Nonlinear physics

Swift–Hohenberg equation

Simplistic pattern formation

$\pd{u}{t}=ru - (1+D\nabla^2)^2u+au^2+bu^3+cu^5$ with periodic boundary conditions

Complex Ginzburg–Landau

A world of complexity

$\pd{\psi}{t}=(D_r+\i D_i)\nabla^2 \psi+(a_r+\i a_i)\psi+(b_r+\i b_i)\psi|\psi|^2$

Cahn–Hilliard equation

Phase separation

$\pd{u}{t}=\nabla^2 (F(u)-g\nabla^2u)+f(u)$

Kuramoto–Sivashinsky equation

Chaos at your fingertips

$\pd{u}{t}=-\nabla^2u-\nabla^4u-|\vnabla u|^2 $

The Gray–Scott model

Intricate reaction–diffusion patterning

$\pd{u}{t}=\nabla^2 u+u^2v - (a+b)u$, $\pd{v}{t}=D\nabla^2v -u^2v + a(1 - v)$

Viscous Burgers' equation

Nonlinear waves

$\pd{u}{t} =-u\pd{u}{x}+\varepsilon \pdd{u}{x}$

Pattern formation & advection

Going with the flow?

Solitons in the Korteweg–De Vries & Zakharov–Kuznetsov equations

Solitary movement through each other

$\pd{\phi}{t}=-\pdn{\phi}{x}{3}-6\phi \pd{\phi}{x}$

The Perona–Malik equation

Image denoising via nonlinear anisotropic diffusion

$\pd{u}{t}=\vnabla \cdot \left (\mathrm{e}^{-D |\vnabla u|^2}\vnabla u\right) $

Hyperbolic reaction–diffusion systems

Turing wave instabilities

$\tau\pdd{u}{t}+\pd{u}{t}=D_u\nabla^2 u+f(u,v)$, $\tau\pdd{v}{t}+\pd{v}{t}=D_v\nabla^2v+ g(u,v)$

Bending in nonlinear beams

Exploring state-dependent stiffness

$\pd{y}{t}=-\pdd{}{x}[E(y)\pdd{y}{x}]$

Stochastic partial differential equations

Randomness in time and space

$\pd{u}{t}=D\nabla^2 u +f(u)+\frac{dW_t}{dt}$