See also Nonlinear partial differential equation and List of partial differential equation topics.
vxt=uxxb+a×vx−2v×(v×b){\displaystyle \displaystyle \mathbf {v} _{xt}=u_{xx}\mathbf {b} +\mathbf {a} \times \mathbf {v} _{x}-2\mathbf {v} \times (\mathbf {v} \times \mathbf {b} )}
φxx+c3φyy=(|u|2)x{\displaystyle \displaystyle \varphi _{xx}+c_{3}\varphi _{yy}=(|u|^{2})_{x}}