Multipeakons and the Classical Moment Problem

Abstract

dedicated to gian-carlo rotaClassical results of Stieltjes are used to obtain explicit formulas for the peakon antipeakon solutions of the Camassa Holm equation. The closed form solution is expressed in terms of the orthogonal polynomials of the related classical moment problem. It is shown that collisions occur only in peakon antipeakon pairs, and the details of the collisions are analyzed using results from the moment problem. A sharp result on the steepening of the slope at the time of collision is given. Asymptotic formulas are given, and the scattering shifts are calculated explicitly. Academic Press

Geometric Integrability of the Camassa–Holm Equation. II

“…Rather, the point is that the local conserved quantities do not guarantee the existence of a priori bounds for the first derivative of the solutions u(x, t) to the CH equation. A careful analysis of the behavior of u x in peakon-antipeakon collisions appears in [5]. Finally, we note that Equations (26) We now classify all first-order nonlocal 7r-symmetries of the CH equation.…”

“…These compatible equations specify the relations between the dependent variables xf and the new dependent variables y b . Conversely, a set of equations of the form (7), which is compatible on solutions to the system S a = 0, determines a covering n = (y b ; Xa,; Di) where the differential operators Ó¿ are defined by means of (5).…”

“…v ' dy so that [V 5 , V e ] = 0 only after the augmented system of Equations (23)- (27) is imposed. This observation reminds us of infernal symmetries, a notion we now recall following [3].…”

“…In other words, we change our viewpoint, from thinking of the CH equation (63) as a differential equation for u(x, t), to considering (63) as an integro-differential equation for m. This approach has proved to be crucial for the analytic study of the CH equation: it has been used to determine whether solutions to CH are global in time or represent breaking waves [13,15,36], and it appears prominently in Lenells' construction of conservation laws [33], in the study of (multi)peakon dynamics and weak solutions [5,10,14], in the scattering/inverse scattering approach to CH [4,5,17], and also in a rigorous proof that the "least action principie" holds for CH [16].…”

Modulation of the Camassa-Holm equation and reciprocal transformations

“…The CH equation possesses soliton solutions, periodic finite-gap solutions [8], [10], [4], real finite-gap solutions [22] and, for ν = 0, multi-peakons [6]. In particular, the algebro-geometric solutions of (CH) are described as Hamiltonian flows on nonlinear subvarieties (strata) of generalized Jacobians.…”