Conservation Laws and $$\tau $$τ-Symmetry Algebra of the Gerdjikov–Ivanov Soliton Hierarchy

Math Methods in App Sciences

Gongye

2022

Self Cite

By the Lagrangian multiplier and constraint variational derivative, a relationship between conserved quantities and multi-Hamiltonian structures is built. Using the relation, a method is founded to prove the infinitedimensional Liouville integrability of evolution equations with continuous variables. As the application, the conservation laws of the Kundu equation are first obtained. Its conserved quantities are deduced for comparing by Fokas' method different from the method used in the existed literature. The integrability of the equation is proved through taking the conservation laws as a starting point.

Section: Introductionmentioning

confidence: 77%

Section: Introductionmentioning

confidence: 99%

The relationship between the conservation laws and multi‐Hamiltonian structures of the Kundu equation

Math Methods in App Sciences

Gongye

2022

Self Cite

“…The systems in Equations (11) and (13) are generalized systems. Many celebrated integrable systems can be derived from them by selecting different values of the arbitrary parameter β.…”

Section: Isospectral and Nonisospcetral Hierarchies Of The Gdnls Equamentioning

confidence: 99%

“…How to obtain a conservation density is very important from the spectral problem of Lax pairs. Actually, CLs can be derived by taking advantage of the generating conservation density and evolution equation of time [13].…”

Section: Introductionmentioning

confidence: 99%

A τ-Symmetry Algebra of the Generalized Derivative Nonlinear Schrödinger Soliton Hierarchy with an Arbitrary Parameter

Gongye

2018

Symmetry

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A matrix spectral problem is researched with an arbitrary parameter. Through zero curvature equations, two hierarchies are constructed of isospectral and nonisospectral generalized derivative nonlinear schrödinger equations. The resulting hierarchies include the Kaup-Newell equation, the Chen-Lee-Liu equation, the Gerdjikov-Ivanov equation, the modified Korteweg-de Vries equation, the Sharma-Tasso-Olever equation and a new equation as special reductions. The integro-differential operator related to the isospectral and nonisospectral hierarchies is shown to be not only a hereditary but also a strong symmetry of the whole isospectral hierarchy. For the isospectral hierarchy, the corresponding τ -symmetries are generated from the nonisospectral hierarchy and form an infinite-dimensional symmetry algebra with the K-symmetries.

“…It should be pointed out that, in the theory of lattice soliton and integrable systems, the integrable families, which possess concise tri-Hamiltonian structure, are very rare [12][13][14][15][16][17][18]. Also, the symmetrical algebraic structure of equations is an important research direction for INDDEs [19][20][21][22][23][24]. Furthermore, the research of nonisospectral INDDEs has been widespread concern, the algebraic structure of isospectral and nonisospectral vector fields is established in [16][17][18], and the key of the theory is to derive the corresponding nonisospectral family of INDDEs.…”

Section: Introductionmentioning

confidence: 99%

A Family of Integrable Differential-Difference Equations: Tri-Hamiltonian Structure and Lie Algebra of Vector Fields

Discrete Dynamics in Nature and Society

Xü

2021

Starting from a novel discrete spectral problem, a family of integrable differential-difference equations is derived through discrete zero curvature equation. And then, tri-Hamiltonian structure of the whole family is established by the discrete trace identity. It is shown that the obtained family is Liouville-integrable. Next, a nonisospectral integrable family associated with the discrete spectral problem is constructed through nonisospectral discrete zero curvature representation. Finally, Lie algebra of isospectral and nonisospectral vector fields is deduced.