Introduction to Nonlinear PDE - Core Definitions

Nonlinear PDEs model phenomena where superposition fails: shock waves, solitons, pattern formation, and geometric flows. They require new techniques beyond linear theory and exhibit rich behaviors including finite-time blowup, multiple solutions, and spontaneous pattern emergence.

DefinitionNonlinear PDEs

A PDE is nonlinear if it involves nonlinear functions of the solution or its derivatives. Examples:

Semilinear: $\Delta u = f(u)$ (linear in highest derivatives, nonlinear in lower)
Quasilinear: $\nabla \cdot (a(u)\nabla u) = 0$ (linear in highest derivatives, coefficients depend on $u$ )
Fully nonlinear: $F(D^2u, Du, u, x) = 0$ (nonlinear in highest derivatives)

ExampleClassic Nonlinear Equations

Burgers equation: $u_t + uu_x = \nu u_{xx}$ (simplest model for shocks)
Korteweg-de Vries: $u_t + 6uu_x + u_{xxx} = 0$ (solitons, integrable)
Nonlinear Schrödinger: $i\psi_t + \Delta\psi + |\psi|^2\psi = 0$ (wave collapse)
Minimal surface equation: $\nabla \cdot \frac{\nabla u}{\sqrt{1 + |\nabla u|^2}} = 0$ (geometric)
Porous medium: $u_t = \Delta(u^m)$ (fast diffusion, finite speed)

DefinitionWeak Solutions and Shocks

For conservation laws $u_t + f(u)_x = 0$ , classical solutions may not exist globally due to shock formation. Weak solutions satisfy: $\int\int (u\phi_t + f(u)\phi_x)\,dx\,dt = 0$ for test functions $\phi$ . Multiple weak solutions exist; entropy conditions select physical ones.

Remark

Nonlinearity destroys uniqueness guarantees and enables new phenomena: bifurcations, pattern selection, finite-time blowup ( $\|u(t)\| \to \infty$ as $t \to T^* < \infty$ ), global vs local existence dichotomy. Theory is less complete than for linear PDEs, making it an active research frontier.