I have been struggling with some linearization argument of the following paper: “M. Weinstein: Modulational stability of ground states of NLS”. In order to give a bit of context to my question, let us consider the NLS equation $ $ 2i\phi_t+\Delta\phi+\vert\phi\vert^{2\sigma}\phi=0, \quad 0<\sigma<\tfrac{2}{n-2}.$ $ This equation has very interesting “localized” solutions of the form: $ $ \phi(t,x)=u(x)e^{it/2},$ $ where $ u(t,x)$ solves $ \Delta u-u+\vert u\vert^{2\sigma}u=0$ . Besides, the latter equation has an even more interesting real, positive and radial $ H^1(\mathbb{R}^n)$ solution called “Ground state” and denoted by $ R(x)$ .

Now let me try to explain my question. Consider the perturbed Initial Valued Problem (IVP): $ $ 2i\phi_t^\varepsilon+\Delta \phi^\varepsilon+\vert \phi^\varepsilon\vert^{2\sigma}\phi^\varepsilon=\varepsilon F(\vert \phi^\varepsilon\vert)\phi^\varepsilon, \quad \phi^\varepsilon(t=0,x)=R(x)+\varepsilon S(x)$ $ We will seek solutions of the previous equation of the form $ $ \phi^\varepsilon(t,x)=(R(x)+\varepsilon w_1+\varepsilon^2 w_2+…)e^{it/2}.$ $ According to Weinstein if you reeplace this function into the perturbed equation and linearize you will get the following IVP for the linearized perturbation $ w$ : $ $ 2iw_t+\Delta w-w+(\sigma+1)R^{2\sigma}w+\sigma R^{2\sigma}\overline{w}=F(R)R, \quad w(0,x)=0.$ $ Now my problem is: I do not really understand how to obtain this linearization, can someone explain a little bit how to do it? Or recommend some references to learn about it. I tried replacing $ \phi^\varepsilon$ (truncated after $ \varepsilon w$ ) and then I took the derivative with respect to $ \varepsilon$ and evaluate at $ \varepsilon=0$ , but I cannot recover the equation claimed by Weinstein, so I think that I’m not undersitanding how to linearize.

Note2: The parameter $ n$ denotes the dimension.