A Riemann-Hilbert Approach for the Novikov Equation

Symmetry Integrability and Geometry-Methods and Applications

article

Anne Boutet de Monvel¹  Dmitry Shepelsky²  Lech Zielinski³ 

[1] Institut de Mathématiques de Jussieu-PRG, Université Paris Diderot;Mathematical Division, Institute for Low Temperature Physics;Université du Littoral Côte d'Opale

关键词: Novikov equation;    Degasperis–Procesi equation;    Camassa–Holm equation;    inverse scattering transform;    Riemann–Hilbert problem;   

DOI  :  10.3842/SIGMA.2016.095

来源: National Academy of Science of Ukraine

【 摘 要 】

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.

【 授权许可】

Unknown