# List of nonlinear partial differential equations

## A–F

Name Dim Equation Applications
Bateman-Burgers equation 1+1 ${\displaystyle \displaystyle u_{t}+uu_{x}=\nu u_{xx}}$ Fluid mechanics
Benjamin–Bona–Mahony 1+1 ${\displaystyle \displaystyle u_{t}+u_{x}+uu_{x}-u_{xxt}=0}$ Fluid mechanics
Benjamin–Ono 1+1 ${\displaystyle \displaystyle u_{t}+Hu_{xx}+uu_{x}=0}$ internal waves in deep water
Boomeron 1+1 ${\displaystyle \displaystyle u_{t}=\mathbf {b} \cdot \mathbf {v} _{x},\quad \displaystyle \mathbf {v} _{xt}=u_{xx}\mathbf {b} +\mathbf {a} \times \mathbf {v} _{x}-2\mathbf {v} \times (\mathbf {v} \times \mathbf {b} )}$ Solitons
Boltzmann equation 1+6 ${\displaystyle {\frac {\partial f_{i}}{\partial t}}+{\frac {\mathbf {p} _{i}}{m_{i}}}\cdot \nabla f_{i}+\mathbf {F} \cdot {\frac {\partial f_{i}}{\partial \mathbf {p} _{i}}}=\left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} },\quad \left({\frac {\partial f_{i}}{\partial t}}\right)_{\mathrm {coll} }=\sum _{j=1}^{n}\iint g_{ij}I_{ij}(g_{ij},\Omega )[f'_{i}f'_{j}-f_{i}f_{j}]\,d\Omega \,d^{3}\mathbf {p'} }$ Statistical mechanics
Born–Infeld 1+1 ${\displaystyle \displaystyle (1-u_{t}^{2})u_{xx}+2u_{x}u_{t}u_{xt}-(1+u_{x}^{2})u_{tt}=0}$ Electrodynamics
Boussinesq 1+1 ${\displaystyle \displaystyle u_{tt}-u_{xx}-u_{xxxx}-3(u^{2})_{xx}=0}$ Fluid mechanics
Boussinesq type equation 1+1 ${\displaystyle \displaystyle u_{tt}-u_{xx}-2\alpha (uu_{x})_{x}-\beta u_{xxtt}=0=0}$ Fluid mechanics
Buckmaster 1+1 ${\displaystyle \displaystyle u_{t}=(u^{4})_{xx}+(u^{3})_{x}}$ Thin viscous fluid sheet flow
Cahn–Hilliard equation Any ${\displaystyle \displaystyle c_{t}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right)}$ Phase separation
Calabi flow Any ${\displaystyle {\frac {\partial g_{ij}}{\partial t}}=(\Delta R)g_{ij}}$ Calabi–Yau manifolds
Camassa–Holm 1+1 ${\displaystyle u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}\,}$ Peakons
Carleman 1+1 ${\displaystyle \displaystyle u_{t}+u_{x}=v^{2}-u^{2}=v_{x}-v_{t}}$
Cauchy momentum any ${\displaystyle \displaystyle \rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=\nabla \cdot \sigma +\rho \mathbf {f} }$ Momentum transport
Chaffee–Infante equation ${\displaystyle u_{t}-u_{xx}+\lambda (u^{3}-u)=0}$
Clairaut equation any ${\displaystyle x\cdot Du+f(Du)=u}$ Differential geometry
Clarke's equation 1+1 ${\displaystyle (\theta _{t}-\gamma e^{\theta })_{tt}=(\theta _{t}-e^{\theta })_{xx}}$ Combustion
Complex Monge–Ampère Any ${\displaystyle \displaystyle \det(\partial _{i{\bar {j}}}\varphi )=}$ lower order terms Calabi conjecture
Davey–Stewartson 1+2 ${\displaystyle \displaystyle iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\varphi _{x},\quad \displaystyle \varphi _{xx}+c_{3}\varphi _{yy}=(|u|^{2})_{x}}$ Finite depth waves
Degasperis–Procesi 1+1 ${\displaystyle \displaystyle u_{t}-u_{xxt}+4uu_{x}=3u_{x}u_{xx}+uu_{xxx}}$ Peakons
Dispersive long wave 1+1 ${\displaystyle \displaystyle u_{t}=(u^{2}-u_{x}+2w)_{x}}$, ${\displaystyle w_{t}=(2uw+w_{x})_{x}}$
Drinfeld–Sokolov–Wilson 1+1 ${\displaystyle \displaystyle u_{t}=3ww_{x},\quad \displaystyle w_{t}=2w_{xxx}+2uw_{x}+u_{x}w}$
Dym equation 1+1 ${\displaystyle \displaystyle u_{t}=u^{3}u_{xxx}.\,}$ Solitons
Eckhaus equation 1+1 ${\displaystyle iu_{t}+u_{xx}+2|u|_{x}^{2}u+|u|^{4}u=0}$ Integrable systems
Eikonal equation any ${\displaystyle \displaystyle |\nabla u(x)|=F(x),\ x\in \Omega }$ optics
Einstein field equations Any ${\displaystyle \displaystyle R_{\mu \nu }-{\textstyle 1 \over 2}R\,g_{\mu \nu }+\Lambda g_{\mu \nu }={\frac {8\pi G}{c^{4}}}T_{\mu \nu }}$ General relativity
Ernst equation 2 ${\displaystyle \displaystyle \Re (u)(u_{rr}+u_{r}/r+u_{zz})=(u_{r})^{2}+(u_{z})^{2}}$
Estevez–Mansfield–Clarkson equation ${\displaystyle U_{tyyy}+\beta U_{y}U_{yt}+\beta U_{yy}U_{t}+U_{tt}=0{\text{ in which }}U=u(x,y,t)}$
Euler equations 1+3 ${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0,\quad \rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\rho \mathbf {f} ,\quad {\frac {\partial s}{\partial t}}+\mathbf {v} \cdot \nabla s=0}$ non-viscous fluids
Fisher's equation 1+1 ${\displaystyle \displaystyle u_{t}=u(1-u)+u_{xx}}$ Gene propagation
Fitzhugh–Nagumo 1+1 ${\displaystyle \displaystyle u_{t}=u_{xx}+u(u-a)(1-u)+w,\quad \displaystyle w_{t}=\varepsilon u}$ Biological neuron model
Föppl–von Kármán equations ${\displaystyle {\frac {Eh^{3}}{12(1-\nu ^{2})}}\nabla ^{4}w-h{\frac {\partial }{\partial x_{\beta }}}\left(\sigma _{\alpha \beta }{\frac {\partial w}{\partial x_{\alpha }}}\right)=P,\quad {\frac {\partial \sigma _{\alpha \beta }}{\partial x_{\beta }}}=0}$ Solid Mechanics

## G–K

Name Dim Equation Applications
G equation 1+3 ${\displaystyle G_{t}+\mathbf {v} \cdot \nabla G=S_{L}(G)|\nabla G|}$ turbulent combustion
Generic scalar transport 1+3 ${\displaystyle \displaystyle \varphi _{t}+\nabla \cdot f(t,x,\varphi ,\nabla \varphi )=g(t,x,\varphi )}$ transport
Ginzburg–Landau 1+3 ${\displaystyle \displaystyle \alpha \psi +\beta |\psi |^{2}\psi +{\tfrac {1}{2m}}\left(-i\hbar \nabla -2e\mathbf {A} \right)^{2}\psi =0}$ Superconductivity
Gross–Pitaevskii 1 + n ${\displaystyle \displaystyle i\partial _{t}\psi =\left(-{\tfrac {1}{2}}\nabla ^{2}+V(x)+g|\psi |^{2}\right)\psi }$ Bose–Einstein condensate
Gyrokinetics equation 1 + 5 ${\displaystyle {\displaystyle {\frac {\partial h_{s}}{\partial t}}+\left(v_{||}{\hat {b}}+{\vec {V}}_{ds}+\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\right)\cdot {\vec {\nabla }}_{\vec {R}}h_{s}-\sum _{s'}\left\langle C\left[h_{s},h_{s'}\right]\right\rangle _{\varphi }={\frac {Z_{s}ef_{s0}}{T_{s}}}{\frac {\partial \left\langle \phi \right\rangle _{\varphi }}{\partial t}}-{\frac {\partial f_{s0}}{\partial \psi }}\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\cdot {\vec {\nabla }}\psi }}$ Microturbulence in plasma
Guzmán 1 + n ${\displaystyle \displaystyle J_{t}+gJ_{x}+1/2\sigma ^{2}J_{xx}-\lambda \sigma ^{2}(J_{x})^{2}+f=0}$ Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation Any ${\displaystyle \displaystyle i\partial _{t}u+\Delta u=\left(\pm |x|^{-n}|u|^{2}\right)u}$
Hasegawa–Mima 1+3 ${\displaystyle \displaystyle 0={\frac {\partial }{\partial t}}\left(\nabla ^{2}\varphi -\varphi \right)-\left[\left(\nabla \varphi \times {\hat {\mathbf {z} }}\right)\cdot \nabla \right]\left[\nabla ^{2}\varphi -\ln \left({\frac {n_{0}}{\omega _{ci}}}\right)\right]}$ Turbulence in plasma
Heisenberg ferromagnet 1+1 ${\displaystyle \displaystyle \mathbf {S} _{t}=\mathbf {S} \wedge \mathbf {S} _{xx}.}$ Magnetism
Hirota–Satsuma 1+1 ${\displaystyle \displaystyle u_{t}={\tfrac {1}{2}}u_{xxx}+3uu_{x}-6ww_{x},\quad w_{t}+w_{xxx}+3uw_{x}=0}$
Hunter–Saxton 1+1 ${\displaystyle \displaystyle \left(u_{t}+uu_{x}\right)_{x}={\tfrac {1}{2}}u_{x}^{2}}$ Liquid crystals
Ishimori equation 1+2 ${\displaystyle \displaystyle \mathbf {S} _{t}=\mathbf {S} \wedge \left(\mathbf {S} _{xx}+\mathbf {S} _{yy}\right)+u_{x}\mathbf {S} _{y}+u_{y}\mathbf {S} _{x},\quad \displaystyle u_{xx}-\alpha ^{2}u_{yy}=-2\alpha ^{2}\mathbf {S} \cdot \left(\mathbf {S} _{x}\wedge \mathbf {S} _{y}\right)}$ Integrable systems
Kadomtsev –Petviashvili 1+2 ${\displaystyle \displaystyle \partial _{x}\left(\partial _{t}u+u\partial _{x}u+\varepsilon ^{2}\partial _{xxx}u\right)+\lambda \partial _{yy}u=0}$ Shallow water waves
Kardar–Parisi–Zhang equation 1+3 ${\displaystyle \displaystyle h_{t}=\nu \nabla ^{2}h+\lambda (\nabla h)^{2}/2+\eta }$ Stochastics
von Karman 2 ${\displaystyle \displaystyle \nabla ^{4}u=E\left(w_{xy}^{2}-w_{xx}w_{yy}\right),\quad \nabla ^{4}w=a+b\left(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy}\right)}$
Kaup 1+1 ${\displaystyle \displaystyle f_{x}=2fgc(x-t)=g_{t}}$
Kaup–Kupershmidt 1+1 ${\displaystyle \displaystyle u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}}$ Integrable systems
Klein–Gordon–Maxwell any ${\displaystyle \displaystyle \nabla ^{2}s=\left(|\mathbf {a} |^{2}+1\right)s,\quad \nabla ^{2}\mathbf {a} =\nabla (\nabla \cdot \mathbf {a} )+s^{2}\mathbf {a} }$
Klein–Gordon (nonlinear) any ${\displaystyle \nabla ^{2}u+\lambda u^{p}=0}$ Relativistic quantum mechanics
Khokhlov–Zabolotskaya 1+2 ${\displaystyle \displaystyle u_{xt}-(uu_{x})_{x}=u_{yy}}$
Korteweg–de Vries (KdV) 1+1 ${\displaystyle \displaystyle \partial _{t}u+\partial _{x}^{3}u+6u\partial _{x}u=0}$ Shallow waves, Integrable systems
KdV (super) 1+1 ${\displaystyle \displaystyle u_{t}=6uu_{x}-u_{xxx}+3ww_{xx},\quad w_{t}=3u_{x}w+6uw_{x}-4w_{xxx}}$
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation 1 + n ${\displaystyle \displaystyle u_{t}+\nabla ^{4}u+\nabla ^{2}u+{\tfrac {1}{2}}|\nabla u|^{2}=0}$ Combustion

## L–Q

Name Dim Equation Applications
Landau–Lifshitz model 1+n ${\displaystyle \displaystyle {\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S} }$ Magnetic field in solids
Lin–Tsien equation 1+2 ${\displaystyle \displaystyle 2u_{tx}+u_{x}u_{xx}-u_{yy}}$
Liouville equation any ${\displaystyle \displaystyle \nabla ^{2}u+e^{\lambda u}=0}$
Liouville–Bratu–Gelfand equation any ${\displaystyle \nabla ^{2}\psi +\lambda e^{\psi }=0}$ combustion, astrophysics
Minimal surface 3 ${\displaystyle \displaystyle \operatorname {div} (Du/{\sqrt {1+|Du|^{2}}})=0}$ minimal surfaces
Monge–Ampère any ${\displaystyle \displaystyle \det(\partial _{ij}\varphi )=}$ lower order terms
Navier–Stokes
(and its derivation)
1+3 ${\displaystyle \displaystyle \rho \left({\frac {\partial v_{i}}{\partial t}}+v_{j}{\frac {\partial v_{i}}{\partial x_{j}}}\right)=-{\frac {\partial p}{\partial x_{i}}}+{\frac {\partial }{\partial x_{j}}}\left[\mu \left({\frac {\partial v_{i}}{\partial x_{j}}}+{\frac {\partial v_{j}}{\partial x_{i}}}\right)+\lambda {\frac {\partial v_{k}}{\partial x_{k}}}\right]+\rho f_{i}}$

+ mass conservation: ${\displaystyle {\frac {\partial \rho }{\partial t}}+{\frac {\partial \left(\rho \,v_{i}\right)}{\partial x_{i}}}=0}$
+ an equation of state to relate p and ρ, e.g. for an incompressible flow: ${\displaystyle {\frac {\partial v_{i}}{\partial x_{i}}}=0}$

Fluid flow, gas flow
Nonlinear Schrödinger (cubic) 1+1 ${\displaystyle \displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi }$ optics, water waves
Nonlinear Schrödinger (derivative) 1+1 ${\displaystyle \displaystyle i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\partial _{x}(i\kappa |\psi |^{2}\psi )}$ optics, water waves
Omega equation 1+3 ${\displaystyle \displaystyle \nabla ^{2}\omega +{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega }{\partial p^{2}}}}$ ${\displaystyle \displaystyle ={\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\mathbf {V} _{g}\cdot \nabla _{p}(\zeta _{g}+f)+{\frac {R}{\sigma p}}\nabla _{p}^{2}(\mathbf {V} _{g}\cdot \nabla _{p}T)}$ atmospheric physics
Plateau 2 ${\displaystyle \displaystyle (1+u_{y}^{2})u_{xx}-2u_{x}u_{y}u_{xy}+(1+u_{x}^{2})u_{yy}=0}$
Pohlmeyer–Lund–Regge 2 ${\displaystyle \displaystyle u_{xx}-u_{yy}\pm \sin u\cos u+{\frac {\cos u}{\sin ^{3}u}}(v_{x}^{2}-v_{y}^{2})=0,\quad \displaystyle (v_{x}\cot ^{2}u)_{x}=(v_{y}\cot ^{2}u)_{y}}$
Porous medium 1+n ${\displaystyle \displaystyle u_{t}=\Delta (u^{\gamma })}$ diffusion
Prandtl 1+2 ${\displaystyle \displaystyle u_{t}+uu_{x}+vu_{y}=U_{t}+UU_{x}+{\frac {\mu }{\rho }}u_{yy}}$, ${\displaystyle \displaystyle u_{x}+v_{y}=0}$ boundary layer

## R–Z, α–ω

Name Dim Equation Applications
Rayleigh 1+1 ${\displaystyle \displaystyle u_{tt}-u_{xx}=\varepsilon (u_{t}-u_{t}^{3})}$
Ricci flow Any ${\displaystyle \displaystyle \partial _{t}g_{ij}=-2R_{ij}}$ Poincaré conjecture
Richards equation 1+3 ${\displaystyle \displaystyle \theta _{t}=\left[K(\theta )\left(\psi _{z}+1\right)\right]_{z}}$ Variably saturated flow in porous media
Rosenau–Hyman equation 1+1 ${\displaystyle u_{t}+a\left(u^{n}\right)_{x}+\left(u^{n}\right)_{xxx}=0}$ compacton solutions
Sawada–Kotera 1+1 ${\displaystyle \displaystyle u_{t}+45u^{2}u_{x}+15u_{x}u_{xx}+15uu_{xxx}+u_{xxxxx}=0}$
Schlesinger Any ${\displaystyle \displaystyle {\partial A_{i} \over \partial t_{j}}{\left[A_{i},\ A_{j}\right] \over t_{i}-t_{j}},\quad i\neq j,\quad {\partial A_{i} \over \partial t_{i}}=-\sum _{j=1 \atop j\neq i}^{n}{\left[A_{i},\ A_{j}\right] \over t_{i}-t_{j}},\quad 1\leq i,j\leq n}$ isomonodromic deformations
Seiberg–Witten 1+3 ${\displaystyle \displaystyle D^{A}\varphi =0,\qquad F_{A}^{+}=\sigma (\varphi )}$ Seiberg–Witten invariants, QFT
Shallow water 1+2 ${\displaystyle \displaystyle \eta _{t}+(\eta u)_{x}+(\eta v)_{y}=0,\ (\eta u)_{t}+\left(\eta u^{2}+{\frac {1}{2}}g\eta ^{2}\right)_{x}+(\eta uv)_{y}=0,\ (\eta v)_{t}+(\eta uv)_{x}+\left(\eta v^{2}+{\frac {1}{2}}g\eta ^{2}\right)_{y}=0}$ shallow water waves
Sine–Gordon 1+1 ${\displaystyle \displaystyle \,\varphi _{tt}-\varphi _{xx}+\sin \varphi =0}$ Solitons, QFT
Sinh–Gordon 1+1 ${\displaystyle \displaystyle u_{xt}=\sinh u}$ Solitons, QFT
Sinh–Poisson 1+n ${\displaystyle \displaystyle \nabla ^{2}u+\sinh u=0}$
Swift–Hohenberg any ${\displaystyle \displaystyle u_{t}=ru-(1+\nabla ^{2})^{2}u+N(u)}$ pattern forming
Thomas equation 2 ${\displaystyle \displaystyle u_{xy}+\alpha u_{x}+\beta u_{y}+\gamma u_{x}u_{y}=0}$
Thirring model 1+1 ${\displaystyle \displaystyle iu_{x}+v+u|v|^{2}=0}$, ${\displaystyle \displaystyle iv_{t}+u+v|u|^{2}=0}$ Dirac field, QFT
Toda lattice any ${\displaystyle \displaystyle \nabla ^{2}\log u_{n}=u_{n+1}-2u_{n}+u_{n-1}}$
Veselov–Novikov equation 1+2 ${\displaystyle \displaystyle (\partial _{t}+\partial _{z}^{3}+\partial _{\bar {z}}^{3})v+\partial _{z}(uv)+\partial _{\bar {z}}(uw)=0}$, ${\displaystyle \displaystyle \partial _{\bar {z}}u=3\partial _{z}v}$, ${\displaystyle \displaystyle \partial _{z}w=3\partial _{\bar {z}}v}$ shallow water waves
Vorticity equation ${\displaystyle {\frac {\partial {\boldsymbol {\omega }}}{\partial t}}+(\mathbf {u} \cdot \nabla ){\boldsymbol {\omega }}=({\boldsymbol {\omega }}\cdot \nabla )\mathbf {u} -{\boldsymbol {\omega }}(\nabla \cdot \mathbf {u} )+{\frac {1}{\rho ^{2}}}\nabla \rho \times \nabla p+\nabla \times \left({\frac {\nabla \cdot \tau }{\rho }}\right)+\nabla \times \left({\frac {\mathbf {f} }{\rho }}\right),\ {\boldsymbol {\omega }}=\nabla \times \mathbf {u} }$ Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu 1+1 ${\displaystyle \displaystyle iu_{t}+((1+|u|^{2})^{-1/2}u)_{xx}=0}$
WDVV equations Any ${\displaystyle \displaystyle \sum _{\sigma ,\tau =1}^{n}\left({\partial ^{3}F \over \partial t^{\alpha }t^{\beta }t^{\sigma }}\eta ^{\sigma \tau }{\partial ^{3}F \over \partial t^{\mu }t^{\nu }t^{\tau }}\right)}$ ${\displaystyle \displaystyle =\sum _{\sigma ,\tau =1}^{n}\left({\partial ^{3}F \over \partial t^{\alpha }t^{\nu }t^{\sigma }}\eta ^{\sigma \tau }{\partial ^{3}F \over \partial t^{\mu }t^{\beta }t^{\tau }}\right)}$ Topological field theory, QFT
WZW model 1+1 ${\displaystyle S_{k}(\gamma )=-\,{\frac {k}{8\pi }}\int _{S^{2}}d^{2}x\,{\mathcal {K}}(\gamma ^{-1}\partial ^{\mu }\gamma \,,\,\gamma ^{-1}\partial _{\mu }\gamma )+2\pi k\,S^{\mathrm {W} Z}(\gamma )}$

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

QFT
Whitham equation 1+1 ${\displaystyle \displaystyle \eta _{t}+\alpha \eta \eta _{x}+\int _{-\infty }^{+\infty }K(x-\xi )\,\eta _{\xi }(\xi ,t)\,{\text{d}}\xi =0}$ water waves
Williams spray equation ${\displaystyle {\frac {\partial f_{j}}{\partial t}}+\nabla _{x}\cdot (\mathbf {v} f_{j})+\nabla _{v}\cdot (F_{j}f_{j})=-{\frac {\partial }{\partial r}}(R_{j}f_{j})-{\frac {\partial }{\partial T}}(E_{j}f_{j})+Q_{j}+\Gamma _{j},\ F_{j}={\dot {\mathbf {v} }},\ R_{j}={\dot {r}},\ E_{j}={\dot {T}},\ j=1,2,...,M}$ Combustion
Yamabe n ${\displaystyle \displaystyle \Delta \varphi +h(x)\varphi =\lambda f(x)\varphi ^{(n+2)/(n-2)}}$ Differential geometry
Yang–Mills equation (source-free) Any ${\displaystyle \displaystyle D_{\mu }F^{\mu \nu }=0,\quad F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },\,A_{\nu }]}$ Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual) 4 ${\displaystyle F_{\alpha \beta }=\pm \varepsilon _{\alpha \beta \mu \nu }F^{\mu \nu },\quad F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },\,A_{\nu }]}$ Instantons, Donaldson theory, QFT
Yukawa equation 1+n ${\displaystyle \displaystyle i\partial _{t}^{}u+\Delta u=-Au,\quad \displaystyle \Box A=m_{}^{2}A+|u|^{2}}$ Meson-nucleon interactions, QFT
Zakharov system 1+3 ${\displaystyle \displaystyle i\partial _{t}^{}u+\Delta u=un,\quad \displaystyle \Box n=-\Delta (|u|_{}^{2})}$ Langmuir waves
Zakharov–Schulman 1+3 ${\displaystyle \displaystyle iu_{t}+L_{1}u=\varphi u,\quad \displaystyle L_{2}\varphi =L_{3}(|u|^{2})}$ Acoustic waves
Zoomeron 1+1 ${\displaystyle \displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx}+2(u^{2})_{xt}=0}$ Solitons
φ4 equation 1+1 ${\displaystyle \displaystyle \varphi _{tt}-\varphi _{xx}-\varphi +\varphi ^{3}=0}$ QFT
σ-model 1+1 ${\displaystyle \displaystyle {\mathbf {v} }_{xt}+({\mathbf {v} }_{x}{\mathbf {v} }_{t}){\mathbf {v} }=0}$ Harmonic maps, integrable systems, QFT