The inverse problem which arises in the Camassa-Holm equation is revisited for the class of discrete densities. The method of solution relies on the use of orthogonal polynomials. The explicit formulas are obtained directly from the analysis on the real axis without any additional transformation to a "string" type boundary value problem known from prior works. Hongyou Wu in memoriam

The peakon inverse problem for the Degasperis-Procesi equation is solved directly on the real line, using Cauchy biorthogonal polynomials, without any additional transformation to a "string" type boundary value problem known from prior works.

In this paper we show that non-smooth functions which are distributional traveling wave solutions to the two component Camassa-Holm equation are distributional traveling wave solutions to the Camassa-Holm equation provided that the set u −1 (c), where c is the speed of the wave, is of measure zero. In particular there are no new peakon or cuspon solutions beyond those already satisfying the Camassa-Holm equation. However, the two component Camassa-Holm equation has distinct from Camassa-Holm equation smooth traveling wave solutions as well as new distributional solutions when the measure of u −1 (c) is not zero. We provide examples of such solutions.Keywords: Camassa-Holm equation; traveling waves; peakons. Mathematics Subject Classification 2000: 35Q35, 35Q53The Camassa-Holm equation [1] arises as a model for the unidirectional propagation of shallow water waves over a flat bottom, u(x, t) representing the water's free surface, and κ ∈ R being a parameter related to the critical shallow water speed (see [13] and [12] for the derivation of this equation). Camassa and Holm [1] discovered that the equation has non-smooth solitary waves that retain their individual characteristics through the interaction and eventually emerge with their original shapes and speeds. The traveling wave solutions of the Camassa-Holm equation have been classified by Lenells [14]. The peaked traveling waves appear to have the characteristic of the traveling waves of greatest height similar to exact traveling wave solutions of the governing equations for water waves with a peak at their crest (see [17,3,4]). However, simpler approximate shallow water models like KdV do not admit traveling wave solutions with this feature (see [18]). Also, it should be noted that the Camassa-Holm peaked traveling waves are stable wave forms (see [5,6,15]). Therefore, they are physically recognizable. An alternative, and useful for generalizations form of this equation iswhere m = u − u xx + 1 2 κ.117

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