Here we consider the generalized Korteweg-de Vries (gKdV) equation with nonlinear term $s u^n \partial{u}/ \partial{x}$ where $n > 0$ and $ s=\pm 1$. Famous kinds of this equation are the Korteweg-de Vries (KdV) equation and modified Korteweg-de Vries (mKdV) equation are integrable and fully investigated. High-order versions of gKdV with $ n > 2$ are appeared in hydrodynamics of stratified fluid (Kurkina et al, 2011). Some versions of gKdV contain non-integer values of $n$, for instance, $n = 1/3$ - the Shamel equation appeared for ion-acoustic waves due to resonant electrons (Schamel, 1973). Also log-KdV equation used for solitary waves in FPU lattices can be mentioned (James and Pelinovsky, 2014). In all papers cited above the main attention is paid to soliton dynamics, its stability and interactions. Dynamics of modulated wave packets in KdV-like systems is less studied. If the wave amplitude is weak (to compare with dispersion) a standard way to investigate the stability of weakly modulated wave train is the deriving the Nonlinear Schrodinger equation (NLS) and determination its type. For the canonic KdV equation the NLS equation has a defocused type and therefore the wave packet is stable (Zakharov and Kuznetsov, 1986).In the case of modified Korteweg-de Vries with $s = +1$ a wave train is modulationally unstable and this leads to generation of rogue waves (Grimshaw et al, 2010). To our knowledge we do not know publications where the NLS equation has been derived from gKdV equation. This task problem is analyzed in our talk.
References:
Kurkina O.E., Kurkin A.A., Soomere, T. Pelinovsky E.N., Ruvinskaya E.A. Higher-order (2+4) Korteweg-de Vries - like equation for interfacial waves in a symmetric three-layer fluid. Physics Fluids. 2011, vol. 23, 116602.

Schame, H. A modified Korteweg-de Vries equation for ion acoustic waves due to resonant electrons. J. Plasma Phys. 1973, vol. 9, 377?387.

James G., Pelinovsky D.E. Gaussian solitary waves and compactons in Fermi-Pasta-Ulam lattices with Hertzian potentials, Proceedings of Royal Society A, 2014, vol. 470, 20130465 (20 pages).

Zakharov V.E., Kuznetsov E.A. Multi-scale expansions in the theory of systems integrable by the inverse scattering transform, Physica D, 1986, vol. 18, 455?463.

Grimshaw R., Pelinovsky E., Talipova T., Sergeeva A. Rogue internal waves in the ocean: long wave model. European Physical Journal Special Topics, 2010, vol. 185, 195 - 208.