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

Name Dim Equation Applications
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 , ( ∂ 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 {\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 )[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
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

Name Dim Equation Applications
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}}
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

Name Dim Equation Applications
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

Name Dim Equation Applications
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