List of nonlinear partial differential equations

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See also Nonlinear partial differential equation, List of partial differential equation topics and List of nonlinear ordinary differential equations.

## Contents

* 1 A–F
* 2 G–K
* 3 L–Q
* 4 R–Z, α–ω
* 5 References

## A–F

Name  | Dim  | Equation  | Applications

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|
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Bateman-Burgers equation | 1+1  | ut+uux=νuxx | Fluid mechanics
Benjamin–Bona–Mahony | 1+1  | ut+ux+uux−uxxt=0 | Fluid mechanics
Benjamin–Ono | 1+1  | ut+Huxx+uux=0 | internal waves in deep water
Boomeron  | 1+1  | ut=𝐛⋅𝐯x,𝐯xt=uxx𝐛+𝐚×𝐯x−2𝐯×(𝐯×𝐛) | Solitons
Boltzmann equation | 1+6  | ∂fi∂t+𝐩imi⋅∇fi+𝐅⋅∂fi∂𝐩i=(∂fi∂t)coll,

(∂fi∂t)coll=∑j=1n∬gijIij(gij,Ω)[f'if'j−fifj]dΩd3𝐩′

| Statistical mechanics
Born–Infeld  | 1+1  | (1−ut2)uxx+2uxutuxt−(1+ux2)utt=0 | Electrodynamics
Boussinesq  | 1+1  | utt−uxx−uxxxx−3(u2)xx=0 | Fluid mechanics
Boussinesq type equation  | 1+1  | utt−uxx−2α(uux)x−βuxxtt=0 | Fluid mechanics
Buckmaster | 1+1  | ut=(u4)xx+(u3)x | Thin viscous fluid sheet flow
Cahn–Hilliard equation | Any  | ct=D∇2(c3−c−γ∇2c) | Phase separation
Calabi flow | Any  | ∂gij∂t=(ΔR)gij | Calabi–Yau manifolds
Camassa–Holm | 1+1  | ut+2κux−uxxt+3uux=2uxuxx+uuxxx | Peakons
Carleman  | 1+1  | ut+ux=v2−u2=vx−vt |
Cauchy momentum | any  | ρ(∂𝐯∂t+𝐯⋅∇𝐯)=∇⋅σ+ρ𝐟 | Momentum transport
Chafee–Infante equation  |  | ut−uxx+λ(u3−u)=0 |
Clairaut equation | any  | x⋅Du+f(Du)=u | Differential geometry
Clarke's equation | 1+1  | (θt−γδeθ)tt=∇2(θt−δeθ) | Combustion
Complex Monge–Ampère  | Any  | det⁡(∂ij¯φ)= lower order terms  | Calabi conjecture
Constant astigmatism  | 1+1  | zyy+(1z)xx+2=0 | Differential geometry
Davey–Stewartson | 1+2  | iut+c0uxx+uyy=c1|u|2u+c2uφx,φxx+c3φyy=(|u|2)x | Finite depth waves
Degasperis–Procesi | 1+1  | ut−uxxt+4uux=3uxuxx+uuxxx | Peakons
Dispersive long wave  | 1+1  | ut=(u2−ux+2w)x, wt=(2uw+wx)x |
Drinfeld–Sokolov–Wilson | 1+1  | ut=3wwx,wt=2wxxx+2uwx+uxw |
Dym equation | 1+1  | ut=u3uxxx. | Solitons
Eckhaus equation | 1+1  | iut+uxx+2|u|x2u+|u|4u=0 | Integrable systems
Eikonal equation | any  | |∇u(x)|=F(x), x∈Ω | optics
Einstein field equations | Any  | Rμν−12Rgμν+Λgμν=8πGc4Tμν | General relativity
Erdogan–Chatwin equation | 1+1  | φt=(φx+aφx3)x | Fluid dynamics
Ernst equation | 2  | ℜ(u)(urr+ur/r+uzz)=(ur)2+(uz)2 |
Estevez–Mansfield–Clarkson equation  |  | Utyyy+βUyUyt+βUyyUt+Utt=0 in which U=u(x,y,t) |
Euler equations  | 1+3  | ∂ρ∂t+∇⋅(ρ𝐮)=0,ρ(∂𝐮∂t+𝐯⋅∇𝐯)=−∇p+ρ𝐟,∂s∂t+𝐯⋅∇s=0 | non-viscous fluids
Fisher's equation | 1+1  | ut=u(1−u)+uxx | Gene propagation
FitzHugh–Nagumo model | 1+1  | ut=uxx+u(u−a)(1−u)+w,wt=εu | Biological neuron model
Föppl–von Kármán equations |  | Eh312(1−ν2)∇4w−h∂∂xβ(σαβ∂w∂xα)=P,∂σαβ∂xβ=0 | Solid Mechanics
Fujita–Storm equation  |  | ut=a(u−2ux)x |

## G–K

Name  | Dim  | Equation  | Applications

|
|
|

G equation | 1+3  | Gt+𝐯⋅∇G=SL(G)|∇G| | turbulent combustion
Generic scalar transport  | 1+3  | φt+∇⋅f(t,x,φ,∇φ)=g(t,x,φ) | transport
Ginzburg–Landau | 1+3  | αψ+β|ψ|2ψ+12m(−iℏ∇−2e𝐀)2ψ=0 | Superconductivity
Gross–Pitaevskii | 1 + n | i∂tψ=(−12∇2+V(x)+g|ψ|2)ψ | Bose–Einstein condensate
Gyrokinetics equation | 1 + 5 | ∂hs∂t+(v||b^+V→ds+⟨V→ϕ⟩φ)⋅∇→R→hs−∑s′⟨C[hs,hs′]⟩φ=Zsefs0Ts∂⟨ϕ⟩φ∂t−∂fs0∂ψ⟨V→ϕ⟩φ⋅∇→ψ | Microturbulence in plasma
Guzmán  | 1 + n | Jt+gJx+1/2σ2Jxx−λσ2(Jx)2+f=0 | Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation | Any  | i∂tu+Δu=(±|x|−n|u|2)u |
Hasegawa–Mima | 1+3  | 0=∂∂t(∇2φ−φ)−[(∇φ×𝐳^)⋅∇][∇2φ−ln⁡(n0ωci)] | Turbulence in plasma
Heisenberg ferromagnet  | 1+1  | 𝐒t=𝐒∧𝐒xx. | Magnetism
Hicks | 1+1  | ψrr−ψr/r+ψzz=r2dH/dψ−ΓdΓ/dψ | Fluid dynamics
Hunter–Saxton | 1+1  | (ut+uux)x=12ux2 | Liquid crystals
Ishimori equation | 1+2  | 𝐒t=𝐒∧(𝐒xx+𝐒yy)+ux𝐒y+uy𝐒x,uxx−α2uyy=−2α2𝐒⋅(𝐒x∧𝐒y) | Integrable systems
Kadomtsev –Petviashvili | 1+2  | ∂x(∂tu+u∂xu+ε2∂xxxu)+λ∂yyu=0 | Shallow water waves
Kardar–Parisi–Zhang equation | 1+3  | ht=ν∇2h+λ(∇h)2/2+η | Stochastics
von Karman  | 2  | ∇4u=E(wxy2−wxxwyy),∇4w=a+b(uyywxx+uxxwyy−2uxywxy) |
Kaup  | 1+1  | fx=2fgc(x−t)=gt |
Kaup–Kupershmidt | 1+1  | ut=uxxxxx+10uxxxu+25uxxux+20u2ux | Integrable systems
Klein–Gordon–Maxwell  | any  | ∇2s=(|𝐚|2+1)s,∇2𝐚=∇(∇⋅𝐚)+s2𝐚 |
Klein–Gordon (nonlinear)  | any  | ∇2u+λup=0 | Relativistic quantum mechanics
Khokhlov–Zabolotskaya  | 1+2  | uxt−(uux)x=uyy |
Kompaneyets | 1+1  | nt=x−2[x4(nx+n2+n)]x | Physical kinetics
Korteweg–de Vries (KdV)  | 1+1  | ut+uxxx−6uux=0 | Shallow waves, Integrable systems
KdV (super)  | 1+1  | ut=6uux−uxxx+3wwxx,wt=3uxw+6uwx−4wxxx |
There are many different variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation | 1 + n | ut+∇4u+∇2u+12|∇u|2=0 | Combustion

## L–Q

Name  | Dim  | Equation  | Applications

|
|
|

Landau–Lifshitz model | 1+n | ∂𝐒∂t=𝐒∧∑i∂2𝐒∂xi2+𝐒∧J𝐒 | Magnetic field in solids
Lin–Tsien equation | 1+2  | 2utx+uxuxx−uyy=0 |
Liouville equation | any  | ∇2u+eλu=0 |
Liouville–Bratu–Gelfand equation | any  | ∇2ψ+λeψ=0 | combustion, astrophysics
Logarithmic Schrödinger equation | any  | i∂ψ∂t+Δψ+ψln⁡|ψ|2=0. | Superfluids, Quantum gravity
Minimal surface  | 3  | div⁡(Du/1+|Du|2)=0 | minimal surfaces
Monge–Ampère | any  | det⁡(∂ijφ)= lower order terms  |
Navier–Stokes
(and its derivation) | 1+3  | ρ(∂vi∂t+vj∂vi∂xj)=−∂p∂xi+∂∂xj[μ(∂vi∂xj+∂vj∂xi)+λ∂vk∂xk]+ρfi

\+ mass conservation: ∂ρ∂t+∂(ρvi)∂xi=0
\+ an equation of state to relate p and ρ, e.g. for an incompressible flow: ∂vi∂xi=0

| Fluid flow, gas flow
Nonlinear Schrödinger (cubic) | 1+1  | i∂tψ=−12∂x2ψ+κ|ψ|2ψ | optics, water waves
Nonlinear Schrödinger (derivative) | 1+1  | i∂tψ=−12∂x2ψ+∂x(iκ|ψ|2ψ) | optics, water waves
Omega equation | 1+3  | ∇2ω+f2σ∂2ω∂p2 =fσ∂∂p𝐕g⋅∇p(ζg+f)+Rσp∇p2(𝐕g⋅∇pT) | atmospheric physics
Plateau  | 2  | (1+uy2)uxx−2uxuyuxy+(1+ux2)uyy=0 | minimal surfaces
Pohlmeyer–Lund–Regge  | 2  | uxx−uyy±sin⁡ucos⁡u+cos⁡usin3u(vx2−vy2)=0,(vxcot2u)x=(vycot2u)y |
Porous medium | 1+n | ut=Δ(uγ) | diffusion
Prandtl | 1+2  | ut+uux+vuy=Ut+UUx+μρuyy, ux+vy=0 | boundary layer

## R–Z, α–ω

Name  | Dim  | Equation  | Applications

|
|
|

Rayleigh | 1+1  | utt−uxx=ε(ut−ut3) |
Ricci flow | Any  | ∂tgij=−2Rij | Poincaré conjecture
Richards equation | 1+3  | θt=[K(θ)(ψz+1)]z | Variably saturated flow in porous media
Rosenau–Hyman | 1+1  | ut+a(un)x+(un)xxx=0 | compacton solutions
Sawada–Kotera  | 1+1  | ut+45u2ux+15uxuxx+15uuxxx+uxxxxx=0 |
Sack–Schamel equation  | 1+1  | V¨+∂η[11−V¨∂η(1−V¨V)]=0 | plasmas
Schamel equation | 1+1  | ϕt+(1+bϕ)ϕx+ϕxxx=0 | plasmas, solitons, optics
Schlesinger  | Any  | ∂Ai∂tj[Ai, Aj]ti−tj,i≠j,∂Ai∂ti=−∑j=1j≠in[Ai, Aj]ti−tj,1≤i,j≤n | isomonodromic deformations
Seiberg–Witten  | 1+3  | DAφ=0,FA+=σ(φ) | Seiberg–Witten invariants, QFT
Shallow water | 1+2  | ηt+(ηu)x+(ηv)y=0, (ηu)t+(ηu2+12gη2)x+(ηuv)y=0, (ηv)t+(ηuv)x+(ηv2+12gη2)y=0 | shallow water waves
Sine–Gordon  | 1+1  | φtt−φxx+sin⁡φ=0 | Solitons, QFT
Sinh–Gordon  | 1+1  | uxt=sinh⁡u | Solitons, QFT
Sinh–Poisson  | 1+n | ∇2u+sinh⁡u=0 | Fluid Mechanics
Swift–Hohenberg | any  | ut=ru−(1+∇2)2u+N(u) | pattern forming
Thomas  | 2  | uxy+αux+βuy+γuxuy=0 |
Thirring | 1+1  | iux+v+u|v|2=0, ivt+u+v|u|2=0 | Dirac field, QFT
Toda lattice | any  | ∇2log⁡un=un+1−2un+un−1 |
Veselov–Novikov | 1+2  | (∂t+∂z3+∂z¯3)v+∂z(uv)+∂z¯(uw)=0, ∂z¯u=3∂zv, ∂zw=3∂z¯v | shallow water waves
Vorticity equation |  | ∂ω∂t+(𝐮⋅∇)ω=(ω⋅∇)𝐮−ω(∇⋅𝐮)+1ρ2∇ρ×∇p+∇×(∇⋅τρ)+∇×(𝐟ρ), ω=∇×𝐮 | Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu  | 1+1  | iut+((1+|u|2)−1/2u)xx=0 |
WDVV equations  | Any  | ∑σ,τ=1n(∂3F∂tαtβtσηστ∂3F∂tμtνtτ) =∑σ,τ=1n(∂3F∂tαtνtσηστ∂3F∂tμtβtτ) | Topological field theory, QFT
WZW model  | 1+1  | Sk(γ)=−k8π∫S2d2x𝒦(γ−1∂μγ,γ−1∂μγ)+2πkSWZ(γ)

SWZ(γ)=−148π2∫B3d3yεijk𝒦(γ−1∂γ∂yi,[γ−1∂γ∂yj,γ−1∂γ∂yk])

| QFT
Whitham equation | 1+1  | ηt+αηηx+∫−∞+∞K(x−ξ)ηξ(ξ,t)dξ=0 | water waves
Williams spray equation |  | ∂fj∂t+∇x⋅(𝐯fj)+∇v⋅(Fjfj)=−∂∂r(Rjfj)−∂∂T(Ejfj)+Qj+Γj, Fj=𝐯˙, Rj=r˙, Ej=T˙, j=1,2,...,M | Combustion
Yamabe | n | Δφ+h(x)φ=λf(x)φ(n+2)/(n−2) | Differential geometry
Yang–Mills (source-free) | Any  | DμFμν=0,Fμν=Aμ,ν−Aν,μ+[Aμ,Aν] | Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual) | 4  | Fαβ=±εαβμνFμν,Fμν=Aμ,ν−Aν,μ+[Aμ,Aν] | Instantons, Donaldson theory, QFT
Yukawa  | 1+n | i∂tu+Δu=−Au,◻A=m2A+|u|2 | Meson-nucleon interactions, QFT
Zakharov system | 1+3  | i∂tu+Δu=un,◻n=−Δ(|u|2) | Langmuir waves
Zakharov–Schulman | 1+3  | iut+L1u=φu,L2φ=L3(|u|2) | Acoustic waves
Zeldovich–Frank-Kamenetskii equation  | 1+3  | ut=D∇2u+β22u(1−u)e−β(1−u) | Combustion
Zoomeron  | 1+1  | (uxt/u)tt−(uxt/u)xx+2(u2)xt=0 | Solitons
φ4 equation | 1+1  | φtt−φxx−φ+φ3=0 | QFT
σ-model | 1+1  | 𝐯xt+(𝐯x𝐯t)𝐯=0 | Harmonic maps, integrable systems, QFT

## References

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