Integrable peakon equations with cubic nonlinearity

By Andrew N. W. Hone, Jing Ping Wang. 2008

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We present a new integrable partial differential equation found by Vladimir Novikov. Like the Camassa–Holm and Degasperis–Procesi equations, this new equation admits peaked soliton (peakon) solutions, but it has nonlinear terms that are cubic, rather than quadratic. We give a matrix Lax pair for V Novikov's equation, and show how it is related by a reciprocal transformation to a negative flow in the Sawada–Kotera hierarchy. Infinitely many conserved quantities are found, as well as a bi-Hamiltonian structure. The latter is used to obtain the Hamiltonian form of the finite-dimensional system for the interaction of N peakons, and the two-body dynamics (N = 2) is explicitly integrated. Finally, all of this is compared with some analogous results for another cubic peakon equation derived by Zhijun Qiao.

Subfields: Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/679ebec20b8c5904cbd0b4b4ffae66cef837179b

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Citations in the corpus

Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation (2014, neutral): “7) have been studied respectively in [26,27,37,42,43].”
Global existence and blow-up phenomena for the weakly dissipative Novikov equation (2012, methodology): “The Novikov equation possesses a matrix Lax pair, many conserved densities, a bi-Hamiltonian structure as well as peakon solutions u(x, t) = ± √ ce−|x−ct−x0|, c > 0, where x0 is a constant [2].”
Well-posedness and global existence for the Novikov equation (2012, background): “1) arises as a zero curvature equation Ft −Gx +[F,G] = 0, this being the compatibility condition for the linear system [13] { !x = F!, !t = G!,”
Equations with Peakon Traveling Wave Solutions (2013, background): “For related results about them, we refer the reader to [2, 5–15, 17–19, 21] and the references therein.”
The periodic Cauchy problem for Novikov's equation (2010, background): “Another representation of this equation as a matrix Lax pair is given in [11] and is shown to be related to a negative flow in the Sawada-Kotera hierarchy.”
The Cauchy problem for the integrable Novikov equation (2012, background): “1) possesses a Lax pair, many conserved densities, a Hamiltonian structure and peakon solutions u(x, t) = ±√ce−|x−ct−x0| , c > 0, where x0 is a constant, as well as the explicit formulas for multipeakon solutions [10,9].”
Global weak solutions for the Novikov equation (2011, background): “1) has the soliton waves with corner at their peaks, discovered in [19].”
Analysis on the blow-up of solutions to a class of integrable peakon equations (2016, background): “Moreover, these equations admit a remarkable variety of the so-called “peakon” solutions – peaked traveling wave solutions with a discontinuous derivative at crest [15, 20, 26].”