List of nonlinear partial differential equations

Template:Short description See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.

A–F

Name	Dim	Equation	Applications
Bateman-Burgers equation	1+1	$u_t+u{u}_x=\nu u_{xx}$	Fluid mechanics
Benjamin–Bona–Mahony	1+1	$u_t+u_x+u{u}_x-u_{xxt}=0$	Fluid mechanics
Benjamin–Ono	1+1	$u_t+H{u}_{xx}+u{u}_x=0$	internal waves in deep water
Boomeron	1+1	$u_t=𝐛\cdot {𝐯}_x,{𝐯}_{xt}=u_{xx}𝐛+𝐚\times {𝐯}_x-2𝐯\times (𝐯\times 𝐛)$	Solitons
Boltzmann equation	1+6	$\frac{\partial f_i}{\partial t}+\frac{{𝐩}_i}{m_i}\cdot \mathrm{\nabla }{f}_i+𝐅\cdot \frac{\partial f_i}{\partial {𝐩}_i}={\left(\frac{\partial f_i}{\partial t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}},$ ${\left(\frac{\partial f_i}{\partial t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}=\sum _{j=1}^n\iint g_{ij}{I}_{ij}(g_{ij},\mathrm{\Omega })[f{\text{'}}_{i}f{\text{'}}_j-f_i{f}_j]d\mathrm{\Omega }{d}^3{𝐩}^{\prime }$	Statistical mechanics
Born–Infeld	1+1	$(1-u_t^2)u_{xx}+2u_x{u}_t{u}_{xt}-(1+u_x^2)u_{tt}=0$	Electrodynamics
Boussinesq	1+1	$u_{tt}-u_{xx}-u_{xxxx}-3(u^2{)}_{xx}=0$	Fluid mechanics
Boussinesq type equation	1+1	$u_{tt}-u_{xx}-2\alpha (u{u}_x{)}_x-\beta u_{xxtt}=0$	Fluid mechanics
Buckmaster	1+1	$u_t=(u^4{)}_{xx}+(u^3{)}_x$	Thin viscous fluid sheet flow
Cahn–Hilliard equation	Any	$c_t=D{\mathrm{\nabla }}^2(c^3-c-\gamma {\mathrm{\nabla }}^{2}c)$	Phase separation
Calabi flow	Any	$\frac{\partial g_{ij}}{\partial t}=(\mathrm{\Delta }R)g_{ij}$	Calabi–Yau manifolds
Camassa–Holm	1+1	$u_t+2\kappa u_x-u_{xxt}+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}$	Peakons
Carleman	1+1	$u_t+u_x=v^2-u^2=v_x-v_t$
Cauchy momentum	any	$\rho (\frac{\partial 𝐯}{\partial t}+𝐯\cdot \mathrm{\nabla }𝐯)=\mathrm{\nabla }\cdot \sigma +\rho 𝐟$	Momentum transport
Chafee–Infante equation		$u_t-u_{xx}+\lambda (u^3-u)=0$
Clairaut equation	any	$x\cdot Du+f(Du)=u$	Differential geometry
Clarke's equation	1+1	$({\theta }_t-\gamma \delta e^{\theta }{)}_{tt}={\mathrm{\nabla }}^2({\theta }_t-\delta e^{\theta })$	Combustion
Complex Monge–Ampère	Any	$\det ({\partial }_{i\overline{j}}\phi )=$ lower order terms	Calabi conjecture
Constant astigmatism	1+1	$z_{yy}+{\left(\frac{1}{z}\right)}_{xx}+2=0$	Differential geometry
Davey–Stewartson	1+2	$i{u}_t+c_0{u}_{xx}+u_{yy}=c_1|u{|}^{2}u+c_{2}u{\phi }_x,{\phi }_{xx}+c_3{\phi }_{yy}=(|u{|}^2{)}_x$	Finite depth waves
Degasperis–Procesi	1+1	$u_t-u_{xxt}+4u{u}_x=3u_x{u}_{xx}+u{u}_{xxx}$	Peakons
Dispersive long wave	1+1	$u_t=(u^2-u_x+2w{)}_x$, $w_t=(2uw+w_x{)}_x$
Drinfeld–Sokolov–Wilson	1+1	$u_t=3w{w}_x,w_t=2w_{xxx}+2u{w}_x+u_{x}w$
Dym equation	1+1	$u_t=u^3{u}_{xxx}.$	Solitons
Eckhaus equation	1+1	$i{u}_t+u_{xx}+2|u{|}_x^{2}u+|u{|}^{4}u=0$	Integrable systems
Eikonal equation	any	$|\mathrm{\nabla }u(x)|=F(x),x\in \mathrm{\Omega }$	optics
Einstein field equations	Any	$R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$	General relativity
Erdogan–Chatwin equation	1+1	${\phi }_t=({\phi }_x+a{\phi }_x^3{)}_x$	Fluid dynamics
Ernst equation	2	$\mathrm{\Re }(u)(u_{rr}+u_r/r+u_{zz})=(u_r{)}^2+(u_z{)}^2$
Estevez–Mansfield–Clarkson equation		$U_{tyyy}+\beta U_y{U}_{yt}+\beta U_{yy}{U}_t+U_{tt}=0\text{ in which }U=u(x,y,t)$
Euler equations	1+3	$\frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho 𝐮)=0,\rho (\frac{\partial 𝐮}{\partial t}+𝐯\cdot \mathrm{\nabla }𝐯)=-\mathrm{\nabla }p+\rho 𝐟,\frac{\partial s}{\partial t}+𝐯\cdot \mathrm{\nabla }s=0$	non-viscous fluids
Fisher's equation	1+1	$u_t=u(1-u)+u_{xx}$	Gene propagation
FitzHugh–Nagumo model	1+1	$u_t=u_{xx}+u(u-a)(1-u)+w,w_t=\epsilon u$	Biological neuron model
Föppl–von Kármán equations		$\frac{E{h}^3}{12(1-{\nu }^2)}{\mathrm{\nabla }}^{4}w-h\frac{\partial }{\partial x_{\beta }}\left({\sigma }_{\alpha \beta }\frac{\partial w}{\partial x_{\alpha }}\right)=P,\frac{\partial {\sigma }_{\alpha \beta }}{\partial x_{\beta }}=0$	Solid Mechanics
Fujita–Storm equation		$$u_t=a(u^{-2}{u}_x{)}_x$$

G–K

Name	Dim	Equation	Applications
G equation	1+3	$G_t+𝐯\cdot \mathrm{\nabla }G=S_L(G)|\mathrm{\nabla }G|$	turbulent combustion
Generic scalar transport	1+3	${\phi }_t+\mathrm{\nabla }\cdot f(t,x,\phi ,\mathrm{\nabla }\phi )=g(t,x,\phi )$	transport
Ginzburg–Landau	1+3	$\alpha \psi +\beta |\psi {|}^2\psi +\frac{1}{2m}{(-i\hslash \mathrm{\nabla }-2e𝐀)}^2\psi =0$	Superconductivity
Gross–Pitaevskii	Template:Math	$i{\partial }_t\psi =(-\frac{1}{2}{\mathrm{\nabla }}^2+V(x)+g|\psi {|}^2)\psi$	Bose–Einstein condensate
Gyrokinetics equation	Template:Math	$\frac{\partial h_s}{\partial t}+(v_{||}\hat{b}+{\overrightarrow{V}}_{ds}+{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi })\cdot {\overrightarrow{\mathrm{\nabla }}}_{\overrightarrow{R}}{h}_s-\sum _{s^{\prime }}{⟨C[h_s,h_{s^{\prime }}]⟩}_{\phi }=\frac{Z_{s}e{f}_{s0}}{T_s}\frac{\partial {⟨\varphi ⟩}_{\phi }}{\partial t}-\frac{\partial f_{s0}}{\partial \psi }{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi }\cdot \overrightarrow{\mathrm{\nabla }}\psi$	Microturbulence in plasma
Guzmán	Template:Math	$J_t+g{J}_x+1/2{\sigma }^2{J}_{xx}-\lambda {\sigma }^2(J_x{)}^2+f=0$	Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation	Any	$i{\partial }_{t}u+\mathrm{\Delta }u=(\pm |x{|}^{-n}|u{|}^2)u$
Hasegawa–Mima	1+3	$0=\frac{\partial }{\partial t}({\mathrm{\nabla }}^2\phi -\phi )-[(\mathrm{\nabla }\phi \times \widehat{𝐳})\cdot \mathrm{\nabla }][{\mathrm{\nabla }}^2\phi -\ln \left(\frac{n_0}{{\omega }_{ci}}\right)]$	Turbulence in plasma
Heisenberg ferromagnet	1+1	${𝐒}_t=𝐒\wedge {𝐒}_{xx}.$	Magnetism
Hicks	1+1	${\psi }_{rr}-{\psi }_r/r+{\psi }_{zz}=r^2\mathrm{d}H/\mathrm{d}\psi -\mathrm{\Gamma }\mathrm{d}\mathrm{\Gamma }/\mathrm{d}\psi$	Fluid dynamics
Hunter–Saxton	1+1	${(u_t+u{u}_x)}_x=\frac{1}{2}{u}_x^2$	Liquid crystals
Ishimori equation	1+2	${𝐒}_t=𝐒\wedge ({𝐒}_{xx}+{𝐒}_{yy})+u_x{𝐒}_y+u_y{𝐒}_x,u_{xx}-{\alpha }^2{u}_{yy}=-2{\alpha }^2𝐒\cdot ({𝐒}_x\wedge {𝐒}_y)$	Integrable systems
Kadomtsev –Petviashvili	1+2	${\partial }_x({\partial }_{t}u+u{\partial }_{x}u+{\epsilon }^2{\partial }_{xxx}u)+\lambda {\partial }_{yy}u=0$	Shallow water waves
Kardar–Parisi–Zhang equation	1+3	$h_t=\nu {\mathrm{\nabla }}^{2}h+\lambda (\mathrm{\nabla }h{)}^2/2+\eta$	Stochastics
von Karman	2	${\mathrm{\nabla }}^{4}u=E(w_{xy}^2-w_{xx}{w}_{yy}),{\mathrm{\nabla }}^{4}w=a+b(u_{yy}{w}_{xx}+u_{xx}{w}_{yy}-2u_{xy}{w}_{xy})$
Kaup	1+1	$f_x=2fgc(x-t)=g_t$
Kaup–Kupershmidt	1+1	$u_t=u_{xxxxx}+10u_{xxx}u+25u_{xx}{u}_x+20u^2{u}_x$	Integrable systems
Klein–Gordon–Maxwell	any	${\mathrm{\nabla }}^{2}s=(|𝐚{|}^2+1)s,{\mathrm{\nabla }}^2𝐚=\mathrm{\nabla }(\mathrm{\nabla }\cdot 𝐚)+s^2𝐚$
Klein–Gordon (nonlinear)	any	${\mathrm{\nabla }}^{2}u+\lambda u^p=0$	Relativistic quantum mechanics
Khokhlov–Zabolotskaya	1+2	$u_{xt}-(u{u}_x{)}_x=u_{yy}$
Kompaneyets	1+1	$n_t=x^{-2}[x^4(n_x+n^2+n){]}_x$	Physical kinetics
Korteweg–de Vries (KdV)	1+1	$u_t+u_{xxx}-6u{u}_x=0$	Shallow waves, Integrable systems
KdV (super)	1+1	$u_t=6u{u}_x-u_{xxx}+3w{w}_{xx},w_t=3u_{x}w+6u{w}_x-4w_{xxx}$
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation	Template:Math	$u_t+{\mathrm{\nabla }}^{4}u+{\mathrm{\nabla }}^{2}u+\frac{1}{2}|\mathrm{\nabla }u{|}^2=0$	Combustion

L–Q

Name	Dim	Equation	Applications
Landau–Lifshitz model	1+n	$\frac{\partial 𝐒}{\partial t}=𝐒\wedge \sum _i\frac{{\partial }^2𝐒}{\partial x_i^2}+𝐒\wedge J𝐒$	Magnetic field in solids
Lin–Tsien equation	1+2	$2u_{tx}+u_x{u}_{xx}-u_{yy}=0$
Liouville equation	any	${\mathrm{\nabla }}^{2}u+e^{\lambda u}=0$
Liouville–Bratu–Gelfand equation	any	${\mathrm{\nabla }}^2\psi +\lambda e^{\psi }=0$	combustion, astrophysics
Logarithmic Schrödinger equation	any	$i\frac{\partial \psi }{\partial t}+\mathrm{\Delta }\psi +\psi \ln |\psi {|}^2=0.$	Superfluids, Quantum gravity
Minimal surface	3	$\mathrm{div}(Du/\sqrt{1+|Du{|}^2})=0$	minimal surfaces
Monge–Ampère	any	$\det ({\partial }_{ij}\phi )=$ lower order terms
Navier–Stokes (and its derivation)	1+3	$\rho (\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j})=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}[\mu (\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})+\lambda \frac{\partial v_k}{\partial x_k}]+\rho f_i$ + mass conservation: $\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho v_i\right)}{\partial x_i}=0$ + an equation of state to relate p and ρ, e.g. for an incompressible flow: $\frac{\partial v_i}{\partial x_i}=0$	Fluid flow, gas flow
Nonlinear Schrödinger (cubic)	1+1	$i{\partial }_t\psi =-\frac{1}{2}\underset{x}{\overset{2}{\partial }}\psi +\kappa |\psi {|}^2\psi$	optics, water waves
Nonlinear Schrödinger (derivative)	1+1	$i{\partial }_t\psi =-\frac{1}{2}\underset{x}{\overset{2}{\partial }}\psi +{\partial }_x(i\kappa |\psi {|}^2\psi )$	optics, water waves
Omega equation	1+3	${\mathrm{\nabla }}^2\omega +\frac{f^2}{\sigma }\frac{{\partial }^2\omega }{\partial p^2}$ $=\frac{f}{\sigma }\frac{\partial }{\partial p}{𝐕}_g\cdot {\mathrm{\nabla }}_p({\zeta }_g+f)+\frac{R}{\sigma p}\underset{p}{\overset{2}{\mathrm{\nabla }}}({𝐕}_g\cdot {\mathrm{\nabla }}_{p}T)$	atmospheric physics
Plateau	2	$(1+u_y^2)u_{xx}-2u_x{u}_y{u}_{xy}+(1+u_x^2)u_{yy}=0$	minimal surfaces
Pohlmeyer–Lund–Regge	2	$u_{xx}-u_{yy}\pm \sin u\cos u+\frac{\cos u}{{\sin }^{3}u}(v_x^2-v_y^2)=0,(v_x{\cot }^{2}u{)}_x=(v_y{\cot }^{2}u{)}_y$
Porous medium	1+n	$u_t=\mathrm{\Delta }(u^{\gamma })$	diffusion
Prandtl	1+2	$u_t+u{u}_x+v{u}_y=U_t+U{U}_x+\frac{\mu }{\rho }{u}_{yy}$, $u_x+v_y=0$	boundary layer

R–Z, α–ω

Name	Dim	Equation	Applications
Rayleigh	1+1	$u_{tt}-u_{xx}=\epsilon (u_t-u_t^3)$
Ricci flow	Any	${\partial }_t{g}_{ij}=-2R_{ij}$	Poincaré conjecture
Richards equation	1+3	${\theta }_t={[K(\theta )({\psi }_z+1)]}_z$	Variably saturated flow in porous media
Rosenau–Hyman	1+1	$u_t+a{\left(u^n\right)}_x+{\left(u^n\right)}_{xxx}=0$	compacton solutions
Sawada–Kotera	1+1	$u_t+45u^2{u}_x+15u_x{u}_{xx}+15u{u}_{xxx}+u_{xxxxx}=0$
Sack–Schamel equation	1+1	$\ddot{V}+{\partial }_{\eta }\left[\frac{1}{1-\ddot{V}}{\partial }_{\eta }\left(\frac{1-\ddot{V}}{V}\right)\right]=0$	plasmas
Schamel equation	1+1	${\varphi }_t+(1+b\sqrt{\varphi }){\varphi }_x+{\varphi }_{xxx}=0$	plasmas, solitons, optics
Schlesinger	Any	$\frac{\partial A_i}{\partial t_j}\frac{[A_i,A_j]}{t_i-t_j},i\ne j,\frac{\partial A_i}{\partial t_i}=-\sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^n\frac{[A_i,A_j]}{t_i-t_j},1\le i,j\le n$	isomonodromic deformations
Seiberg–Witten	1+3	$D^A\phi =0,F_A^{+}=\sigma (\phi )$	Seiberg–Witten invariants, QFT
Shallow water	1+2	${\eta }_t+(\eta u{)}_x+(\eta v{)}_y=0,(\eta u{)}_t+{(\eta u^2+\frac{1}{2}g{\eta }^2)}_x+(\eta uv{)}_y=0,(\eta v{)}_t+(\eta uv{)}_x+{(\eta v^2+\frac{1}{2}g{\eta }^2)}_y=0$	shallow water waves
Sine–Gordon	1+1	${\phi }_{tt}-{\phi }_{xx}+\sin \phi =0$	Solitons, QFT
Sinh–Gordon	1+1	$u_{xt}=\sinh u$	Solitons, QFT
Sinh–Poisson	1+n	${\mathrm{\nabla }}^{2}u+\sinh u=0$	Fluid Mechanics
Swift–Hohenberg	any	$u_t=ru-(1+{\mathrm{\nabla }}^2{)}^{2}u+N(u)$	pattern forming
Thomas	2	$u_{xy}+\alpha u_x+\beta u_y+\gamma u_x{u}_y=0$
Thirring	1+1	$i{u}_x+v+u|v{|}^2=0$, $i{v}_t+u+v|u{|}^2=0$	Dirac field, QFT
Toda lattice	any	${\mathrm{\nabla }}^2\log u_n=u_{n+1}-2u_n+u_{n-1}$
Veselov–Novikov	1+2	$({\partial }_t+\underset{z}{\overset{3}{\partial }}+\underset{\overline{z}}{\overset{3}{\partial }})v+{\partial }_z(uv)+{\partial }_{\overline{z}}(uw)=0$, ${\partial }_{\overline{z}}u=3{\partial }_{z}v$, ${\partial }_{z}w=3{\partial }_{\overline{z}}v$	shallow water waves
Vorticity equation		$\frac{\partial \mathit{\omega }}{\partial t}+(𝐮\cdot \mathrm{\nabla })\mathit{\omega }=(\mathit{\omega }\cdot \mathrm{\nabla })𝐮-\mathit{\omega }(\mathrm{\nabla }\cdot 𝐮)+\frac{1}{{\rho }^2}\mathrm{\nabla }\rho \times \mathrm{\nabla }p+\mathrm{\nabla }\times \left(\frac{\mathrm{\nabla }\cdot \tau }{\rho }\right)+\mathrm{\nabla }\times \left(\frac{𝐟}{\rho }\right),\mathit{\omega }=\mathrm{\nabla }\times 𝐮$	Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu	1+1	$i{u}_t+((1+|u{|}^2{)}^{-1/2}u{)}_{xx}=0$
WDVV equations	Any	$\sum _{\sigma ,\tau =1}^n\left(\frac{{\partial }^{3}F}{\partial t^{\alpha }{t}^{\beta }{t}^{\sigma }}{\eta }^{\sigma \tau }\frac{{\partial }^{3}F}{\partial t^{\mu }{t}^{\nu }{t}^{\tau }}\right)$ $=\sum _{\sigma ,\tau =1}^n\left(\frac{{\partial }^{3}F}{\partial t^{\alpha }{t}^{\nu }{t}^{\sigma }}{\eta }^{\sigma \tau }\frac{{\partial }^{3}F}{\partial t^{\mu }{t}^{\beta }{t}^{\tau }}\right)$	Topological field theory, QFT
WZW model	1+1	$S_k(\gamma )=-\frac{k}{8\pi }\underset{S^2}{\int }{d}^{2}x𝒦({\gamma }^{-1}{\partial }^{\mu }\gamma ,{\gamma }^{-1}{\partial }_{\mu }\gamma )+2\pi k{S}^{\mathrm{W}Z}(\gamma )$ $S^{\mathrm{W}Z}(\gamma )=-\frac{1}{48{\pi }^2}\underset{B^3}{\int }{d}^{3}y{\epsilon }^{ijk}𝒦({\gamma }^{-1}\frac{\partial \gamma }{\partial y^i},[{\gamma }^{-1}\frac{\partial \gamma }{\partial y^j},{\gamma }^{-1}\frac{\partial \gamma }{\partial y^k}])$	QFT
Whitham equation	1+1	${\eta }_t+\alpha \eta {\eta }_x+\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}K(x-\xi ){\eta }_{\xi }(\xi ,t)\text{d}\xi =0$	water waves
Williams spray equation		$\frac{\partial f_j}{\partial t}+{\mathrm{\nabla }}_x\cdot (𝐯f_j)+{\mathrm{\nabla }}_v\cdot (F_j{f}_j)=-\frac{\partial }{\partial r}(R_j{f}_j)-\frac{\partial }{\partial T}(E_j{f}_j)+Q_j+{\mathrm{\Gamma }}_j,F_j=\dot{𝐯},R_j=\dot{r},E_j=\dot{T},j=1,2,...,M$	Combustion
Yamabe	n	$\mathrm{\Delta }\phi +h(x)\phi =\lambda f(x){\phi }^{(n+2)/(n-2)}$	Differential geometry
Yang–Mills (source-free)	Any	$D_{\mu }{F}^{\mu \nu }=0,F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },A_{\nu }]$	Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual)	4	$F_{\alpha \beta }=\pm {\epsilon }_{\alpha \beta \mu \nu }{F}^{\mu \nu },F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },A_{\nu }]$	Instantons, Donaldson theory, QFT
Yukawa	1+n	$i\underset{t}{\overset{}{\partial }}u+\mathrm{\Delta }u=-Au,◻A=m_{}^{2}A+|u{|}^2$	Meson-nucleon interactions, QFT
Zakharov system	1+3	$i\underset{t}{\overset{}{\partial }}u+\mathrm{\Delta }u=un,◻n=-\mathrm{\Delta }(|u{|}_{}^2)$	Langmuir waves
Zakharov–Schulman	1+3	$i{u}_t+L_{1}u=\phi u,L_2\phi =L_3(|u{|}^2)$	Acoustic waves
Zeldovich–Frank-Kamenetskii equation	1+3	$u_t=D{\mathrm{\nabla }}^{2}u+\frac{{\beta }^2}{2}u(1-u)e^{-\beta (1-u)}$	Combustion
Zoomeron	1+1	$(u_{xt}/u{)}_{tt}-(u_{xt}/u{)}_{xx}+2(u^2{)}_{xt}=0$	Solitons
φ4 equation	1+1	${\phi }_{tt}-{\phi }_{xx}-\phi +{\phi }^3=0$	QFT
σ-model	1+1	${𝐯}_{xt}+({𝐯}_x{𝐯}_t)𝐯=0$	Harmonic maps, integrable systems, QFT