We develop in this paper a new method to construct two explicit Lie algebras E and F.By using aloop algebra E of the Lie algebra E and the reduced self-dual Yang-Mills equations,we obtain an expanding integrablemodel ...We develop in this paper a new method to construct two explicit Lie algebras E and F.By using aloop algebra E of the Lie algebra E and the reduced self-dual Yang-Mills equations,we obtain an expanding integrablemodel of the Giachetti-Johnson (GJ)hierarchy whose Hamiltonian structure can also be derived by using the traceidentity.This provides a much simplier construction method in comparing with the tedious variational identity approach.Furthermore,the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra g_N.As an application,we apply the loop algebra E of the Lie algebra E to obtain a kind of expanding integrable model ofthe Kaup-Newell (KN)hierarchy which,consisting of two arbitrary parameters a and f3,can be reduced to two nonlinearevolution equations.In addition,we use a loop algebra F of the Lie algebra F to obtain an expanding integrable model ofthe BT hierarchy whose Hamiltonian structure is the same as using the trace identity.Finally,we deduce five integrablesystems in R3 based on the self-dual Yang-Mills equations,which include Poisson structures,irregular lines,and thereduced equations.展开更多
Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced,for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the disp...Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced,for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained.From the reductions of the coupling integrable couplings,the corresponding coupled integrable couplings of the BK equation,the DLW equation,and the TB equation are obtained,respectively.Especially,the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation.The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out,respectively,by employing the variational identity.Finally, we decompose the BK hierarchy of evolution equations into x-constrained Hows and t_n-constrained Hows whose adjoint representations and the Lax pairs are given.展开更多
We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov–Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical ...We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov–Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the(2+1)-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of(2+1)-dimensional expanding dynamical model of the(2+1)-dimensional dynamical system is generated as well.展开更多
In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1a...In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained,respectively.A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy.Finally,via two different enlarging Lie algebras of the Lie algebra A_1,we derive two resulting differential-difference integrable couplings of the Toda hierarchy,of course,they are all various discrete expanding integrable models of the Toda hierarchy.When the introduced spectral matrices are higher degrees,the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple.展开更多
基金Supported by a Research Grant from the CityU Strategic Research under Grant No. 7002564
文摘We develop in this paper a new method to construct two explicit Lie algebras E and F.By using aloop algebra E of the Lie algebra E and the reduced self-dual Yang-Mills equations,we obtain an expanding integrablemodel of the Giachetti-Johnson (GJ)hierarchy whose Hamiltonian structure can also be derived by using the traceidentity.This provides a much simplier construction method in comparing with the tedious variational identity approach.Furthermore,the nonlinear integrable coupling of the GJ hierarchy is readily obtained by introducing the Lie algebra g_N.As an application,we apply the loop algebra E of the Lie algebra E to obtain a kind of expanding integrable model ofthe Kaup-Newell (KN)hierarchy which,consisting of two arbitrary parameters a and f3,can be reduced to two nonlinearevolution equations.In addition,we use a loop algebra F of the Lie algebra F to obtain an expanding integrable model ofthe BT hierarchy whose Hamiltonian structure is the same as using the trace identity.Finally,we deduce five integrablesystems in R3 based on the self-dual Yang-Mills equations,which include Poisson structures,irregular lines,and thereduced equations.
基金Supported by the National Science Foundation of China under Grant No.10971031the Natural Science Foundation of Shandong Province under Grant No.ZR2009AL021
文摘Two kinds of higher-dimensional Lie algebras and their loop algebras are introduced,for which a few expanding integrable models including the coupling integrable couplings of the Broer-Kaup (BK) hierarchy and the dispersive long wave (DLW) hierarchy as well as the TB hierarchy are obtained.From the reductions of the coupling integrable couplings,the corresponding coupled integrable couplings of the BK equation,the DLW equation,and the TB equation are obtained,respectively.Especially,the coupling integrable coupling of the TB equation reduces to a few integrable couplings of the well-known mKdV equation.The Hamiltonian structures of the coupling integrable couplings of the three kinds of soliton hierarchies are worked out,respectively,by employing the variational identity.Finally, we decompose the BK hierarchy of evolution equations into x-constrained Hows and t_n-constrained Hows whose adjoint representations and the Lax pairs are given.
基金Supported by the Fundamental Research Funds for the Central University under Grant No.2017XKZD11
文摘We extend two known dynamical systems obtained by Blaszak, et al. via choosing Casimir functions and utilizing Novikov–Lax equation so that a series of novel dynamical systems including generalized Burgers dynamical system, heat equation, and so on, are followed to be generated. Then we expand some differential operators presented in the paper to deduce two types of expanding dynamical models. By taking the generalized Burgers dynamical system as an example, we deform its expanding model to get a half-expanding system, whose recurrence operator is derived from Lax representation, and its Hamiltonian structure is also obtained by adopting a new way. Finally, we expand the generalized Burgers dynamical system to the(2+1)-dimensional case whose Hamiltonian structure is derived by Poisson tensor and gradient of the Casimir function. Besides, a kind of(2+1)-dimensional expanding dynamical model of the(2+1)-dimensional dynamical system is generated as well.
基金Supported by the National Natural Science Foundation of China under Grant No.11371361the Innovation Team of Jiangsu Province hosted by China University of Mining and Technology(2014)+1 种基金Hong Kong Research Grant Council under Grant No.HKBU202512the Natural Science Foundation of Shandong Province under Grant No.ZR2013AL016
文摘In the paper,we introduce the Lie algebras and the commutator equations to rewrite the Tu-d scheme for generating discrete integrable systems regularly.By the approach the various loop algebras of the Lie algebra A_1are defined so that the well-known Toda hierarchy and a novel discrete integrable system are obtained,respectively.A reduction of the later hierarchy is just right the famous Ablowitz-Ladik hierarchy.Finally,via two different enlarging Lie algebras of the Lie algebra A_1,we derive two resulting differential-difference integrable couplings of the Toda hierarchy,of course,they are all various discrete expanding integrable models of the Toda hierarchy.When the introduced spectral matrices are higher degrees,the way presented in the paper is more convenient to generate discrete integrable equations than the Tu-d scheme by using the software Maple.
