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Similarity reductions of peakon equations: the \(b\)-family

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Abstract

The \(b\)-family is a one-parameter family of Hamiltonian partial differential equations of nonevolutionary 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 \(b=2\) and \(b=3\) (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 nonintegrable. After a discussion of traveling 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 \(b=3\), 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.

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Funding

L. E. Barnes was supported by a PhD studentship from SMSAS, Kent. The research of A. N. W. Hone was supported by Fellowship EP/M004333/1 from the Engineering & Physical Sciences Research Council, UK, and is currently funded by grant IEC\R3\193024 from the Royal Society.

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  1. School of Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury, United Kingdom

    L. E. Barnes & A. N. W. Hone

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  1. L. E. Barnes
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Correspondence to A. N. W. Hone.

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Prepared from an English manuscript submitted by the author; for the Russian version, see Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 212, pp. 303–324 https://doi.org/10.4213/tmf10238.

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Barnes, L.E., Hone, A.N.W. Similarity reductions of peakon equations: the \(b\)-family. Theor Math Phys 212, 1149–1167 (2022). https://doi.org/10.1134/S0040577922080104

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