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Reyes, Enríque G.

doi:10.1023/a:1014933316169

Cited by 113 publications

(26 citation statements)

References 21 publications

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“…Based on the work of Chern & Tenenblat [40] and the subsequent works [41,42], a differential equation for a real-valued function u(x, t) is said to describe pseudo-spherical surfaces if it is the necessary and sufficient condition for the existence of smooth functions f ij , i = 1, 2, 3, j = 1, 2, depending on x, t, u and its derivatives, such that the one-forms ω i = f i1 dx + f i2 dt satisfy the structure equations of a surface of constant Gaussian curvature equal to −1 with metric ω 2 1 + ω 2 2 and connection one-form ω 3 , namely dω 1 = ω 3 ∧ ω 2 , dω 2 = ω 1 ∧ ω 3 and dω 3 = ω 1 ∧ ω 2 .…”

Section: Geometric Integrabilitymentioning

confidence: 99%

A new two-component integrable system with peakon solutions

Xia

Zhang²

2015

Proc. R. Soc. A.

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where b is an arbitrary real constant, is proposed in this paper. This system is shown integrable with its Lax pair, bi-Hamiltonian structure and infinitely many conservation laws. Geometrically, this system describes a non-trivial one-parameter family of pseudo-spherical surfaces. In the case b = 0, the peaked soliton (peakon) and multipeakon solutions to this two-component system are derived. In particular, the two-peakon dynamical system is explicitly solved and their interactions are investigated in details. Moreover, a new integrable cubic nonlinear equation with linear dispersionis obtained by imposing the complex conjugate reduction v = u * to the two-component system. The complex-valued N-peakon solution and kink wave solution to this complex equation are also derived.

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Section: Geometric Integrabilitymentioning

confidence: 99%

A new two-component integrable system with peakon solutions

Xia

Zhang²

2015

Proc. R. Soc. A.

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“…21 It is also completely integrable, possesses bi-Hamiltonian form and infinite sequence of conservation laws. 4,8,9,22,32 The soliton solution has the form…”

Section: Camassa-holm Equationmentioning

confidence: 99%

Poisson Structures of Equations Associated With Groups of Diffeomorphisms

Ivanov¹

2009

Trends in Differential Geometry, Complex Analysis and Mathematical Physics

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“…Thus, for ω = 0, CH has various conformal properties [15]. CH is also completely integrable, possesses bi-Hamiltonian form and infinite sequence of conservation laws [2,6,12,16,24]. The Lax pair is…”

Section: Motion Of a Rigid Body With A Fixed Pointthe So(3) Examplementioning

confidence: 99%

THE CAMASSA-HOLM EQUATION AS A GEODESIC FLOW FOR THE H¹ RIGHT-INVARIANT METRIC

Constantin

Ivanov

2007

Topics in Contemporary Differential Geometry, Complex Analysis and Mathematical Physics

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The fundamental role played by the Lie groups in mechanics, and especially by the dual space of the Lie algebra of the group and the coadjoint action are illustrated through the Camassa-Holm equation (CH). In 1996 Misio lek observed that CH is a geodesic flow equation on the group of diffeomorphisms, preserving the H 1 metric. This example is analogous to the Euler equations in hydrodynamics, which describe geodesic flow for a right-invariant metric on the infinite-dimensional group of diffeomorphisms preserving the volume element of the domain of fluid flow and to the Euler equations of rigid body whith a fixed point, describing geodesics for a left-invariant metric on SO(3).The momentum map and an explicit parametrization of the Virasoro group, related to recently obtained solutions for the CH equation are presented.

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Cited by 113 publications

References 21 publications

A new two-component integrable system with peakon solutions

A new two-component integrable system with peakon solutions

Poisson Structures of Equations Associated With Groups of Diffeomorphisms

THE CAMASSA-HOLM EQUATION AS A GEODESIC FLOW FOR THE H¹ RIGHT-INVARIANT METRIC

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Untitled

Cited by 113 publications

References 21 publications

A new two-component integrable system with peakon solutions

A new two-component integrable system with peakon solutions

Poisson Structures of Equations Associated With Groups of Diffeomorphisms

THE CAMASSA-HOLM EQUATION AS A GEODESIC FLOW FOR THE H1 RIGHT-INVARIANT METRIC

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THE CAMASSA-HOLM EQUATION AS A GEODESIC FLOW FOR THE H¹ RIGHT-INVARIANT METRIC