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.

## 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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Bateman-Burgers equation | 1+1  | [math]\displaystyle{ \displaystyle u_t+uu_x=\nu u_{xx} }[/math] | Fluid mechanics
Benjamin–Bona–Mahony | 1+1  | [math]\displaystyle{ \displaystyle u_t+u_x+uu_x-u_{xxt}=0 }[/math] | Fluid mechanics
Benjamin–Ono | 1+1  | [math]\displaystyle{ \displaystyle u_t+Hu_{xx}+uu_x=0 }[/math] | internal waves in deep water
Boomeron  | 1+1  | [math]\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}) }[/math] | Solitons
Boltzmann equation | 1+6  | [math]\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_if_j] \,d\Omega\,d^3\mathbf{p'} }[/math] | Statistical mechanics
Born–Infeld  | 1+1  | [math]\displaystyle{ \displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0 }[/math] | Electrodynamics
Boussinesq  | 1+1  | [math]\displaystyle{ \displaystyle u_{tt} - u_{xx} - u_{xxxx} - 3(u^2)_{xx} = 0 }[/math] | Fluid mechanics
Boussinesq type equation  | 1+1  | [math]\displaystyle{ \displaystyle u_{tt}-u_{xx}-2 \alpha (u u_x)_{x}-\beta u_{xxtt}=0 }[/math] | Fluid mechanics
Buckmaster | 1+1  | [math]\displaystyle{ \displaystyle u_t=(u^4)_{xx}+(u^3)_x }[/math] | Thin viscous fluid sheet flow
Cahn–Hilliard equation | Any  | [math]\displaystyle{ \displaystyle c_t = D\nabla^2\left(c^3-c-\gamma\nabla^2 c\right) }[/math] | Phase separation
Calabi flow | Any  | [math]\displaystyle{ \frac{\partial g_{ij}}{\partial t}=(\Delta R)g_{ij} }[/math] | Calabi–Yau manifolds
Camassa–Holm | 1+1  | [math]\displaystyle{ u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}\, }[/math] | Peakons
Carleman  | 1+1  | [math]\displaystyle{ \displaystyle u_t+u_x=v^2-u^2=v_x-v_t }[/math] |
Cauchy momentum | any  | [math]\displaystyle{ \displaystyle \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \nabla \cdot \sigma + \rho\mathbf{f} }[/math] | Momentum transport
Chafee–Infante equation  |  | [math]\displaystyle{ u_t-u_{xx}+\lambda(u^3-u)=0 }[/math] |
Clairaut equation  | any  | [math]\displaystyle{ x\cdot Du+f(Du)=u }[/math] | Differential geometry
Clarke's equation | 1+1  | [math]\displaystyle{ (\theta_t-\gamma e^{\theta})_{tt}=(\theta_t-e^\theta)_{xx} }[/math] | Combustion
Complex Monge–Ampère  | Any  | [math]\displaystyle{ \displaystyle \det(\partial_{i\bar j}\varphi) = }[/math] lower order terms  | Calabi conjecture
Constant astigmatism  | 1+1  | [math]\displaystyle{ z_{yy} + \left(\frac{1}{z}\right)_{xx} + 2 = 0 }[/math] | Differential geometry
Davey–Stewartson | 1+2  | [math]\displaystyle{ \displaystyle i u_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 }[/math] | Finite depth waves
Degasperis–Procesi | 1+1  | [math]\displaystyle{ \displaystyle u_t - u_{xxt} + 4u u_x = 3 u_x u_{xx} + u u_{xxx} }[/math] | Peakons
Dispersive long wave  | 1+1  | [math]\displaystyle{ \displaystyle u_t=(u^2-u_x+2w)_x }[/math], [math]\displaystyle{ w_t=(2uw+w_x)_x }[/math] |
Drinfeld–Sokolov–Wilson | 1+1  | [math]\displaystyle{ \displaystyle u_t=3ww_x, \quad \displaystyle w_t=2w_{xxx}+2uw_x+u_xw }[/math] |
Dym equation | 1+1  | [math]\displaystyle{ \displaystyle u_t = u^3u_{xxx}.\, }[/math] | Solitons
Eckhaus equation | 1+1  | [math]\displaystyle{ iu_t+u_{xx}+2|u|^2_xu+|u|^4u=0 }[/math] | Integrable systems
Eikonal equation | any  | [math]\displaystyle{ \displaystyle |\nabla u(x)|=F(x), \ x\in \Omega }[/math] | optics
Einstein field equations | Any  | [math]\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} }[/math] | General relativity
Ernst equation | 2  | [math]\displaystyle{ \displaystyle \Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2 }[/math] |
Estevez–Mansfield–Clarkson equation  |  | [math]\displaystyle{ U_{tyyy}+\beta U_y U_{yt}+\beta U_{yy} U_t+U_{tt}=0 \text{ in which } U=u(x,y,t) }[/math] |
Euler equations  | 1+3  | [math]\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 }[/math] | non-viscous fluids
Fisher's equation | 1+1  | [math]\displaystyle{ \displaystyle u_t=u(1-u)+u_{xx} }[/math] | Gene propagation
FitzHugh–Nagumo model | 1+1  | [math]\displaystyle{ \displaystyle u_t=u_{xx}+u(u-a)(1-u)+w, \quad \displaystyle w_t=\varepsilon u }[/math] | Biological neuron model
Föppl–von Kármán equations |  | [math]\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 }[/math] | Solid Mechanics

## G–K

Name  | Dim  | Equation  | Applications

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G equation | 1+3  | [math]\displaystyle{ G_t + \mathbf{v}\cdot\nabla G = S_L(G) |\nabla G| }[/math] | turbulent combustion
Generic scalar transport  | 1+3  | [math]\displaystyle{ \displaystyle \varphi_t + \nabla \cdot f(t,x,\varphi,\nabla\varphi) = g(t,x,\varphi) }[/math] | transport
Ginzburg–Landau | 1+3  | [math]\displaystyle{ \displaystyle \alpha \psi + \beta |\psi|^2 \psi + \tfrac{1}{2m} \left(-i\hbar\nabla - 2e\mathbf{A} \right)^2 \psi = 0 }[/math] | Superconductivity
Gross–Pitaevskii | 1 + n | [math]\displaystyle{ \displaystyle i\partial_t\psi = \left (-\tfrac12\nabla^2 + V(x) + g|\psi|^2 \right ) \psi }[/math] | Bose–Einstein condensate
Gyrokinetics equation | 1 + 5 | [math]\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 } }[/math] | Microturbulence in plasma
Guzmán  | 1 + n | [math]\displaystyle{ \displaystyle J_t+gJ_x+1/2\sigma^2J_{xx}-\lambda\sigma^2(J_x)^2+f=0 }[/math] | Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation | Any  | [math]\displaystyle{ \displaystyle i\partial_tu + \Delta u= \left (\pm |x|^{-n} |u|^2 \right) u }[/math] |
Hasegawa–Mima | 1+3  | [math]\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] }[/math] | Turbulence in plasma
Heisenberg ferromagnet  | 1+1  | [math]\displaystyle{ \displaystyle \mathbf{S}_t=\mathbf{S}\wedge \mathbf{S}_{xx}. }[/math] | Magnetism
Hicks | 1+1  | [math]\displaystyle{ \psi_{rr} - \psi_r/r + \psi_{zz} = r^2 \mathrm{d}H/\mathrm{d} \psi - \Gamma \mathrm{d} \Gamma/\mathrm{d}\psi }[/math] | Fluid dynamics
Hunter–Saxton | 1+1  | [math]\displaystyle{ \displaystyle \left (u_t + u u_x \right )_x = \tfrac{1}{2} u_x^2 }[/math] | Liquid crystals
Ishimori equation | 1+2  | [math]\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) }[/math] | Integrable systems
Kadomtsev –Petviashvili | 1+2  | [math]\displaystyle{ \displaystyle \partial_x \left (\partial_t u+u \partial_x u+\varepsilon^2\partial_{xxx}u \right )+\lambda\partial_{yy}u=0 }[/math] | Shallow water waves
Kardar–Parisi–Zhang equation | 1+3  | [math]\displaystyle{ \displaystyle h_t=\nu \nabla^2 h + \lambda (\nabla h)^2 /2+ \eta }[/math] | Stochastics
von Karman  | 2  | [math]\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) }[/math] |
Kaup  | 1+1  | [math]\displaystyle{ \displaystyle f_x=2fgc(x-t)=g_t }[/math] |
Kaup–Kupershmidt | 1+1  | [math]\displaystyle{ \displaystyle u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x }[/math] | Integrable systems
Klein–Gordon–Maxwell  | any  | [math]\displaystyle{ \displaystyle \nabla^2s= \left (|\mathbf a|^2+1 \right )s, \quad \nabla^2\mathbf a =\nabla(\nabla\cdot\mathbf a)+s^2\mathbf a }[/math] |
Klein–Gordon (nonlinear)  | any  | [math]\displaystyle{ \nabla^2u+\lambda u^p=0 }[/math] | Relativistic quantum mechanics
Khokhlov–Zabolotskaya  | 1+2  | [math]\displaystyle{ \displaystyle u_{xt} -(uu_x)_x =u_{yy} }[/math] |
Korteweg–de Vries (KdV)  | 1+1  | [math]\displaystyle{ \displaystyle u_{t}+u_{xxx}-6u u_{x}=0 }[/math] | Shallow waves, Integrable systems
KdV (super)  | 1+1  | [math]\displaystyle{ \displaystyle u_t=6uu_x-u_{xxx}+3ww_{xx}, \quad w_t=3u_xw+6uw_x-4w_{xxx} }[/math] |
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation | 1 + n | [math]\displaystyle{ \displaystyle u_t+\nabla^4u+\nabla^2u+ \tfrac{1}{2}|\nabla u|^2=0 }[/math] | Combustion

## L–Q

Name  | Dim  | Equation  | Applications

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|
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Landau–Lifshitz model | 1+n | [math]\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} }[/math] | Magnetic field in solids
Lin–Tsien equation | 1+2  | [math]\displaystyle{ \displaystyle 2u_{tx}+u_xu_{xx}-u_{yy}=0 }[/math] |
Liouville equation  | any  | [math]\displaystyle{ \displaystyle \nabla^2u+e^{\lambda u}=0 }[/math] |
Liouville–Bratu–Gelfand equation | any  | [math]\displaystyle{ \nabla^2 \psi + \lambda e^\psi=0 }[/math] | combustion, astrophysics
Logarithmic Schrödinger equation | any  | [math]\displaystyle{ i \frac{\partial \psi}{\partial t} + \Delta \psi + \psi \ln |\psi|^2 = 0. }[/math] | Superfluids, Quantum gravity
Minimal surface  | 3  | [math]\displaystyle{ \displaystyle \operatorname{div}(Du/\sqrt{1+|Du|^2})=0 }[/math] | minimal surfaces
Monge–Ampère | any  | [math]\displaystyle{ \displaystyle \det(\partial_{ij}\varphi) = }[/math] lower order terms  |
Navier–Stokes
(and its derivation) | 1+3  | [math]\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 }[/math]

\+ mass conservation: [math]\displaystyle{ \frac{\partial \rho}{\partial t} + \frac{\partial \left( \rho\, v_i \right)}{\partial x_i} = 0 }[/math]
\+ an equation of state to relate p and ρ, e.g. for an incompressible flow: [math]\displaystyle{ \frac{\partial v_i}{\partial x_i} = 0 }[/math]

| Fluid flow, gas flow
Nonlinear Schrödinger (cubic) | 1+1  | [math]\displaystyle{ \displaystyle i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi }[/math] | optics, water waves
Nonlinear Schrödinger (derivative) | 1+1  | [math]\displaystyle{ \displaystyle i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\partial_x(i\kappa|\psi|^2 \psi) }[/math] | optics, water waves
Omega equation | 1+3  | [math]\displaystyle{ \displaystyle \nabla^2\omega + \frac{f^2}{\sigma}\frac{\partial^2\omega}{\partial p^2} }[/math] [math]\displaystyle{ \displaystyle = \frac{f}{\sigma}\frac{\partial}{\partial p}\mathbf{V}_g\cdot\nabla_p (\zeta_g + f) + \frac{R}{\sigma p}\nabla^2_p(\mathbf{V}_g\cdot\nabla_p T) }[/math] | atmospheric physics
Plateau  | 2  | [math]\displaystyle{ \displaystyle (1+u_y^2)u_{xx} -2u_xu_yu_{xy} +(1+u_x^2)u_{yy}=0 }[/math] | minimal surfaces
Pohlmeyer–Lund–Regge  | 2  | [math]\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^2u)_x = (v_y\cot^2 u)_y }[/math] |
Porous medium | 1+n | [math]\displaystyle{ \displaystyle u_t=\Delta(u^\gamma) }[/math] | diffusion
Prandtl  | 1+2  | [math]\displaystyle{ \displaystyle u_t+uu_x+vu_y=U_t+UU_x+\frac{\mu}{\rho}u_{yy} }[/math], [math]\displaystyle{ \displaystyle u_x+v_y=0 }[/math] | boundary layer

## R–Z, α–ω

Name  | Dim  | Equation  | Applications

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|
|

Rayleigh  | 1+1  | [math]\displaystyle{ \displaystyle u_{tt}-u_{xx} = \varepsilon(u_t-u_t^3) }[/math] |
Ricci flow | Any  | [math]\displaystyle{ \displaystyle \partial_t g_{ij}=-2 R_{ij} }[/math] | Poincaré conjecture
Richards equation | 1+3  | [math]\displaystyle{ \displaystyle \theta_t=\left[ K(\theta) \left (\psi_z + 1 \right) \right]_z }[/math] | Variably saturated flow in porous media
Rosenau–Hyman | 1+1  | [math]\displaystyle{ u_t + a \left(u^n\right)_x + \left(u^n\right)_{xxx} = 0 }[/math] | compacton solutions
Sawada–Kotera  | 1+1  | [math]\displaystyle{ \displaystyle u_t+45u^2u_x+15u_xu_{xx}+15uu_{xxx}+u_{xxxxx}=0 }[/math] |
Schlesinger  | Any  | [math]\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 }[/math] | isomonodromic deformations
Seiberg–Witten  | 1+3  | [math]\displaystyle{ \displaystyle D^A\varphi=0, \qquad F^+_A=\sigma(\varphi) }[/math] | Seiberg–Witten invariants, QFT
Shallow water | 1+2  | [math]\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 }[/math] | shallow water waves
Sine–Gordon  | 1+1  | [math]\displaystyle{ \displaystyle \, \varphi_{tt}- \varphi_{xx} + \sin\varphi = 0 }[/math] | Solitons, QFT
Sinh–Gordon  | 1+1  | [math]\displaystyle{ \displaystyle u_{xt}= \sinh u }[/math] | Solitons, QFT
Sinh–Poisson  | 1+n | [math]\displaystyle{ \displaystyle \nabla^2u+\sinh u=0 }[/math] | Fluid Mechanics
Swift–Hohenberg | any  | [math]\displaystyle{ \displaystyle u_t = r u - (1+\nabla^2)^2u + N(u) }[/math] | pattern forming
Thomas  | 2  | [math]\displaystyle{ \displaystyle u_{xy}+\alpha u_x+\beta u_y+\gamma u_xu_y=0 }[/math] |
Thirring | 1+1  | [math]\displaystyle{ \displaystyle iu_x+v+u|v|^2=0 }[/math], [math]\displaystyle{ \displaystyle iv_t+u+v|u|^2=0 }[/math] | Dirac field, QFT
Toda lattice | any  | [math]\displaystyle{ \displaystyle \nabla^2\log u_n = u_{n+1}-2u_n+u_{n-1} }[/math] |
Veselov–Novikov | 1+2  | [math]\displaystyle{ \displaystyle (\partial_t+\partial_z^3+\partial_{\bar z}^3)v+\partial_z(uv)+\partial_{\bar z}(uw) =0 }[/math], [math]\displaystyle{ \displaystyle \partial_{\bar z}u=3\partial_zv }[/math], [math]\displaystyle{ \displaystyle \partial_zw=3\partial_{\bar z} v }[/math] | shallow water waves
Vorticity equation |  | [math]\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} }[/math] | Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu  | 1+1  | [math]\displaystyle{ \displaystyle iu_t+((1+|u|^2)^{-1/2}u)_{xx}=0 }[/math] |
WDVV equations  | Any  | [math]\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) }[/math] [math]\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) }[/math] | Topological field theory, QFT
WZW model  | 1+1  | [math]\displaystyle{ S_k(\gamma)= - \, \frac {k}{8\pi} \int_{S^2} d^2x\, \mathcal{K} (\gamma^{-1} \partial^\mu \gamma \, , \, \gamma^{-1} \partial_\mu \gamma) + 2\pi k\, S^{\mathrm WZ}(\gamma) }[/math]

[math]\displaystyle{ S^{\mathrm WZ}(\gamma) = - \, \frac{1}{48\pi^2} \int_{B^3} d^3y\, \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) }[/math]

| QFT
Whitham equation | 1+1  | [math]\displaystyle{ \displaystyle \eta_t + \alpha \eta \eta_x + \int_{-\infty}^{+\infty} K(x-\xi)\, \eta_\xi(\xi,t)\, \text{d}\xi = 0 }[/math] | water waves
Williams spray equation |  | [math]\displaystyle{ \frac{\partial f_j}{\partial t} + \nabla_x\cdot(\mathbf{v}f_j) + \nabla_v\cdot(F_jf_j) =- \frac{\partial }{\partial r}(R_jf_j) - \frac{\partial }{\partial T}(E_jf_j) + Q_j + \Gamma_j,\ F_j = \dot{\mathbf{v}},\ R_j = \dot{r},\ E_j = \dot{T},\ j = 1,2,...,M }[/math] | Combustion
Yamabe | n | [math]\displaystyle{ \displaystyle\Delta \varphi+h(x)\varphi = \lambda f(x)\varphi^{(n+2)/(n-2)} }[/math] | Differential geometry
Yang–Mills (source-free) | Any  | [math]\displaystyle{ \displaystyle D_\mu F^{\mu\nu}=0, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu] }[/math] | Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual) | 4  | [math]\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] }[/math] | Instantons, Donaldson theory, QFT
Yukawa  | 1+n | [math]\displaystyle{ \displaystyle i \partial_t^{}u + \Delta u = -A u,\quad \displaystyle\Box A = m^2_{} A + |u|^2 }[/math] | Meson-nucleon interactions, QFT
Zakharov system | 1+3  | [math]\displaystyle{ \displaystyle i \partial_t^{} u + \Delta u = un,\quad \displaystyle \Box n = -\Delta (|u|^2_{}) }[/math] | Langmuir waves
Zakharov–Schulman | 1+3  | [math]\displaystyle{ \displaystyle iu_t + L_1u = \varphi u,\quad \displaystyle L_2 \varphi = L_3( | u |^2) }[/math] | Acoustic waves
Zeldovich–Frank-Kamenetskii equation  | 1+3  | [math]\displaystyle{ \displaystyle u_t = D\nabla^2 u + \frac{\beta^2}{2}u(1-u) e^{-\beta(1-u)} }[/math] | Combustion
Zoomeron  | 1+1  | [math]\displaystyle{ \displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx} +2(u^2)_{xt}=0 }[/math] | Solitons
φ4 equation | 1+1  | [math]\displaystyle{ \displaystyle \varphi_{tt}-\varphi_{xx}-\varphi+\varphi^3=0 }[/math] | QFT
σ-model | 1+1  | [math]\displaystyle{ \displaystyle {\mathbf v}_{xt}+({\mathbf v}_x{\mathbf v}_t){\mathbf v}=0 }[/math] | Harmonic maps, integrable systems, QFT

## References

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