Discrete Moving Frames and Discrete Integrable Systems

Mansfield, Elizabeth L.; Beffa, Gloria Maŕı; Wang, Jing Ping

doi:10.1007/s10208-013-9153-0

Cited by 32 publications

(68 citation statements)

References 37 publications

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“…We refer the reader to Refs. for the theoretical foundations and further examples. Definition Let G act (locally) freely and regularly on J [ n ] .…”

Section: Equivariant Moving Framesmentioning

confidence: 99%

“…As it is well‐known from the differential theory of equivariant moving frames, the exterior derivative and the invariantization map ι do not commute,

d \circ ι \neq ι \circ d .

Similarly, the shift operators and the invariantization map ι do not commute,

{boldS}_{i} \circ ι \neq ι \circ {boldS}_{i}, i = 1, \dots, p .

Example To illustrate the noncommutativity relations (18), let us revisit Example . Based on the cross‐section ,

d \circ ι_{n} false(x_{n} false) = d false(0 false) = 0 .

On the other hand,

ι_{n} \circ d false(x_{n} false) = ϖ_{n}^{n} = \frac{Δ x_{n} d x_{n} + Δ u_{n} d u_{n}}{\sqrt{{false(normalΔ x_{n} false)}^{2} + {false(normalΔ u_{n} false)}^{2}}} .

Similarly, for the shift operator

boldS

, which maps n to

n + 1

, we have

S \circ ι_{n} false(x_{n} false) = S false(0 false) = 0,

while

ι_{n} \circ S false(x_{n} false) = ι_{n} false(x_{n + 1} false) = K_{n} .

…”

Section: Recurrence Relationsmentioning

confidence: 99%

“…For the shift operators, the recurrence relations were recently introduced in Ref. . These recurrence relations are now introduced.…”

Section: Recurrence Relationsmentioning

confidence: 99%

“…As illustrated in Example , given a discrete moving frame ρ, the invariantization of a differential form

ω \in {normalΩ}^{*} false({normalJ}^{false[\infty false]} false)

with respect to

ι_{N}

and

ι_{K}

, where

N \neq K

, will, in general, produce different invariant differential forms. While different, these differential forms are related by certain recurrence relations for the shift operators. These recurrence relations involve a distinguished set of invariants, called the Maurer‐Cartan invariants.…”

Section: Recurrence Relationsmentioning

confidence: 99%

“…Similarly, the shift operators (3) and the invariantization map do not commute 17,28,31 , On the other hand,…”

Section: Recurrence Relationsmentioning

confidence: 99%

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Invariant discrete flows

Benson

Valiquette

2019

Stud Appl Math

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In this paper, we investigate the evolution of joint invariants under invariant geometric flows using the theory of equivariant moving frames and the induced invariant discrete variational complex. For certain arc length preserving planar curve flows invariant under the special Euclidean group SE(2), the special linear group SL(2), and the semidirect group R⋉double-struckR2, we find that the induced evolution of the discrete curvature satisfies the differential‐difference mKdV, KdV, and Burgers' equations, respectively. These three equations are completely integrable, and we show that a recursion operator can be constructed by precomposing the characteristic operator of the curvature by a certain invariant difference operator. Finally, we derive the constraint for the integrability of the discrete curvature evolution to lift to the evolution of the discrete curve itself.

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“…We refer the reader to Refs. for the theoretical foundations and further examples. Definition Let G act (locally) freely and regularly on J [ n ] .…”

Section: Equivariant Moving Framesmentioning

confidence: 99%

“…As it is well‐known from the differential theory of equivariant moving frames, the exterior derivative and the invariantization map ι do not commute,

d \circ ι \neq ι \circ d .

Similarly, the shift operators and the invariantization map ι do not commute,

{boldS}_{i} \circ ι \neq ι \circ {boldS}_{i}, i = 1, \dots, p .

Example To illustrate the noncommutativity relations (18), let us revisit Example . Based on the cross‐section ,

d \circ ι_{n} false(x_{n} false) = d false(0 false) = 0 .

On the other hand,

ι_{n} \circ d false(x_{n} false) = ϖ_{n}^{n} = \frac{Δ x_{n} d x_{n} + Δ u_{n} d u_{n}}{\sqrt{{false(normalΔ x_{n} false)}^{2} + {false(normalΔ u_{n} false)}^{2}}} .

Similarly, for the shift operator

boldS

, which maps n to

n + 1

, we have

S \circ ι_{n} false(x_{n} false) = S false(0 false) = 0,

while

ι_{n} \circ S false(x_{n} false) = ι_{n} false(x_{n + 1} false) = K_{n} .

…”

Section: Recurrence Relationsmentioning

confidence: 99%

“…For the shift operators, the recurrence relations were recently introduced in Ref. . These recurrence relations are now introduced.…”

Section: Recurrence Relationsmentioning

confidence: 99%

“…As illustrated in Example , given a discrete moving frame ρ, the invariantization of a differential form

ω \in {normalΩ}^{*} false({normalJ}^{false[\infty false]} false)

with respect to

ι_{N}

and

ι_{K}

, where

N \neq K

Section: Recurrence Relationsmentioning

confidence: 99%

“…Similarly, the shift operators (3) and the invariantization map do not commute 17,28,31 , On the other hand,…”

Section: Recurrence Relationsmentioning

confidence: 99%

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Invariant discrete flows

Benson

Valiquette

2019

Stud Appl Math

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Recursive Moving Frames for Lie Pseudo-Groups

Olver

Valiquette

2018

Results Math

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Rational Recursion Operators for Integrable Differential–Difference Equations

Carpentier

Михайлов

Wang

2019

Commun. Math. Phys.

Self Cite

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In this paper we introduce preHamiltonian pairs of difference operators and study their connections with Nijenhuis operators and the existence of weakly non-local inverse recursion operators for differential-difference equations. We begin with a rigorous setup of the problem in terms of the skew field Q of rational (pseudo-difference) operators over a difference field F with a zero characteristic subfield of constants k ⊂ F and the principal ideal ring M n (Q) of matrix rational (pseudo-difference) operators. In particular, we give a criteria for a rational operator to be weakly non-local. A difference operator H is called preHamiltonian, if its image is a Lie k-subalgebra with respect the the Lie bracket on F. Two preHamiltonian operators form a preHamiltonian pair if any k-linear combination of them is preHamiltonian. Then we show that a preHamiltonian pair naturally leads to a Nijenhuis operator, and a Nijenhuis operator can be represented in terms of a preHamiltonian pair. This provides a systematical method to check whether a rational operator is Nijenhuis. As an application, we construct a preHamiltonian pair and thus a Nijenhuis recursion operator for the differential-difference equation recently discovered by Adler & Postnikov. The Nijenhuis operator obtained is not weakly non-local. We prove that it generates an infinite hierarchy of local commuting symmetries. We also illustrate our theory on the well known examples including the Toda, the Ablowitz-Ladik and the Kaup-Newell differential-difference equations.

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Discrete Moving Frames and Discrete Integrable Systems

Cited by 32 publications

References 37 publications

Invariant discrete flows

Invariant discrete flows

Recursive Moving Frames for Lie Pseudo-Groups

Rational Recursion Operators for Integrable Differential–Difference Equations

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