So I have this nonlinear PDE, the sine-Gordon Equation, $u_{tt}-c^{2}u_{xx}+\omega_{p}^{2}\sin u=0$ whose linearized solution is given by $u_0$. ($c$ and $\omega_p$ are constant.)
My reference tells me that to get the actual solution, we need the solution to have the form $$ u=\epsilon\left(u_{0}+\epsilon u_{1}+\epsilon^{2}u_{2}+\cdots\right), $$ where the $u_i$'s contribute the nonlinear components.
How did this come about?
