A generalized Tu formula and Hamiltonian structures of fractional AKNS hierarchy

Abstract

With the modified Riemann-Liouville fractional derivative, a fractional Tu formula is presented to investigate generalized Hamilton structure of fractional soliton equations. The obtained results can be reduced to the classical Hamilton hierachy of ordinary calculus.

The Fractional Quadratic-Form Identity and Hamiltonian Structure of an Integrable Coupling of the Fractional Broer-Kaup Hierarchy

“…In the previous derivation, and are fractional differentiable functions with respect to . The fractional closed condition = 0 admits the fractional Hamilton's equations [40]…”

“…Later, many integrable systems and their Hamiltonian structures were worked out [36][37][38][39]. Recently, Wu and Zhang proposed the generalized Tu formula and searched for the Hamiltonian structure of fractional AKNS hierarchy [40]. In [41], a generalized Hamiltonian structure of the fractional soliton equation hierarchy was presented.…”

Fractional isospectral and non-isospectral AKNS hierarchies and their analytic methods for N-fractal solutions with Mittag-Leffler functions

“…Recently, some nonlocal integrable evolution equations [11,12] have been found from the symmetry reductions of the isAKNS hierarchy. In 2011, Wu and Zhang [13] constructed Hamiltonian structures of a fractional AKNS hierarchy by a generalized Tu formula. In 2020, Gao et al [14] solved the (2 + 1)dimensional AKNS equation with conformable derivatives and a perturbation parameter by the sine-Gordon expansion method.…”

Integrable couplings of fractional L‐hierarchy and its Hamiltonian structures

“…However, since Riewe [16,17] proposed a concept of nonconservation mechanics, fractional conservation laws [18], Lie symmetries [19], and fractional Hamiltonian systems [20][21][22][23][24][25][26][27][28][29][30] have been receiving more and more attention. In [31], Wu proposed the generalized Tu formula and research for the Hamiltonian structures of fractional AKNS hierarchy.In [32,33], Wang and Xia obtained the fractional supersoliton hierarchies and their super-Hamiltonian structures by using fractional supertrace identity.In this paper, we first introduce a fractional bilinear form variational identity.Then,we consider a kind of explicit Lie algebra for constructing the fractional L-hierarchy. At last, we obtained the Hamiltonian structures of the integrable couplings of fractional L-hierarchy by using the fractional bilinear form variational identity.…”