Geometric Integrability of the Camassa–Holm Equation. II

Heredero, Rafael Hernández; Reyes, Enríque G.

doi:10.1093/imrn/rnr120

Cited by 25 publications

(26 citation statements)

References 51 publications

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“…Subtracting This is the transformation Schiff found in [74]. Yet another Darboux transformation, and a non-linear superposition rule for ACH, appear in [33,34].…”

Section: Equation (411) Is Not Invariant Under This Change Insteadmentioning

confidence: 86%

“…(3.12)-its bi-Hamiltonian character, the existence of a Lax pair, the existence of a recursion operator-have been discussed by Camassa and Holm [8], Camassa et al [9], and by Fokas and Fuchssteiner [23,[26][27][28] (see also [24]); well-posedness, weak solutions and the existence of wave breaking have been analyzed by Li and Olver [43] and also by Constantin and Escher [13,14]; and, it has been proven that the Camassa-Holm equation (3.12) can be interpreted as a geodesic flow on the Virasoro group, see Misio lek [46]. Finally, the case κ = 0 has been analyzed from the geometric point of view advocated in this paper in [33,34,60,63].…”

Section: The Camassa-holm Equationmentioning

confidence: 87%

“…Bäcklund transformations of hierarchies of pseudo-spherical type can be also considered, see [59]. A newer example of a Darboux transformation is furnished by the associated Camassa-Holm equation [33,34]:…”

Section: 4) Take a Solution U Of Kdv And Compute γ From (A) Equatiomentioning

confidence: 99%

“…Theorem 6.9 has been found via extensive symbolic computations carried out with the help of Mathematica software written by R. Hernández Heredero (Universidad Politécnica de Madrid). It is based on earlier classification results appearing in [33,34]. The function A(λ) appearing in (6.49) is of importance…”

Section: The Kaup-kupershmidt Sawada-kotera and 5th Order Korteweg-dmentioning

confidence: 99%

“…Specifically, it is shown how to construct non-local symmetries of some interesting equations of mathematical physics (the KaupKupershmidt, Sawada-Kotera, 5th-order KdV and Camassa-Holm equations), how non-local symmetries can be used to single out integrable equations, and how the Virasoro algebra appears in the theory of non-local symmetries. This last part builds up on work reported in the papers [31,33,34,[60][61][62][63].Guide: Section 2 contains basic definitions (Sect. 2.1), characterization results (Sect.…”

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confidence: 99%

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Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

Reyes

2011

Results. Math.

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This paper surveys some recent developments around the notion of a scalar partial differential equation describing pseudo-spherical surfaces due to Chern and Tenenblat. It is shown how conservation laws, pseudo-potentials, and linear problems arise naturally from geometric considerations, and it is also explained how Darboux and Bäcklund transformations can be constructed starting from geometric data. Classification results for equations in this class are stated, and hierarchies of equations of pseudo-spherical type are introduced, providing a connection between differential geometry and the study of hierarchies of equations which are the integrability condition of sl(2, R)-valued linear problems. Furthermore, the existence of correspondences between any two solutions to equations of pseudo-spherical type is reviewed, and a correspondence theorem for hierarchies is also mentioned. As applications, an elementary immersion result for pseudo-spherical metrics arising from the Chern-Tenenblat construction is proven, and non-local symmetries of the Kaup-Kupershmidt, Sawada-Kotera, fifth order Korteweg-de Vries and Camassa-Holm (CH) equation with non-zero critical wave speed are considered. It is shown that the existence of a non-local symmetry of a particular type is enough to single the first three equations out from a whole family of equations describing pseudo-spherical surfaces while, in the CH case, it is shown that it admits an infinite-dimensional Lie algebra of non-local symmetries which includes the Virasoro algebra.Mathematics Subject Classification (2010). Primary 53B20; Secondary 37K10, 76M60. Equations of Pseudo-Spherical Type 55and a finite number of its derivatives with respect to x) are usually members of infinite hierarchies of evolution equations u τn = F n such that (see for instance [19]): (a) the flows generated by the equations u τn = F n commute, and (b) each equation u τn = F n is the integrability condition of a linear problem of the form v x = X v, v τn = T n v, hierarchies of evolution equations describing pseudo-spherical surfaces were defined in [59]; it was proven that they do possess characteristics (a) and (b), and moreover, that there exist correspondences between (suitably generic) solutions of any two hierarchies of equations of pseudo-spherical type. Thus, the Chern-Tenenblat structure is of interest for integrability, analysis and certainly geometry, since it is quite natural to ask which equations (besides the classical sine-Gordon equation [22]) allow one to construct pseudospherical metrics. With respect to analysis, one can trace back the relevance of the class of equations of pseudo-spherical type to the local uniqueness of surfaces of constant Gaussian curvature; with respect to integrability, one would say that the relevance of the Chern-Tenenblat structure arises from the fact that the Chern-Tenenblat construction [12] encodes in differential geometric terms the idea of a Wahlquist-Eastbrook quadratic pseudopotential [80]. This is the reason behind the fact that one can unc...

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“…Subtracting This is the transformation Schiff found in [74]. Yet another Darboux transformation, and a non-linear superposition rule for ACH, appear in [33,34].…”

Section: Equation (411) Is Not Invariant Under This Change Insteadmentioning

confidence: 86%

Section: The Camassa-holm Equationmentioning

confidence: 87%

Section: 4) Take a Solution U Of Kdv And Compute γ From (A) Equatiomentioning

confidence: 99%

Section: The Kaup-kupershmidt Sawada-kotera and 5th Order Korteweg-dmentioning

confidence: 99%

mentioning

confidence: 99%

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Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

Reyes

2011

Results. Math.

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A new kind of nonlocal symmetry for the μ‐Camassa‐Holm equation with linear dispersion

Shi

2018

Math Methods in App Sciences

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Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

Kang¹,

Liu²,

Olver

et al. 2015

J Nonlinear Sci

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Motivated by the dynamics of neuronal responses, we analyze the dynamics of the FitzHugh-Nagumo slow-fast system with delayed self-coupling. This system provides a canonical example of a canard explosion for sufficiently small delays. Beyond this regime, delays significantly enrich the dynamics, leading to mixed-mode oscillations, bursting and chaos. These behaviors emerge from a delay-induced subcritical Bogdanov-Takens instability arising at the fold points of the S-shaped critical manifold. Underlying the transition from canardinduced to delay-induced dynamics is an abrupt switch in the nature of the Hopf bifurcation.

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Geometric Integrability of the Camassa–Holm Equation. II

Cited by 25 publications

References 51 publications

Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

Equations of Pseudo-Spherical Type (After S.S. Chern and K. Tenenblat)

A new kind of nonlocal symmetry for the μ‐Camassa‐Holm equation with linear dispersion

Liouville Correspondence Between the Modified KdV Hierarchy and Its Dual Integrable Hierarchy

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