List of nonlinear partial differential equations

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

## A–F

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Name  | Dim  | Equation  | Applications

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Bateman-Burgers equation | 1+1  |  u t + u u x = ν u x x {\displaystyle \displaystyle u_{t}+uu_{x}=\nu u_{xx}} | Fluid mechanics
Benjamin–Bona–Mahony | 1+1  |  u t + u x + u u x − u x x t = 0 {\displaystyle \displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0} | Fluid mechanics
Benjamin–Ono | 1+1  |  u t + H u x x + u u x = 0 {\displaystyle \displaystyle u_{t}+Hu_{xx}+uu_{x}=0} | internal waves in deep water
Boomeron | 1+1  |  u t = b ⋅ v x , v x t = u x x b + a × v x − 2 v × ( v × b ) {\displaystyle \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  |  ∂ f i ∂ t + p i m i ⋅ ∇ f i + F ⋅ ∂ f i ∂ p i = ( ∂ f i ∂ t ) c o l l , {\displaystyle {\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} },}

( ∂ f i ∂ t ) c o l l = ∑ j = 1 n ∬ g i j I i j ( g i j , Ω ) [ f i ′ f j ′ − f i f j ] d Ω d 3 p ′ {\displaystyle \left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )[f'_{i}f'_{j}-f_{i}f_{j}]\,d\Omega \,d^{3}\mathbf {p'} }

| Statistical mechanics
Born–Infeld | 1+1  |  ( 1 − u t 2 ) u x x + 2 u x u t u x t − ( 1 + u x 2 ) u t t = 0 {\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0} | Electrodynamics
Boussinesq | 1+1  |  u t t − u x x − u x x x x − 3 ( u 2 ) x x = 0 {\displaystyle \displaystyle u_{tt}-u_{xx}-u_{xxxx}-3(u^{2})_{xx}=0} | Fluid mechanics
Boussinesq type equation | 1+1  |  u t t − u x x − 2 α ( u u x ) x − β u x x t t = 0 {\displaystyle \displaystyle u_{tt}-u_{xx}-2\alpha (uu_{x})_{x}-\beta u_{xxtt}=0} | Fluid mechanics
Buckmaster | 1+1  |  u t = ( u 4 ) x x + ( u 3 ) x {\displaystyle \displaystyle u_{t}=(u^{4})_{xx}+(u^{3})_{x}} | Thin viscous fluid sheet flow
Cahn–Hilliard equation | Any  |  c t = D ∇ 2 ( c 3 − c − γ ∇ 2 c ) {\displaystyle \displaystyle c_{t}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right)} | Phase separation
Calabi flow | Any  |  ∂ g i j ∂ t = ( Δ R ) g i j {\displaystyle {\frac {\partial g_{ij}}{\partial t}}=(\Delta R)g_{ij}} | Calabi–Yau manifolds
Camassa–Holm | 1+1  |  u t + 2 κ u x − u x x t + 3 u u x = 2 u x u x x + u u x x x {\displaystyle u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}\,} | Peakons
Carleman | 1+1  |  u t + u x = v 2 − u 2 = v x − v t {\displaystyle \displaystyle u_{t}+u_{x}=v^{2}-u^{2}=v_{x}-v_{t}} |
Cauchy momentum | any  |  ρ ( ∂ v ∂ t + v ⋅ ∇ v ) = ∇ ⋅ σ + ρ f {\displaystyle \displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=\nabla \cdot \sigma +\rho \mathbf {f} } | Momentum transport
Chafee–Infante equation |  |  u t − u x x + λ ( u 3 − u ) = 0 {\displaystyle u_{t}-u_{xx}+\lambda (u^{3}-u)=0} |
Clairaut equation | any  |  x ⋅ D u + f ( D u ) = u {\displaystyle x\cdot Du+f(Du)=u} | Differential geometry
Clarke's equation | 1+1  |  ( θ t − γ δ e θ ) t t = ∇ 2 ( θ t − δ e θ ) {\displaystyle (\theta _{t}-\gamma \delta e^{\theta })_{tt}=\nabla ^{2}(\theta _{t}-\delta e^{\theta })} | Combustion
Complex Monge–Ampère | Any  |  det ( ∂ i j ¯ φ ) = {\displaystyle \displaystyle \det(\partial _{i{\bar {j}}}\varphi )=} lower order terms  | Calabi conjecture
Constant astigmatism  | 1+1  |  z y y + ( 1 z ) x x + 2 = 0 {\displaystyle z_{yy}+\left({\frac {1}{z}}\right)_{xx}+2=0} | Differential geometry
Davey–Stewartson | 1+2  |  i u t + c 0 u x x + u y y = c 1 | u | 2 u + c 2 u φ x , φ x x + c 3 φ y y = ( | u | 2 ) x {\displaystyle \displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\varphi _{x},\quad \displaystyle \varphi _{xx}+c_{3}\varphi _{yy}=(|u|^{2})_{x}} | Finite depth waves
Degasperis–Procesi | 1+1  |  u t − u x x t + 4 u u x = 3 u x u x x + u u x x x {\displaystyle \displaystyle u_{t}-u_{xxt}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}} | Peakons
Dispersive long wave | 1+1  |  u t = ( u 2 − u x + 2 w ) x {\displaystyle \displaystyle u_{t}=(u^{2}-u_{x}+2w)_{x}} ,  w t = ( 2 u w + w x ) x {\displaystyle w_{t}=(2uw+w_{x})_{x}} |
Drinfeld–Sokolov–Wilson | 1+1  |  u t = 3 w w x , w t = 2 w x x x + 2 u w x + u x w {\displaystyle \displaystyle u_{t}=3ww_{x},\quad \displaystyle w_{t}=2w_{xxx}+2uw_{x}+u_{x}w} |
Dym equation | 1+1  |  u t = u 3 u x x x . {\displaystyle \displaystyle u_{t}=u^{3}u_{xxx}.\,} | Solitons
Eckhaus equation | 1+1  |  i u t + u x x + 2 | u | x 2 u + | u | 4 u = 0 {\displaystyle iu_{t}+u_{xx}+2|u|_{x}^{2}u+|u|^{4}u=0} | Integrable systems
Eikonal equation | any  |  | ∇ u ( x ) | = F ( x ) , x ∈ Ω {\displaystyle \displaystyle |\nabla u(x)|=F(x),\ x\in \Omega } | optics
Einstein field equations | Any  |  R μ ν − 1 2 R g μ ν + Λ g μ ν = 8 π G c 4 T μ ν {\displaystyle \displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }} | General relativity
Erdogan–Chatwin equation | 1+1  |  φ t = ( φ x + a φ x 3 ) x {\displaystyle \varphi _{t}=(\varphi _{x}+a\varphi _{x}^{3})_{x}} | Fluid dynamics
Ernst equation | 2  |  ℜ ( u ) ( u r r + u r / r + u z z ) = ( u r ) 2 + ( u z ) 2 {\displaystyle \displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}} |
Estevez–Mansfield–Clarkson equation |  |  U t y y y + β U y U y t + β U y y U t + U t t = 0 in which U = u ( x , y , t ) {\displaystyle 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  |  ∂ ρ ∂ t + ∇ ⋅ ( ρ u ) = 0 , ρ ( ∂ u ∂ t + v ⋅ ∇ v ) = − ∇ p + ρ f , ∂ s ∂ t + v ⋅ ∇ s = 0 {\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0,\quad \rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\rho \mathbf {f} ,\quad {\frac {\partial s}{\partial t}}+\mathbf {v} \cdot \nabla s=0} | non-viscous fluids
Fisher's equation | 1+1  |  u t = u ( 1 − u ) + u x x {\displaystyle \displaystyle u_{t}=u(1-u)+u_{xx}} | Gene propagation
FitzHugh–Nagumo model | 1+1  |  u t = u x x + u ( u − a ) ( 1 − u ) + w , w t = ε u {\displaystyle \displaystyle u_{t}=u_{xx}+u(u-a)(1-u)+w,\quad \displaystyle w_{t}=\varepsilon u} | Biological neuron model
Föppl–von Kármán equations |  |  E h 3 12 ( 1 − ν 2 ) ∇ 4 w − h ∂ ∂ x β ( σ α β ∂ w ∂ x α ) = P , ∂ σ α β ∂ x β = 0 {\displaystyle {\frac {Eh^{3}}{12(1-\nu ^{2})}}\nabla ^{4}w-h{\frac {\partial }{\partial x_{\beta }}}\left(\sigma _{\alpha \beta }{\frac {\partial w}{\partial x_{\alpha }}}\right)=P,\quad {\frac {\partial \sigma _{\alpha \beta }}{\partial x_{\beta }}}=0} | Solid Mechanics
Fujita–Storm equation |  |  u t = a ( u − 2 u x ) x {\displaystyle u_{t}=a(u^{-2}u_{x})_{x}} |

## G–K

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Name  | Dim  | Equation  | Applications

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G equation | 1+3  |  G t + v ⋅ ∇ G = S L ( G ) | ∇ G | {\displaystyle G_{t}+\mathbf {v} \cdot \nabla G=S_{L}(G)|\nabla G|} | turbulent combustion
Generic scalar transport  | 1+3  |  φ t + ∇ ⋅ f ( t , x , φ , ∇ φ ) = g ( t , x , φ ) {\displaystyle \displaystyle \varphi _{t}+\nabla \cdot f(t,x,\varphi ,\nabla \varphi )=g(t,x,\varphi )} | transport
Ginzburg–Landau | 1+3  |  α ψ + β | ψ | 2 ψ + 1 2 m ( − i ℏ ∇ − 2 e A ) 2 ψ = 0 {\displaystyle \displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\tfrac {1}{2m}}\left(-i\hbar \nabla -2e\mathbf {A} \right)^{2}\psi =0} | Superconductivity
Gross–Pitaevskii | 1 + n |  i ∂ t ψ = ( − 1 2 ∇ 2 + V ( x ) + g | ψ | 2 ) ψ {\displaystyle \displaystyle i\partial _{t}\psi =\left(-{\tfrac {1}{2}}\nabla ^{2}+V(x)+g|\psi |^{2}\right)\psi } | Bose–Einstein condensate
Gyrokinetics equation | 1 + 5 |  ∂ h s ∂ t + ( v | | b ^ + V -> d s + ⟨ V -> ϕ ⟩ φ ) ⋅ ∇ -> R -> h s − ∑ s ′ ⟨ C [ h s , h s ′ ] ⟩ φ = Z s e f s 0 T s ∂ ⟨ ϕ ⟩ φ ∂ t − ∂ f s 0 ∂ ψ ⟨ V -> ϕ ⟩ φ ⋅ ∇ -> ψ {\displaystyle {\displaystyle {\frac {\partial h_{s}}{\partial t}}+\left(v_{||}{\hat {b}}+{\vec {V}}_{ds}+\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\right)\cdot {\vec {\nabla }}_{\vec {R}}h_{s}-\sum _{s'}\left\langle C\left[h_{s},h_{s'}\right]\right\rangle _{\varphi }={\frac {Z_{s}ef_{s0}}{T_{s}}}{\frac {\partial \left\langle \phi \right\rangle _{\varphi }}{\partial t}}-{\frac {\partial f_{s0}}{\partial \psi }}\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\cdot {\vec {\nabla }}\psi }} | Microturbulence in plasma
Guzmán | 1 + n |  J t + g J x + 1 / 2 σ 2 J x x − λ σ 2 ( J x ) 2 + f = 0 {\displaystyle \displaystyle J_{t}+gJ_{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 ∂ t u + Δ u = ( ± | x | − n | u | 2 ) u {\displaystyle \displaystyle i\partial _{t}u+\Delta u=\left(\pm |x|^{-n}|u|^{2}\right)u} |
Hasegawa–Mima | 1+3  |  0 = ∂ ∂ t ( ∇ 2 φ − φ ) − [ ( ∇ φ × z ^ ) ⋅ ∇ ] [ ∇ 2 φ − ln ⁡ ( n 0 ω c i ) ] {\displaystyle \displaystyle 0={\frac {\partial }{\partial t}}\left(\nabla ^{2}\varphi -\varphi \right)-\left[\left(\nabla \varphi \times {\hat {\mathbf {z} }}\right)\cdot \nabla \right]\left[\nabla ^{2}\varphi -\ln \left({\frac {n_{0}}{\omega _{ci}}}\right)\right]} | Turbulence in plasma
Heisenberg ferromagnet | 1+1  |  S t = S ∧ S x x . {\displaystyle \displaystyle \mathbf {S} _{t}=\mathbf {S} \wedge \mathbf {S} _{xx}.} | Magnetism
Hicks | 1+1  |  ψ r r − ψ r / r + ψ z z = r 2 d H / d ψ − Γ d Γ / d ψ {\displaystyle \psi _{rr}-\psi _{r}/r+\psi _{zz}=r^{2}\mathrm {d} H/\mathrm {d} \psi -\Gamma \mathrm {d} \Gamma /\mathrm {d} \psi } | Fluid dynamics
Hunter–Saxton | 1+1  |  ( u t + u u x ) x = 1 2 u x 2 {\displaystyle \displaystyle \left(u_{t}+uu_{x}\right)_{x}={\tfrac {1}{2}}u_{x}^{2}} | Liquid crystals
Ishimori equation | 1+2  |  S t = S ∧ ( S x x + S y y ) + u x S y + u y S x , u x x − α 2 u y y = − 2 α 2 S ⋅ ( S x ∧ S y ) {\displaystyle \displaystyle \mathbf {S} _{t}=\mathbf {S} \wedge \left(\mathbf {S} _{xx}+\mathbf {S} _{yy}\right)+u_{x}\mathbf {S} _{y}+u_{y}\mathbf {S} _{x},\quad \displaystyle u_{xx}-\alpha ^{2}u_{yy}=-2\alpha ^{2}\mathbf {S} \cdot \left(\mathbf {S} _{x}\wedge \mathbf {S} _{y}\right)} | Integrable systems
Kadomtsev –Petviashvili | 1+2  |  ∂ x ( ∂ t u + u ∂ x u + ε 2 ∂ x x x u ) + λ ∂ y y u = 0 {\displaystyle \displaystyle \partial _{x}\left(\partial _{t}u+u\partial _{x}u+\varepsilon ^{2}\partial _{xxx}u\right)+\lambda \partial _{yy}u=0} | Shallow water waves
Kardar–Parisi–Zhang equation | 1+3  |  h t = ν ∇ 2 h + λ ( ∇ h ) 2 / 2 + η {\displaystyle \displaystyle h_{t}=\nu \nabla ^{2}h+\lambda (\nabla h)^{2}/2+\eta } | Stochastics
von Karman | 2  |  ∇ 4 u = E ( w x y 2 − w x x w y y ) , ∇ 4 w = a + b ( u y y w x x + u x x w y y − 2 u x y w x y ) {\displaystyle \displaystyle \nabla ^{4}u=E\left(w_{xy}^{2}-w_{xx}w_{yy}\right),\quad \nabla ^{4}w=a+b\left(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy}\right)} |
Kaup | 1+1  |  f x = 2 f g c ( x − t ) = g t {\displaystyle \displaystyle f_{x}=2fgc(x-t)=g_{t}} |
Kaup–Kupershmidt | 1+1  |  u t = u x x x x x + 10 u x x x u + 25 u x x u x + 20 u 2 u x {\displaystyle \displaystyle u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}} | Integrable systems
Klein–Gordon–Maxwell | any  |  ∇ 2 s = ( | a | 2 + 1 ) s , ∇ 2 a = ∇ ( ∇ ⋅ a ) + s 2 a {\displaystyle \displaystyle \nabla ^{2}s=\left(|\mathbf {a} |^{2}+1\right)s,\quad \nabla ^{2}\mathbf {a} =\nabla (\nabla \cdot \mathbf {a} )+s^{2}\mathbf {a} } |
Klein–Gordon (nonlinear) | any  |  ∇ 2 u + λ u p = 0 {\displaystyle \nabla ^{2}u+\lambda u^{p}=0} | Relativistic quantum mechanics
Khokhlov–Zabolotskaya | 1+2  |  u x t − ( u u x ) x = u y y {\displaystyle \displaystyle u_{xt}-(uu_{x})_{x}=u_{yy}} |
Kompaneyets | 1+1  |  n t = x − 2 [ x 4 ( n x + n 2 + n ) ] x {\displaystyle \displaystyle n_{t}=x^{-2}[x^{4}(n_{x}+n^{2}+n)]_{x}} | Physical kinetics
Korteweg–de Vries (KdV)  | 1+1  |  u t + u x x x − 6 u u x = 0 {\displaystyle \displaystyle u_{t}+u_{xxx}-6uu_{x}=0} | Shallow waves, Integrable systems
KdV (super) | 1+1  |  u t = 6 u u x − u x x x + 3 w w x x , w t = 3 u x w + 6 u w x − 4 w x x x {\displaystyle \displaystyle u_{t}=6uu_{x}-u_{xxx}+3ww_{xx},\quad w_{t}=3u_{x}w+6uw_{x}-4w_{xxx}} |
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation | 1 + n |  u t + ∇ 4 u + ∇ 2 u + 1 2 | ∇ u | 2 = 0 {\displaystyle \displaystyle u_{t}+\nabla ^{4}u+\nabla ^{2}u+{\tfrac {1}{2}}|\nabla u|^{2}=0} | Combustion

## L–Q

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Name  | Dim  | Equation  | Applications

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Landau–Lifshitz model | 1+n |  ∂ S ∂ t = S ∧ ∑ i ∂ 2 S ∂ x i 2 + S ∧ J S {\displaystyle \displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} } | Magnetic field in solids
Lin–Tsien equation | 1+2  |  2 u t x + u x u x x − u y y = 0 {\displaystyle \displaystyle 2u_{tx}+u_{x}u_{xx}-u_{yy}=0} |
Liouville equation | any  |  ∇ 2 u + e λ u = 0 {\displaystyle \displaystyle \nabla ^{2}u+e^{\lambda u}=0} |
Liouville–Bratu–Gelfand equation | any  |  ∇ 2 ψ + λ e ψ = 0 {\displaystyle \nabla ^{2}\psi +\lambda e^{\psi }=0} | combustion, astrophysics
Logarithmic Schrödinger equation | any  |  i ∂ ψ ∂ t + Δ ψ + ψ ln ⁡ | ψ | 2 = 0. {\displaystyle i{\frac {\partial \psi }{\partial t}}+\Delta \psi +\psi \ln |\psi |^{2}=0.} | Superfluids, Quantum gravity
Minimal surface | 3  |  div ⁡ ( D u / 1 + | D u | 2 ) = 0 {\displaystyle \displaystyle \operatorname {div} (Du/{\sqrt {1+|Du|^{2}}})=0} | minimal surfaces
Monge–Ampère | any  |  det ( ∂ i j φ ) = {\displaystyle \displaystyle \det(\partial _{ij}\varphi )=} lower order terms  |
Navier–Stokes
(and its derivation) | 1+3  |  ρ ( ∂ v i ∂ t + v j ∂ v i ∂ x j ) = − ∂ p ∂ x i + ∂ ∂ x j [ μ ( ∂ v i ∂ x j + ∂ v j ∂ x i ) + λ ∂ v k ∂ x k ] + ρ f i {\displaystyle \displaystyle \rho \left({\frac {\partial v_{i}}{\partial t}}+v_{j}{\frac {\partial v_{i}}{\partial x_{j}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}\left[\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)+\lambda {\frac {\partial v_{k}}{\partial x_{k}}}\right]+\rho f_{i}}

\+ mass conservation:  ∂ ρ ∂ t + ∂ ( ρ v i ) ∂ x i = 0 {\displaystyle {\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:  ∂ v i ∂ x i = 0 {\displaystyle {\frac {\partial v_{i}}{\partial x_{i}}}=0}

| Fluid flow, gas flow
Nonlinear Schrödinger (cubic) | 1+1  |  i ∂ t ψ = − 1 2 ∂ x 2 ψ + κ | ψ | 2 ψ {\displaystyle \displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi } | optics, water waves
Nonlinear Schrödinger (derivative) | 1+1  |  i ∂ t ψ = − 1 2 ∂ x 2 ψ + ∂ x ( i κ | ψ | 2 ψ ) {\displaystyle \displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\partial _{x}(i\kappa |\psi |^{2}\psi )} | optics, water waves
Omega equation | 1+3  |  ∇ 2 ω + f 2 σ ∂ 2 ω ∂ p 2 {\displaystyle \displaystyle \nabla ^{2}\omega +{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega }{\partial p^{2}}}} = f σ ∂ ∂ p V g ⋅ ∇ p ( ζ g + f ) + R σ p ∇ p 2 ( V g ⋅ ∇ p T ) {\displaystyle \displaystyle ={\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\mathbf {V} _{g}\cdot \nabla _{p}(\zeta _{g}+f)+{\frac {R}{\sigma p}}\nabla _{p}^{2}(\mathbf {V} _{g}\cdot \nabla _{p}T)} | atmospheric physics
Plateau | 2  |  ( 1 + u y 2 ) u x x − 2 u x u y u x y + ( 1 + u x 2 ) u y y = 0 {\displaystyle \displaystyle (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 x x − u y y ± sin ⁡ u cos ⁡ u + cos ⁡ u sin 3 ⁡ u ( v x 2 − v y 2 ) = 0 , ( v x cot 2 ⁡ u ) x = ( v y cot 2 ⁡ u ) y {\displaystyle \displaystyle u_{xx}-u_{yy}\pm \sin u\cos u+{\frac {\cos u}{\sin ^{3}u}}(v_{x}^{2}-v_{y}^{2})=0,\quad \displaystyle (v_{x}\cot ^{2}u)_{x}=(v_{y}\cot ^{2}u)_{y}} |
Porous medium | 1+n |  u t = Δ ( u γ ) {\displaystyle \displaystyle u_{t}=\Delta (u^{\gamma })} | diffusion
Prandtl | 1+2  |  u t + u u x + v u y = U t + U U x + μ ρ u y y {\displaystyle \displaystyle u_{t}+uu_{x}+vu_{y}=U_{t}+UU_{x}+{\frac {\mu }{\rho }}u_{yy}} ,  u x + v y = 0 {\displaystyle \displaystyle u_{x}+v_{y}=0} | boundary layer

## R–Z, α–ω

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Name  | Dim  | Equation  | Applications

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Rayleigh | 1+1  |  u t t − u x x = ε ( u t − u t 3 ) {\displaystyle \displaystyle u_{tt}-u_{xx}=\varepsilon (u_{t}-u_{t}^{3})} |
Ricci flow | Any  |  ∂ t g i j = − 2 R i j {\displaystyle \displaystyle \partial _{t}g_{ij}=-2R_{ij}} | Poincaré conjecture
Richards equation | 1+3  |  θ t = [ K ( θ ) ( ψ z + 1 ) ] z {\displaystyle \displaystyle \theta _{t}=\left[K(\theta )\left(\psi _{z}+1\right)\right]_{z}} | Variably saturated flow in porous media
Rosenau–Hyman | 1+1  |  u t + a ( u n ) x + ( u n ) x x x = 0 {\displaystyle u_{t}+a\left(u^{n}\right)_{x}+\left(u^{n}\right)_{xxx}=0} | compacton solutions
Sawada–Kotera | 1+1  |  u t + 45 u 2 u x + 15 u x u x x + 15 u u x x x + u x x x x x = 0 {\displaystyle \displaystyle u_{t}+45u^{2}u_{x}+15u_{x}u_{xx}+15uu_{xxx}+u_{xxxxx}=0} |
Sack–Schamel equation | 1+1  |  V ¨ + ∂ η [ 1 1 − V ¨ ∂ η ( 1 − V ¨ V ) ] = 0 {\displaystyle {\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  |  ϕ t + ( 1 + b ϕ ) ϕ x + ϕ x x x = 0 {\displaystyle \phi _{t}+(1+b{\sqrt {\phi }})\phi _{x}+\phi _{xxx}=0} | plasmas, solitons, optics
Schlesinger | Any  |  ∂ A i ∂ t j [ A i , A j ] t i − t j , i ≠ j , ∂ A i ∂ t i = − ∑ j = 1 j ≠ i n [ A i , A j ] t i − t j , 1 ≤ i , j ≤ n {\displaystyle \displaystyle {\partial A_{i} \over \partial t_{j}}{\left[A_{i},\ A_{j}\right] \over t_{i}-t_{j}},\quad i\neq j,\quad {\partial A_{i} \over \partial t_{i}}=-\sum _{j=1 \atop j\neq i}^{n}{\left[A_{i},\ A_{j}\right] \over t_{i}-t_{j}},\quad 1\leq i,j\leq n} | isomonodromic deformations
Seiberg–Witten | 1+3  |  D A φ = 0 , F A + = σ ( φ ) {\displaystyle \displaystyle D^{A}\varphi =0,\qquad F_{A}^{+}=\sigma (\varphi )} | Seiberg–Witten invariants, QFT
Shallow water | 1+2  |  η t + ( η u ) x + ( η v ) y = 0 , ( η u ) t + ( η u 2 + 1 2 g η 2 ) x + ( η u v ) y = 0 , ( η v ) t + ( η u v ) x + ( η v 2 + 1 2 g η 2 ) y = 0 {\displaystyle \displaystyle \eta _{t}+(\eta u)_{x}+(\eta v)_{y}=0,\ (\eta u)_{t}+\left(\eta u^{2}+{\frac {1}{2}}g\eta ^{2}\right)_{x}+(\eta uv)_{y}=0,\ (\eta v)_{t}+(\eta uv)_{x}+\left(\eta v^{2}+{\frac {1}{2}}g\eta ^{2}\right)_{y}=0} | shallow water waves
Sine–Gordon | 1+1  |  φ t t − φ x x + sin ⁡ φ = 0 {\displaystyle \displaystyle \,\varphi _{tt}-\varphi _{xx}+\sin \varphi =0} | Solitons, QFT
Sinh–Gordon | 1+1  |  u x t = sinh ⁡ u {\displaystyle \displaystyle u_{xt}=\sinh u} | Solitons, QFT
Sinh–Poisson | 1+n |  ∇ 2 u + sinh ⁡ u = 0 {\displaystyle \displaystyle \nabla ^{2}u+\sinh u=0} | Fluid Mechanics
Swift–Hohenberg | any  |  u t = r u − ( 1 + ∇ 2 ) 2 u + N ( u ) {\displaystyle \displaystyle u_{t}=ru-(1+\nabla ^{2})^{2}u+N(u)} | pattern forming
Thomas | 2  |  u x y + α u x + β u y + γ u x u y = 0 {\displaystyle \displaystyle 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 {\displaystyle \displaystyle iu_{x}+v+u|v|^{2}=0} ,  i v t + u + v | u | 2 = 0 {\displaystyle \displaystyle iv_{t}+u+v|u|^{2}=0} | Dirac field, QFT
Toda lattice | any  |  ∇ 2 log ⁡ u n = u n + 1 − 2 u n + u n − 1 {\displaystyle \displaystyle \nabla ^{2}\log u_{n}=u_{n+1}-2u_{n}+u_{n-1}} |
Veselov–Novikov | 1+2  |  ( ∂ t + ∂ z 3 + ∂ z ¯ 3 ) v + ∂ z ( u v ) + ∂ z ¯ ( u w ) = 0 {\displaystyle \displaystyle (\partial _{t}+\partial _{z}^{3}+\partial _{\bar {z}}^{3})v+\partial _{z}(uv)+\partial _{\bar {z}}(uw)=0} ,  ∂ z ¯ u = 3 ∂ z v {\displaystyle \displaystyle \partial _{\bar {z}}u=3\partial _{z}v} ,  ∂ z w = 3 ∂ z ¯ v {\displaystyle \displaystyle \partial _{z}w=3\partial _{\bar {z}}v} | shallow water waves
Vorticity equation |  |  ∂ ω ∂ t + ( u ⋅ ∇ ) ω = ( ω ⋅ ∇ ) u − ω ( ∇ ⋅ u ) + 1 ρ 2 ∇ ρ × ∇ p + ∇ × ( ∇ ⋅ τ ρ ) + ∇ × ( f ρ ) , ω = ∇ × u {\displaystyle {\frac {\partial {\boldsymbol {\omega }}}{\partial t}}+(\mathbf {u} \cdot \nabla ){\boldsymbol {\omega }}=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {u} -{\boldsymbol {\omega }}(\nabla \cdot \mathbf {u} )+{\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p+\nabla \times \left({\frac {\nabla \cdot \tau }{\rho }}\right)+\nabla \times \left({\frac {\mathbf {f} }{\rho }}\right),\ {\boldsymbol {\omega }}=\nabla \times \mathbf {u} } | Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu | 1+1  |  i u t + ( ( 1 + | u | 2 ) − 1 / 2 u ) x x = 0 {\displaystyle \displaystyle iu_{t}+((1+|u|^{2})^{-1/2}u)_{xx}=0} |
WDVV equations | Any  |  ∑ σ , τ = 1 n ( ∂ 3 F ∂ t α t β t σ η σ τ ∂ 3 F ∂ t μ t ν t τ ) {\displaystyle \displaystyle \sum _{\sigma ,\tau =1}^{n}\left({\partial ^{3}F \over \partial t^{\alpha }t^{\beta }t^{\sigma }}\eta ^{\sigma \tau }{\partial ^{3}F \over \partial t^{\mu }t^{\nu }t^{\tau }}\right)} = ∑ σ , τ = 1 n ( ∂ 3 F ∂ t α t ν t σ η σ τ ∂ 3 F ∂ t μ t β t τ ) {\displaystyle \displaystyle =\sum _{\sigma ,\tau =1}^{n}\left({\partial ^{3}F \over \partial t^{\alpha }t^{\nu }t^{\sigma }}\eta ^{\sigma \tau }{\partial ^{3}F \over \partial t^{\mu }t^{\beta }t^{\tau }}\right)} | Topological field theory, QFT
WZW model | 1+1  |  S k ( γ ) = − k 8 π ∫ S 2 d 2 x K ( γ − 1 ∂ μ γ , γ − 1 ∂ μ γ ) + 2 π k S W Z ( γ ) {\displaystyle S_{k}(\gamma )=-\,{\frac {k}{8\pi }}\int _{S^{2}}d^{2}x\,{\mathcal {K}}(\gamma ^{-1}\partial ^{\mu }\gamma \,,\,\gamma ^{-1}\partial _{\mu }\gamma )+2\pi k\,S^{\mathrm {W} Z}(\gamma )}

S W Z ( γ ) = − 1 48 π 2 ∫ B 3 d 3 y ε i j k K ( γ − 1 ∂ γ ∂ y i , [ γ − 1 ∂ γ ∂ y j , γ − 1 ∂ γ ∂ y k ] ) {\displaystyle S^{\mathrm {W} Z}(\gamma )=-\,{\frac {1}{48\pi ^{2}}}\int _{B^{3}}d^{3}y\,\varepsilon ^{ijk}{\mathcal {K}}\left(\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{i}}}\,,\,\left[\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{j}}}\,,\,\gamma ^{-1}\,{\frac {\partial \gamma }{\partial y^{k}}}\right]\right)}

| QFT
Whitham equation | 1+1  |  η t + α η η x + ∫ − ∞ + ∞ K ( x − ξ ) η ξ ( ξ , t ) d ξ = 0 {\displaystyle \displaystyle \eta _{t}+\alpha \eta \eta _{x}+\int _{-\infty }^{+\infty }K(x-\xi )\,\eta _{\xi }(\xi ,t)\,{\text{d}}\xi =0} | water waves
Williams spray equation |  |  ∂ f j ∂ t + ∇ x ⋅ ( v f j ) + ∇ v ⋅ ( F j f j ) = − ∂ ∂ r ( R j f j ) − ∂ ∂ T ( E j f j ) + Q j + Γ j , F j = v ˙ , R j = r ˙ , E j = T ˙ , j = 1 , 2 , . . . , M {\displaystyle {\frac {\partial f_{j}}{\partial t}}+\nabla _{x}\cdot (\mathbf {v} f_{j})+\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}+\Gamma _{j},\ F_{j}={\dot {\mathbf {v} }},\ R_{j}={\dot {r}},\ E_{j}={\dot {T}},\ j=1,2,...,M} | Combustion
Yamabe | n |  Δ φ + h ( x ) φ = λ f ( x ) φ ( n + 2 ) / ( n − 2 ) {\displaystyle \displaystyle \Delta \varphi +h(x)\varphi =\lambda f(x)\varphi ^{(n+2)/(n-2)}} | Differential geometry
Yang–Mills (source-free) | Any  |  D μ F μ ν = 0 , F μ ν = A μ , ν − A ν , μ + [ A μ , A ν ] {\displaystyle \displaystyle D_{\mu }F^{\mu \nu }=0,\quad 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 α β = ± ε α β μ ν F μ ν , F μ ν = A μ , ν − A ν , μ + [ A μ , A ν ] {\displaystyle F_{\alpha \beta }=\pm \varepsilon _{\alpha \beta \mu \nu }F^{\mu \nu },\quad F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },\,A_{\nu }]} | Instantons, Donaldson theory, QFT
Yukawa | 1+n |  i ∂ t u + Δ u = − A u , ◻ A = m 2 A + | u | 2 {\displaystyle \displaystyle i\partial _{t}^{}u+\Delta u=-Au,\quad \displaystyle \Box A=m_{}^{2}A+|u|^{2}} | Meson-nucleon interactions, QFT
Zakharov system | 1+3  |  i ∂ t u + Δ u = u n , ◻ n = − Δ ( | u | 2 ) {\displaystyle \displaystyle i\partial _{t}^{}u+\Delta u=un,\quad \displaystyle \Box n=-\Delta (|u|_{}^{2})} | Langmuir waves
Zakharov–Schulman | 1+3  |  i u t + L 1 u = φ u , L 2 φ = L 3 ( | u | 2 ) {\displaystyle \displaystyle iu_{t}+L_{1}u=\varphi u,\quad \displaystyle L_{2}\varphi =L_{3}(|u|^{2})} | Acoustic waves
Zeldovich–Frank-Kamenetskii equation | 1+3  |  u t = D ∇ 2 u + β 2 2 u ( 1 − u ) e − β ( 1 − u ) {\displaystyle \displaystyle u_{t}=D\nabla ^{2}u+{\frac {\beta ^{2}}{2}}u(1-u)e^{-\beta (1-u)}} | Combustion
Zoomeron | 1+1  |  ( u x t / u ) t t − ( u x t / u ) x x + 2 ( u 2 ) x t = 0 {\displaystyle \displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx}+2(u^{2})_{xt}=0} | Solitons
φ4 equation | 1+1  |  φ t t − φ x x − φ + φ 3 = 0 {\displaystyle \displaystyle \varphi _{tt}-\varphi _{xx}-\varphi +\varphi ^{3}=0} | QFT
σ-model | 1+1  |  v x t + ( v x v t ) v = 0 {\displaystyle \displaystyle {\mathbf {v} }_{xt}+({\mathbf {v} }_{x}{\mathbf {v} }_{t}){\mathbf {v} }=0} | Harmonic maps, integrable systems, QFT

## References

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