Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 095, 22 pages      arXiv:1603.08842      https://doi.org/10.3842/SIGMA.2016.095
Contribution to the Special Issue on Asymptotics and Universality in Random Matrices, Random Growth Processes, Integrable Systems and Statistical Physics in honor of Percy Deift and Craig Tracy

A Riemann-Hilbert Approach for the Novikov Equation

Anne Boutet de Monvel a, Dmitry Shepelsky b and Lech Zielinski c
a) Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot, 75205 Paris Cedex 13, France
b) Mathematical Division, Institute for Low Temperature Physics, 47 Nauki Avenue, 61103 Kharkiv, Ukraine
c) LMPA, Université du Littoral Côte d'Opale, 50 rue F. Buisson, CS 80699, 62228 Calais, France

Received June 08, 2016, in final form September 14, 2016; Published online September 24, 2016

Abstract
We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2u_x=3u u_xu_{xx}+u^2u_{xxx}$ considered on the line $x\in(-\infty,\infty)$ in the case of non-zero constant background. The approach is based on the analysis of an associated Riemann-Hilbert (RH) problem, which in this case is a $3\times 3$ matrix problem. The structure of this RH problem shares many common features with the case of the Degasperis-Procesi (DP) equation having quadratic nonlinear terms (see [Boutet de Monvel A., Shepelsky D., Nonlinearity 26 (2013), 2081-2107, arXiv:1107.5995]) and thus the Novikov equation can be viewed as a ''modified DP equation'', in analogy with the relationship between the Korteweg-de Vries (KdV) equation and the modified Korteweg-de Vries (mKdV) equation. We present parametric formulas giving the solution of the Cauchy problem for the Novikov equation in terms of the solution of the RH problem and discuss the possibilities to use the developed formalism for further studying of the Novikov equation.

Key words: Novikov equation; Degasperis-Procesi equation; Camassa-Holm equation; inverse scattering transform; Riemann-Hilbert problem.

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