Dynamics of Partial Differential Equations

Volume 6 (2009)

Number 3

Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa–Holm type equation

Pages: 253 – 289

DOI: http://dx.doi.org/10.4310/DPDE.2009.v6.n3.a3

Authors

Andrew N.W. Hone (Institute of Mathematics, Statistics & Actuarial Science, University of Kent, Canterbury, United Kingdom)

Hans Lundmark (Department of Mathematics Linköping University, Sweden)

Jacek Szmigielski (Department of Mathematics and Statistics, University of Saskatchewan, Canada)

Abstract

Recently Vladimir Novikov found a new integrable analogue of the Cmassa–Holm equation which has nonlinear terms that are cubis, rather than quadratic, an which admits peaked soliton solutions (peakons). In this paper, the explicit formulas for multipeakon solutions of Novikov’s cubically nonlinear equation are calculated, using the matrix Lax pair found by Hone and Wang. By a transformation of the Liouville type, the associated spectral problem is related to a cubic string equation, which is dual to the cubic string that was previously found in the work of Lundmark and Szmigielski on the multipeakons of the Degasperis–Procesi equation.

Keywords

Peakons, cubic string, Novikov’s equation, Degasperis–Procesi equation, distributional Lax pair, sum of minors

2010 Mathematics Subject Classification

37K15

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