Erdogan–Chatwin equation

In fluid dynamics, Erdogan–Chatwin equation refers to a nonlinear diffusion equation for the scalar field, that accounts for shear-induced dispersion due to horizontal buoyancy forces. The equation was named after M. Emin Erdogan and Phillip C. Chatwin, who derived the equaiton in 1967. The equation for the scalar field $\phi (x,t)$ reads

${\phi }_t=({\phi }_x+a{\phi }_x^3{)}_x,$

where $a$ is a positive constant. For $a\ll 1$, the equation reduces to the linear heat equation, ${\phi }_t={\phi }_{xx}$ and for $a\gg 1$, the equation reduces to ${\phi }_t=3a{\phi }_x^2{\phi }_{xx}$.