andrew n.w. hone and michael v. irle- on the non-integrability of the popowicz peakon system

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DISCRETE AND CONTINUOUS Website: www.aimSciences.org DYNAMICAL SYSTEMS Supplement 2009 pp. 359–366 ON THE NON-INTEGRABILITY OF THE POPOWICZ PEAKON SYSTEM Andrew N.W. Hone Inst itute of Mathe matics , Statis tics & Actua rial Scien ce University of Kent, Cant erbury CT2 7NF, UK Michael V. Irle Inst itute of Mathe matics , Statis tics & Actua rial Scien ce University of Kent, Cant erbury CT2 7NF, UK Abstract. We consider a coupled system of Hamiltonian partial differential equa tions introduced by Popowicz, which has the appearance of a two -field coup ling between the Camass a-Holm and Degasperis-Procesi equations. The latter equations are both known to be integrable, and admit peaked soliton (peakon) solutions with discontinuous derivatives at the peaks. A combination of a reciprocal transformation with Painlev´ e analysis provides strong evidence that the Popo wicz system is non-in tegrable. Nev erthe less, we are able to con- struct exact travelling wave solutions in terms of an elliptic integral, together with a degenerat e travelling wave correspondin g to a single peakon. We also describe the dynamics of N -peakon solutions, which is given in terms of an Hamiltonian system on a phase space of dimension 3N . 1. Introduction. The members of a one-parameter family of partial differential equations, namely m t + um x + bu x m = 0, m = u − u xx (1) with parameter b, have been studied recently. The case b = 2 is the Camassa-Holm equation [1], while b = 3 is the Degasperis-Procesi equation [3], and it is known that (with the possible excep tion of b = 0) these are the only integrable cases [ 14], while all of these equations (apart from b = −1) arise as a shallow water approximation to the Euler equations [6]. All of the equations have at least one Hamiltonian structure [12], this being given by m t = B δH δm , B = −b 2 m 1−1/b ∂ x m 1/b ˆ Gm 1/b ∂ x m 1−1/b , (2) with ˆ G = (∂ x − ∂ 3 x ) −1 and the Hamiltonian H = ( b − 1) −1 m dx for b = 0, 1 (and the latter special cases admit a similar expression). One of the most interesting features of these equations is that their soliton solutions are not smooth, but rather the field u has a discontinuous derivative at one or more peaks (hence the name peakons), while the corresponding field m is measure valued. More precisely for the single peakon the solution has the form u = c exp(−|x − ct − x 0 |), with m = 2 c δ(x − ct − x 0 ) 2000 Mathematics Subject Classification. Primary: 37K05, 37K10; Secondary: 37J99 . Key words and phrases. Camassa-Holm equation, Degasperis-Procesi equation, peakons. The second author was supported by an IMSAS studentship at the University of Kent. 359

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Page 1: Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

8/3/2019 Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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Page 2: Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

8/3/2019 Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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Page 3: Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

8/3/2019 Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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Page 4: Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

8/3/2019 Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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Page 5: Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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8/3/2019 Andrew N.W. Hone and Michael V. Irle- On the Non-Integrability of the Popowicz Peakon System

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