In their paper [1] (full text at [2]) Bethuel et al on page 249 (bottom) linearize the moving frame Gross-Pitaevskii-Equation

$0=-ic \partial_{x_1} \widetilde{v} - \Delta \widetilde{v} - \widetilde{v}(1-\vert \widetilde{v}\vert^2)$

as follows: Setting $\widetilde{v}= V+1$ and $V=f+ig$ one gets

$0=-ic \partial_{x_1} (f+ig) - \Delta (f+ig) - (f+ig+1) (1 - (f+1)^2 +g^2)$

and therefore

$0 = \left( \begin{array}{cc} -\Delta+2 & c \partial_{x_1} \\\ -c \partial_{x_1} & -\Delta \end{array} \right) \left( \begin{array}{l} f \\\ g \end{array} \right) - \left( \begin{array}{l} fg^2-g^2-f^3-3f^2 \\\ -g^3-f^2 g - 2 fg \end{array} \right)$

Usually - in order to linearize the LHS - one would proceed by taking the derivative of the last term. But Bethuel et al just factorize and write

$0 = \left( \begin{array}{cc} -\Delta+2 & c \partial_{x_1} \\\ -c \partial_{x_1} & -\Delta \end{array} \right) \left( \begin{array}{l} f \\\ g \end{array} \right) + \left( \begin{array}{cc} g^2+f^2+3f & g\\\ 2g & f^2+g^2 \end{array} \right) \left( \begin{array}{l} f \\\ g \end{array} \right)$

In which way is that a linearization of the equation? Would this linearization be suitable for a stability analysis?