Nonlinear physics


Explore an example

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
$\pd{u}{t}=D\nabla^2 u +f(u)+\frac{\mathrm{d}W_t}{\mathrm{d}t}$
Shallow water equations
Water waves and ripples
$\pd{h}{t} = - \vnabla \cdot \left( \v{u} \left(h+H_{e}\right)\right)$, $\pd{\v{u}}{t} = \nu \nabla^2 {\v{u}} -g \vnabla h - k {\v{u}}-({\v{u}}\cdot \vnabla){\v{u}}-\v{f}\times \v{u}$