Integrable Evolution Equations on Spaces of Tensor Densities and Their Peakon Solutions

Abstract

Abstract. We study a family of equations defined on the space of tensor densities of weight λ on the circle and introduce two integrable PDE. One of the equations turns out to be closely related to the inviscid Burgers equation while the other has not been identified in any form before. We present their Lax pair formulations and describe their bihamiltonian structures. We prove local wellposedness of the corresponding Cauchy problem and include results on blow-up as well as global existence of solutions. Moreover, we construct "peakon" and "multi-peakon" solutions for all λ = 0, 1, and "shock-peakons" for λ = 3. We argue that there is a natural geometric framework for these equations that includes other well-known integrable equations and which is based on V. Arnold's approach to Euler equations on Lie groups.

Conservative weak solutions of the periodic Cauchy problem for μHS equation

“…In order to carry out the analysis of solutions of the system (9)- (12) and obtain a semigroup of solutions, we first prove the existence and uniqueness of solutions for a short time using a Banach contraction argument. For our purposes, the appropriate spaces for the variables y and V are not Banach spaces, hence we introduce a set of transient variables living in Banach spaces.…”

“…Equation (1) admits peaked traveling waves that are analogues of solitons. 10,12 Like the Camassa-Holm Eq. (2) and the Hunter-Saxton Eq.…”

“…They can be written as Euler-Arnold equations on the regular dual of the Lie algebra T e Diff(S 1 ) = vect(S 1 ) of smooth vector fields on the circle. 12 Classical solutions of (1) are studied and some preliminary results are established, concerning local and global (in time) existence and uniqueness in Sobolev spaces as well as breakdown. 10 The periodic Cauchy problem is locally well posed in H s for s > 3/2.…”

Blow-up phenomena and global existence for the weakly dissipative generalized periodic Degasperis-Procesi equation

“…All authors read and approved the final manuscript. 1 Department of Mathematics, Tianshui Normal University, Tianshui, Gansu 741001, P.R. China.…”

On the Cauchy problem for the two-component Euler–Poincaré equations

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