Adjoint Symmetry Constraints Leading to Binary Nonlinearization

Ma, Wen‐Xiu; Zhou, Ruguang

doi:10.2991/jnmp.2002.9.s1.10

Cited by 64 publications

(45 citation statements)

References 20 publications

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“…Moreover, our results pave a way to construct conservation laws from symmetries for self-adjoint discrete evolution equations or difference equations, since symmetries are also adjoint symmetries of self-adjoint discrete evolution equations or difference equations. We also point out that our idea of constructing conservation laws by symmetries and adjoint symmetries is quite similar to that of carrying out binary nonlinearization under symmetry constraints [12,13]. In the theory of binary nonlinearization [12,13], adjoint symmetries are used to establish a balance with non-Lie adjoint symmetries generated from both spectral problems and adjoint spectral problems.…”

Section: Discussionmentioning

confidence: 99%

“…Moreover, Lax pairs are used to make Riccati type equations that ratios of eigenfunctions need to satisfy, and series expansions of solutions of the resulting Riccati type equations around the spectral parameter yield conservation laws (see, e.g., [17,18]). For continuous evolution equations, adjoint symmetries are also called conserved covariants [19], and it is recognized that the product of a symmetry and an adjoint symmetry presents a conserved density (see, e.g., [19][20][21]) and that a functional I is conserved if and only if its variational derivative δI δu is an adjoint symmetry (see, e.g., [13,20,21]). There exists a geometrical theory to deal with adjoint symmetries of the second-order ODEs [22].…”

Section: Discussionmentioning

confidence: 99%

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Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

2015

Symmetry

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A direct approach is proposed for constructing conservation laws of discrete evolution equations, regardless of the existence of a Lagrangian. The approach utilizes pairs of symmetries and adjoint symmetries, in which adjoint symmetries make up for the disadvantage of non-Lagrangian structures in presenting a correspondence between symmetries and conservation laws. Applications are made for the construction of conservation laws of the Volterra lattice equation.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

2015

Symmetry

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“…Adjoint symmetry constraints or equivalently symmetry constraints separate the four-component AKNS equations into two commuting finite-dimensional Liouville integrable Hamiltonian systems [50].…”

Section: Four-component Akns Hierarchymentioning

confidence: 99%

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

2017

Proc. R. Soc. A.

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Using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, and determine Dubrovin type equations for zeros and poles of meromorphic functions defined as ratios of the Baker-Akhiezer functions. We straighten out all flows in soliton hierarchies under the Abel-Jacobi coordinates associated with Lax pairs, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviors of the Baker-Akhiezer functions. We analyze the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.

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“…The symmetry constraints in the case of = 0 is called a Bargmann constraint(see [9]). If taking = 0, = = (24) where we use the following notation…”

Section: Bargmann Symmetry Constraint Of Super Guo Hierarchymentioning

confidence: 99%

“…It is known that a crucial idea in carrying out symmetry constraints is the nonlinearization of Lax pairs for soliton hierarchies. The nonlinearization of Lax pairs is classified into mono-nonlinearization [5]- [7] and binary nonlinearization [8], [9].…”

Section: Introductionmentioning

confidence: 99%

Bargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

Tao¹

2013

IJAPM

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Abstract-An explicit Bargmann symmetry constraint is computed and its associated binary nonlinearization of Lax pairs is carried out for the super Guo hierarchy. Under the obtained symmetry constraint, the n-th flow of the super Guo hierarchy is decomposed into two super finite -dimensional integrable Hamiltonian systems, which defined over the super symmetry manifold R 4N|2N with the corresponding dynamical variables x and t n . The integrals of motion required for Liouville integrability are explicitly given.Index Terms-Symmetry constraint, binary nonlinearization, super guo hierarchy, super finitee-dimensional integrable hamiltonian systems. I. INTRODUCTIONFor almost twenty years, much attention has been paid to the construction of finite-dimensional integrable systems from soliton equations by using symmetry constraints. Either [19]. In this paper, we would like to consider the binary nonlinearization of the super Guo hierarchy. This paper is organized as follows. In the next section, we will recall the super Guo soliton hierarchy and its super Hamiltonian structure. Then in Section III, we compute a Bargmann symmetry constraint for the potential of the super Guo hierarchy. In Section IV, we apply the binary nonlinearization method to super Guo hierarchy, and then obtain super finite-dimensional integrable Hamiltonian hierarchy on the super symmetry manifold ℝ 4 |2 , whose integrals of motion are explicitly given. II. THE SUPER GUO HIERARCHYwhere is a spectral parameter, q and r are even variables, and and are odd variables(see [20] ). Taking If we setthen (2) Si-Xing TaoBargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

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Adjoint Symmetry Constraints Leading to Binary Nonlinearization

Cited by 64 publications

References 20 publications

Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

Conservation Laws of Discrete Evolution Equations by Symmetries and Adjoint Symmetries

Trigonal curves and algebro-geometric solutions to soliton hierarchies I

Bargmann Symmetry Constraint and Binary Nonlinearization of Super Guo Hierarchy

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