A Riemann-Hilbert Approach for the Novikov Equation

TL;DR

This paper develops a Riemann-Hilbert based inverse scattering transform method for solving the Novikov equation on the real line with non-zero background, linking it to the Degasperis-Procesi equation and providing a framework for future analysis.

Contribution

It introduces a novel RH problem approach for the Novikov equation, extending inverse scattering techniques and connecting it to known integrable equations.

Findings

Formulation of a 3x3 matrix RH problem for the Novikov equation.

Parametric formulas for solving the Cauchy problem.

Potential for further analytical and numerical studies using the formalism.

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.