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	[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
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
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
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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