Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

Qu, Changzheng; Song, Junfeng; Yao, Ruoxia

doi:10.3842/sigma.2013.001

Cited by 18 publications

(20 citation statements)

References 61 publications

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“…S. Anco and J. Szmigielski are supported in part by NSERC. The authors would like to thank Z. Qiao for pointing out pertinent references, the referees for stimulating and helpful comments including references, and X.B. Hu and K. Tian for helpful conversations.…”

Section: Acknowledgmentsmentioning

confidence: 99%

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The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

2018

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The tri‐Hamiltonian splitting method applied to the Hirota hierarchy has yielded two U(1)‐invariant nonlinear Partial Differential Equations (PDEs) that admit peakons (nonsmooth solitons). In the present paper, these two peakon PDEs are generalized to a family of U(1)‐invariant peakon PDEs parameterized by the real projective line RboldP1. All equations in this family are shown to possess conservative peakon solutions (whose Sobolev H1false(boldRfalse) norm is time invariant). The Hamiltonian structure for the sector of conservative peakons is identified and the peakon Ordinary Differential Equations (ODEs) are shown to be Hamiltonian with respect to several Poisson structures. As the main result, it is shown that inverse spectral methods allow one to solve explicitly the dynamics of conservative peakons using explicit solutions to a certain interpolation problem. The graphs of multipeakon solutions confirm the existence of multipeakon breathers as well as asymptotic formation of pairs of two peakon bound states in the nonperiodic time domain.

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Section: Acknowledgmentsmentioning

confidence: 99%

“…Remark Equation () can also be derived in a geometric setup (Ref. , equation 44) as a certain type of curve flow in the hyperbolic space

H^{2}

.…”

Section: Introductionmentioning

confidence: 99%

The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

2018

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“…For example, if we take a 1 = 1, = = a a 0 2 8 and c 6 = −1 in (47), we get the two-component CH system (26) in [31] as follows:…”

Section: Remark 5 Upon Taking the Linear Combinationsmentioning

confidence: 99%

“…Similar results to theorem 6 and corollary 3 have been obtained in [27] from Frobenius algebras, but in that context the D x 2 terms cannot appear. We now work on to find the bi-Hamiltonian structure for system (31) in theorem 3 for arbitrary α and β. There is a known quadratic Hamiltonian operator related to the AKNS system, which is not included in our theorem 6.…”

Section: Theorem 6 Let the Operators H And J Be Given By (54) And (3mentioning

confidence: 99%

Two-component generalizations of the Camassa–Holm equation

2017

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Recommended by Tamara Grava AbstractA classification of integrable two-component systems of non-evolutionary partial differential equations that are analogous to the Camassa-Holm equation is carried out via the perturbative symmetry approach. Independently, a classification of compatible pairs of Hamiltonian operators of specific forms is carried out, in order to obtain bi-Hamiltonian structures for the same systems of equations. Using reciprocal transformations, some exact solutions and Lax pairs are also constructed for the systems considered.

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“…Strachan and Zuo provide several methods to extend Frobenius-Virasoro algebra system 23,24 ; Qu provided a way to study multicomponent integrable systems. 25 Recently, we study Z n -Painlevé IV equation, Frobenius Painlevé I equation, and Frobenius Painlevé III equation. 26 In Yang and Li, 27 we construct a Frobenius-valued Sine-Gordon systems that generate some new coupled integrable equations.…”

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confidence: 99%

Affine Weyl group symmetries of Frobenius Painlevé equations

Wang

2019

Math Methods in App Sciences

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In this paper, we introduce a Frobenius Painlevé IV equation and the corresponding Hamilton system, and we give the symmetric form of the Frobenius Painlevé IV equation. Then, we construct the Lax pair of the Frobenius Painlevé IV equation. Furthermore, we recall the Frobenius modified KP hierarchy and the Frobenius KP hierarchy by bilinear equations, then we show how to get Frobenius Painlevé IV equation from the Frobenius modified KP hierarchy. In order to study the different aspects of the Frobenius Painlevé IV equation, we give the similarity reduction and affine Weyl group symmetry of the equation. Similarly, we introduce a Frobenius Painlevé II equation and show the connection between the Frobenius modified KP hierarchy and the Frobenius Painlevé II equation.

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Multi-Component Integrable Systems and Invariant Curve Flows in Certain Geometries

Cited by 18 publications

References 61 publications

The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

The dynamics of conservative peakons in a family of U(1)‐invariant integrable equations of NLS‐Hirota type

Two-component generalizations of the Camassa–Holm equation

Affine Weyl group symmetries of Frobenius Painlevé equations

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