A Riemann-Hilbert Approach for the Novikov Equation

Abstract

We develop the inverse scattering transform method for the Novikov equation $u_t-u_{txx}+4u^2{u}_x=3u{u}_x{u}_{xx}+u^2{u}_{xxx}$ 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.

The analytic tools proposed in arXiv:1107.5995 are applied and further developed in a new context

References in corpus (5)

• Inverse Scattering Transform for the Camassa-Holm equation
• Long-Time Asymptotics for the Camassa-Holm Equation
• Inverse Scattering Transform for the Degasperis-Procesi Equation
• Generalisations of the Camassa-Holm equation
• Smooth multisoliton solutions and their peakon limit of Novikov's Camassa-Holm type equation with cubic nonlinearity

Cited by in corpus (7)

• A Riemann-Hilbert approach to the modified Camassa-Holm equation with nonzero boundary conditions
• A view of the peakon world through the lens of approximation theory
• The coupled Fokas-Lenells equations by a Riemann-Hilbert approach
• A Riemann-Hilbert Approach to the Kundu-Eckhaus Equation on the half-Line
• Riemann-Hilbert approach for multi-soliton solutions of a fifth-order nonlinear Schrodinger equation
• An initial-boundary value problem for the coupled focusing-defocusing complex short pulse equation with a $4\times 4$ Lax pair
• Long-time asymptotic behavior for the Novikov equation in solitonic regions of space time