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WAVEBREAKING AND PEAKONS FOR A MODIFIED CAMASSA–HOLM EQUATION
"... ABSTRACT. In this paper, we investigate the formation of singularities and the existence of peaked travelingwave solutions for a modified CamassaHolm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multipeakon solutions, ..."
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ABSTRACT. In this paper, we investigate the formation of singularities and the existence of peaked travelingwave solutions for a modified CamassaHolm equation with cubic nonlinearity. The equation is known to be integrable, and is shown to admit a single peaked soliton and multipeakon solutions, of a different character than those of the CamassaHolm equation. Singularities of the solutions can occur only in the form of wavebreaking, and a new wavebreaking mechanism for solutions with certain initial profiles is described in detail.
Moving frames and differential invariants in centro–affine geometry
, 2009
"... Explicit formulas for the generating differential invariants and invariant differential operators for curves in two and threedimensional centroequiaffine and centroaffine geometry and surfaces in threedimensional centroequiaffine geometry are constructed using the equivariant method of movi ..."
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Cited by 6 (2 self)
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Explicit formulas for the generating differential invariants and invariant differential operators for curves in two and threedimensional centroequiaffine and centroaffine geometry and surfaces in threedimensional centroequiaffine geometry are constructed using the equivariant method of moving frames. In particular, the algebra of centroequiaffine surface differential invariants is shown to be generated by a single second order invariant.
Poisson Structures for Geometric Curve Flows in Semisimple Homogeneous Spaces
"... Abstract. We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semisimple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated d ..."
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Abstract. We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semisimple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples. 1. Introduction. In 1972, Hasimoto, [6], showed how the evolution of curvature and torsion of space curves under the vortex filament flow is governed by the completely integrable nonlinear Schrödinger equation. Since then, a large variety of integrable soliton equations, including all of the most familiar examples (Korteweg–deVries, modified Korteweg–deVries, Sawada–
Differential Invariant Algebras
"... Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including ..."
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Abstract. The equivariant method of moving frames provides an algorithmic procedure for determining and analyzing the structure of algebras of differential invariants for both finitedimensional Lie groups and infinitedimensional Lie pseudogroups. This paper surveys recent developments, including a few surprises and several open questions. 1. Introduction. Differential invariants are the fundamental building blocks for constructing invariant differential equations and invariant variational problems, as well as determining their explicit solutions and conservation laws. The equivalence, symmetry and rigidity properties of submanifolds are all governed by their differential invariants. Additional applications
Recursive Moving Frames
, 2011
"... A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of ..."
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A recursive algorithm for the equivariant method of moving frames, for both finitedimensional Lie group actions and Lie pseudogroups, is developed and illustrated by several examples of interest. The recursive method enables one avoid unwieldy symbolic expressions that complicate the treatment of large scale applications of the equivariant moving frame method.
HAMILTONIAN EVOLUTIONS OF CURVES IN CLASSICAL AFFINE GEOMETRIES
"... Abstract. In this paper we study geometric Poisson brackets and we show that, if M = (G n IRn)/G endowed with an affine geometry (in the Klein sense), and if G is a classical Lie group, then the geometric Poisson bracket for parametrized curves is a trivial extension of the one for unparametrized cu ..."
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Abstract. In this paper we study geometric Poisson brackets and we show that, if M = (G n IRn)/G endowed with an affine geometry (in the Klein sense), and if G is a classical Lie group, then the geometric Poisson bracket for parametrized curves is a trivial extension of the one for unparametrized curves, except for the case G = GL(n, IR). This trivial extension does not exist in other nonaffine cases (projective, conformal, etc). 1.
Integrable systems and invariant curve flows in centroequiaffine symplectic geometry, Phys
 D
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Recent Advances in the Theory and Application of Lie Pseudo–Groups
"... Abstract. This paper surveys several new developments in the analysis of Lie pseudogroups and their actions on submanifolds. The main themes are direct construction of Maurer–Cartan forms and structure equations, and the use of equivariant moving frames to analyze the algebra of differential invaria ..."
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Abstract. This paper surveys several new developments in the analysis of Lie pseudogroups and their actions on submanifolds. The main themes are direct construction of Maurer–Cartan forms and structure equations, and the use of equivariant moving frames to analyze the algebra of differential invariants and invariant differential forms, including generators, commutation relations, and syzygies. 1. Introduction. Inspired by Galois ’ introduction of group theory to solve polynomial equations, Lie founded his remarkable theory of transformation groups for the purpose of analyzing and solving differential equations. In Lie’s time, abstract groups were as yet unknown, and hence he made no significant distinction between finitedimensional and infinitedimensional
MultiComponent Integrable Systems and Invariant Curve Flows in Certain Geometries
, 2013
"... In this paper, multicomponent generalizations to the Camassa–Holm equation, the modified Camassa–Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa–Holm equation and the multicomponent modified Camassa–H ..."
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In this paper, multicomponent generalizations to the Camassa–Holm equation, the modified Camassa–Holm equation with cubic nonlinearity are introduced. Geometric formulations to the dual version of the Schrödinger equation, the complex Camassa–Holm equation and the multicomponent modified Camassa–Holm equation are provided. It is shown that these equations arise from nonstreching invariant curve flows respectively in the threedimensional Euclidean geometry, the twodimensional Möbius sphere and ndimensional sphere Sn (1). Integrability to these systems is also studied.
Contact Geometry of Curves
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2009
"... Cartan’s method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equiaffine geometry is then made explicit. This delivers the complete set of invariant data which solves the Gequival ..."
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Cartan’s method of moving frames is briefly recalled in the context of immersed curves in the homogeneous space of a Lie group G. The contact geometry of curves in low dimensional equiaffine geometry is then made explicit. This delivers the complete set of invariant data which solves the Gequivalence problem via a straightforward procedure, and which is, in some sense a supplement to the equivariant method of Fels and Olver. Next, the contact geometry of curves in general Riemannian manifolds (M, g) is described. For the special case in which the isometries of (M, g) act transitively, it is shown that the contact geometry provides an explicit algorithmic construction of the differential invariants for curves in M. The inputs required for the construction consist only of the metric g and a parametrisation of structure group SO(n); the group action is not required and no integration is involved. To illustrate the algorithm we explicitly construct complete sets of differential invariants for curves in the Poincaré halfspace H3 and in a family of constant curvature 3metrics. It is conjectured that similar results are possible in other Cartan geometries.