Picture below is from
Weinstein, Michael I., Lyapunov stability of ground states of nonlinear dispersive evolution equations, Commun. Pure Appl. Math. 39, 51-67 (1986). ZBL0594.35005.
I don't know how to get the (3.1) and (3.2). The (2.7) is $$ Q(\phi)=||\nabla \phi(\cdot+x_0,t )e^{i\gamma}-\nabla R(\cdot)||_{L^2}^2+E||\phi(\cdot+x_0,t)e^{i\gamma}-R(\cdot)||_{L^2}^2 $$ where $E$ is a constant. And $R$ is solution of $$ \Delta R - ER + |R|^{2\sigma}R=0. $$ $\phi$ is solution of $$ i\phi_t(x,t) + \Delta \phi(x,t)+ |\phi(x,t)|^{2\sigma}\phi(x,t)=0. $$ Besides, we have $$ \phi(x+x_0, t) e^{i\gamma} \equiv R（x） + w, ~~~~w=u+iv. $$ In fact, I think I should do perturbation over $\phi$ in (2.7)， evenly $$ \frac{d}{dt}|_{t=0}Q(\phi+ t\varphi)=0 $$ then, consider the real part and imaginary part I should get it. But obviously, in my way, I am far from (3.1) and (3.2).
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