The periodic Cauchy problem for Novikov's equation

By Feride Tiglay. 2010

119 citations recorded by the source corpus.

We study the periodic Cauchy problem for an integrable equation with cubic nonlinearities introduced by V. Novikov. Like the Camassa-Holm and Degasperis-Procesi equations, Novikov's equation has Lax pair representations and admits peakon solutions, but it has nonlinear terms that are cubic, rather than quadratic. We show the local well-posedness of the problem in Sobolev spaces and existence and uniqueness of solutions for all time using orbit invariants. Furthermore we prove a Cauchy-Kowalevski type theorem for this equation, that establishes the existence and uniqueness of real analytic solutions.

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/31aeca3c88507ea2c8c85e44e0a6e45e84c62b3b

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In Applied Mathematics

In Mathematical Physics

In Statistical and Nonlinear Physics

Citations in the corpus

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.”
Wave-Breaking and Peakons for a Modified Camassa–Holm Equation (2013, background): “The integrability, peaked solitons, well-posedness and blow up phenomena to the Novikov equation have been studied extensively [34,35,45,56,57].”
Global weak solutions for the Novikov equation (2011, background): “Recently, the Cauchy problem of the periodic Novikov equation on the line and on the circle was investigated in [25, 27].”
The Cauchy problem for the integrable Novikov equation (2012, background): “Tiğlay [15] investigated the Cauchy problem for the periodic Novikov equation.”
Well-posedness, wave breaking and peakons for a modified μ-Camassa-Holm equation (2014, unknown): “7) have been studied respectively in [26,27,37,42,43].”
Analysis on the blow-up of solutions to a class of integrable peakon equations (2016, background): “One can refer to [6, 8, 9, 10, 11, 12, 16, 19, 22, 28] and the references therein for details.”