Similarity reductions of peakon equations: the $b$-family

Abstract

The b-family is a one-parameter family of Hamiltonian partial differential equations of non-evolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases b = 2, 3 (the Camassa-Holm and Degasperis-Procesi equations, respectively) the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter b it is non-integrable. After a discussion of travelling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the b-family, and show that when b = 2 or 3 this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of b the resulting ordinary differential equation is not of Painlevé type.

Similarity reductions of peakon equations: integrable cubic equations

“…The main point is that, as described in [3] (and originally derived in [25]), the ODE (2.57) arises as the equation for scaling similarity reductions of the negative KdV flow (2.3), so by applying the result of Proposition 2.1 to these reductions, a link with (2.52) follows. Under the reduction (2.43) applied to (2.15) with v(X, T ) = 1 2 T…”

“…Their results included numerically stable "ramp-cliff" solutions for −1 < b < 1, looking like the ramp (1.14) in a compact region, joined to a rapidly decaying cliff. The results in [3] show that the scaling similarity reductions of (1.13) are related via a transformation of hodograph type to a non-autonomous ODE of second order; but this ODE only has the Painlevé property in the integrable cases b = 2, 3, when it is equivalent to two different versions of the Painlevé III equation (1.12): the reduction already found for the Camassa-Holm equation in [25], and another set of values of α, β, γ, δ for the reduction of the Degasperis-Procesi equation.…”

“…The equation (2.33) is solved in terms of elliptic functions. A shortcut to deriving the explicit form of these solutions is provided by Proposition 2.1: there is a Miura map between the solutions of (2.33) and the travelling wave solutions of (2.3), as presented in [3] (see also [25]). Identifying c with the wave velocity d in [3], the travelling waves of (2.3) correspond to a KdV field V (Z) = −2℘(Z) − ℘(W ) (up to the freedom to replace Z → Z+ const), where W is an arbitrary constant, so that the Miura formula (2.14) requires that dṽ dZ…”

“…The solution (2.37) can also be obtained more directly from the travelling wave reduction of (2.3), which is given by p(X, T ) = P (Z) with [3]), by applying this reduction to the formula (2.16). This gives ṽ = − dP dZ − 1 2P , and then substituting the explicit form of P as above yields the required expression for ṽ.…”

“…In recent work [3], we studied similarity reductions of the so-called b-family of equations, given by…”