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The interest in the singular solutions (peakons) has been inspired by the Camassa-Holm (CH) equation and its peakons. An integrable peakon equation with cubic nonlinearities was first discovered by Qiao. Another integrable equation with cubic nonlinearities was introduced by V. Novikov . We investigate the peakon and soliton solutions of the Qiao equation.
R.Ivanov and T. Lyons, On the Peakon and Soliton Solutions of an Integrable PDE with Cubic Nonlinearities, BGSIAM'12 proceedings (7-th Annual Meeting of the Bulgarian Section of SIAM, Sofia, 19-20 December 2012), pp 78-87.