Why minimization of $||\nabla \phi(\cdot+x_0)e^{i\gamma} - \nabla R(\cdot)||_{L^2}^2 + E||\phi(\cdot + x_0)e^{i\gamma}-R(\cdot)||_{L^2}^2$ over $x_0(t),\gamma(t)$ implies $$ \int R^{2\sigma}(x)\frac{\partial R(x)}{\partial x_j} u(x,t) dx =0 $$ and $$ \int R^{2\sigma+1}(x)v(x,t)dx=0 $$ where $\phi$ is ground state solitary waves of the nonlinear Schrodinger equation $$ i\phi_t+\Delta\phi+f(|\phi|^2)\phi=0 $$ And $R$ is solution of $$ \Delta R -ER+f(|R|^2)R=0 $$ where $E$ is a constant.
This question is from 57th page of Lyapunov stability of ground states of nonlinear dispersive evolution equations. Seemly, I should derivative the $||\nabla \phi(\cdot+x_0)e^{i\gamma} - \nabla R(\cdot)||_{L^2}^2 + E||\phi(\cdot + x_0)e^{i\gamma}-R(\cdot)||_{L^2}^2$ , but I can't deal the calculation.
Thanks for any hint or answer.