possesses zero-curvature representation and infinitely many conservation laws.This equation includes two undetermined functions,which satisfy a system of constraint conditions and may be reduced to a lot of known multicomponent peakon equations.We find a method to construct constraint condition and thus obtain many novel matrix CH equations.For the trivial reduction matrix CH equation we construct its N-peakon solutions.
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.
文摘In this paper,the translation of the Lax pairs of the Levi equations is pre- sented.Then a symmetry constraint for the Levi equations is given by means of binary nonlinearization method. The spatial part and the temporal parts of the translated Lax pairs and its adjoint Lax pairs of the Levi equations are all constrainted as finite dimensional Liouville integrable Hamiltonian systems. Finally,the involutive solutions of the Levi equations are presented.
