I am going through Yuen and Lake's (1981) "Nonlinear Dynamics of Deep Water Gravity Waves" and I am trying to understand how they carry out a step in the derivation of the non-linear Schrodinger equation (NLSE). They start from
$\omega=(gk)^{1/2}(1+\frac{1}{2}k^2a^2)\tag{1}$
which is the nonlinear dispersion equation for a deep water gravity wave with amplitude $a$, wavenumber $k$ and angular frequency $\omega$. They then state:
"We now allow small perturbations on the wavenumber $k$ and expand the expression about a constant $k_0$, keeping terms to second order in the perturbation and the nonlinearity. We obtain for the perturbed frequency $\omega'$ and perturbed wavenumber $k'$ that
$\omega'-\frac{\omega_0}{2k_0}k' +\frac{\omega_0}{8k_0^2}k'^2+\frac{1}{2}\omega_0k_0^2a^2 = 0\tag{2}$."
So here is my question: how is this step carried out? Surely it doesn't mean that I replace $k$ in equation (1) with $k_0+k'$, because I've tries and it doesn't work out...
Based on Remoissenet (1994), starting with the third order dispersion equation:
$$\omega= (gk(1+a^2k^2))^{\frac{1}{2}}$$
we need to consider small changes in $\omega$, i.e. $\omega'=\omega-\omega_0$. From the dispersion equation, changes in $\omega$ come about from changes in $k$ and $a$. Therefore,
$$\omega-\omega_0=\frac{\partial\omega}{\partial k}(k-k_0)+\frac{1}{2}\frac{\partial^2\omega}{\partial k^2}(k-k_0)^2+\frac{\partial\omega}{\partial a^2}(a^2-a_0^2)\tag{1}\label{1}$$
The derivatives can be calculated from the dispersion equation and found to be
$$\frac{\partial\omega}{\partial k}=\frac{\omega}{2k}$$
$$\frac{\partial^2\omega}{\partial k^2}=-\frac{\omega}{4k^2}$$
$$\frac{\partial\omega}{\partial a^2}=\frac{\omega k^2}{2}$$
Substituting the above into \ref{1}, and replacing the small changes with the dash notation, leads to the result of Yuen & Lake (1982).