Quasilinear

A quasilinear equation is an equation of the form

$F(u,Du,\ldots ,D^{k}u)=0$

which is linear (and nontrivial) in the top order terms $D^{k}u$. Thus a quasilinear equation takes the schematic form

$F(u,Du,\ldots ,D^{k-1}u)D^{k}u=G(u,Du,\ldots ,D^{k-1}u).$

By differentiating this equation up to $k-1$ times and working with the system of fields $v:=(u,Du,\ldots ,D^{k}u)$, one can place such equations in the slightly simpler form

$F(v)D^{k}v=G(v,Dv,\ldots ,D^{k-1}v)$

though this trick comes at the cost of lowering the regularity of the fields.

Quasilinear equations are more nonlinear than semilinear ones, but less nonlinear than fully nonlinear equations.

See also

Quasilinear nonlinear wave (QNLW)