List of nonlinear partial differential equations

A–F

Name	Dim	Equation	Applications
Bateman-Burgers equation	1+1	$\displaystyle u_{t}+uu_{x}=\nu u_{xx}$	Fluid mechanics
Benjamin–Bona–Mahony	1+1	$\displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0$	Fluid mechanics
Benjamin–Ono	1+1	$\displaystyle u_{t}+Hu_{xx}+uu_{x}=0$	internal waves in deep water
Boomeron	1+1	$\displaystyle u_{t}=\mathbf {b} \cdot \mathbf {v} _{x},\quad \displaystyle \mathbf {v} _{xt}=u_{xx}\mathbf {b} +\mathbf {a} \times \mathbf {v} _{x}-2\mathbf {v} \times (\mathbf {v} \times \mathbf {b} )$	Solitons
Boltzmann equation	1+6	${\frac {\partial f_{i}}{\partial t}}+{\frac {\mathbf {p} _{i}}{m_{i}}}\cdot \nabla f_{i}+\mathbf {F} \cdot {\frac {\partial f_{i}}{\partial \mathbf {p} _{i}}}=\left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} },\quad \left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )\,d\Omega \,d^{3}\mathbf {p'}$	Statistical mechanics
Born–Infeld	1+1	$\displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0$	Electrodynamics
Boussinesq	1+1	$\displaystyle u_{tt}-u_{xx}-u_{xxxx}-3(u^{2})_{xx}=0$	Fluid mechanics
Boussinesq type equation	1+1	$\displaystyle u_{tt}-u_{xx}-2\alpha (uu_{x})_{x}-\beta u_{xxtt}=0$	Fluid mechanics
Buckmaster	1+1	$\displaystyle u_{t}=(u^{4})_{xx}+(u^{3})_{x}$	Thin viscous fluid sheet flow
Cahn–Hilliard equation	Any	$\displaystyle c_{t}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right)$	Phase separation
Calabi flow	Any	${\frac {\partial g_{ij}}{\partial t}}=(\Delta R)g_{ij}$