For Fisher-KPP $u_t = u_{xx} + u(1-u)$, seek $u(x,t) = U(x - ct)$. Substituting gives ODE: $-cU' = U'' + U(1-U)$ Solutions with $U(-\infty) = 1$, $U(\infty) = 0$ exist for $c \geq 2$, giving traveling wave fronts propagating with speed $c$.

For semilinear $-\Delta u = f(u)$ with $f$ subcritical growth, define operator $T: u \mapsto w$ where $-\Delta w = f(u)$. Under suitable conditions, $T$ is a contraction on appropriate spaces, so Banach fixed point theorem gives existence/uniqueness via $u = T(u)$.

For $-\Delta u + u = u^p$ on bounded $\Omega$ with $u = 0$ on $\partial\Omega$, seek critical points of: $J[u] = \frac{1}{2}\int(|\nabla u|^2 + u^2) - \frac{1}{p+1}\int|u|^{p+1}$

Mountain pass theorem (when $J$ lacks compactness) finds saddle point solutions even without uniqueness or minimizers.

Newton's method: Linearize $F(u) = 0$ as $F(u^n) + F'(u^n)(u^{n+1} - u^n) = 0$, iterate.

Continuation methods: Start from solvable $F_0(u) = 0$, follow solution branch as parameter varies to target $F_1(u) = 0$.

Splitting methods: For $u_t = A(u) + B(u)$, alternate solving $u_t = A(u)$ and $u_t = B(u)$ separately (operator splitting, ADI).