  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.
