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.

Name	Dim	Equation	Applications
Bateman-Burgers equation	1+1	${\displaystyle {\displaystyle u_t+u{u}_x=\nu u_{xx}}}$	Fluid mechanics
Benjamin–Bona–Mahony	1+1	${\displaystyle {\displaystyle u_t+u_x+u{u}_x-u_{xxt}=0}}$	Fluid mechanics
Benjamin–Ono	1+1	${\displaystyle {\displaystyle u_t+H{u}_{xx}+u{u}_x=0}}$	internal waves in deep water
Boomeron	1+1	${\displaystyle {\displaystyle u_t=\mathbf{𝐛}\cdot {\mathbf{𝐯}}_x,{\displaystyle {\mathbf{𝐯}}_{xt}=u_{xx}\mathbf{𝐛}+\mathbf{𝐚}\times {\mathbf{𝐯}}_x-2\mathbf{𝐯}\times (\mathbf{𝐯}\times \mathbf{𝐛})}}}$	Solitons
Boltzmann equation	1+6	${\displaystyle \frac{\partial f_i}{\partial t}+\frac{{\mathbf{𝐩}}_i}{m_i}\cdot \mathrm{\nabla }{f}_i+\mathbf{𝐅}\cdot \frac{\partial f_i}{\partial {\mathbf{𝐩}}_i}={\left(\frac{\partial f_i}{\partial t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}},{\left(\frac{\partial f_i}{\partial t}\right)}_{\mathrm{c}\mathrm{o}\mathrm{l}\mathrm{l}}={\displaystyle \sum _{j=1}^n}\iint g_{ij}{I}_{ij}(g_{ij},\mathrm{\Omega })[f{\text{'}}_{i}f{\text{'}}_j-f_i{f}_j]d\mathrm{\Omega }{d}^3{\mathbf{𝐩}}^{\prime }}$	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	${\displaystyle {\displaystyle u_{tt}-u_{xx}-2\alpha (u{u}_x{)}_x-\beta u_{xxtt}=0}}$	Fluid mechanics
Buckmaster	1+1	${\displaystyle {\displaystyle u_t=(u^4{)}_{xx}+(u^3{)}_x}}$	Thin viscous fluid sheet flow
Cahn–Hilliard equation	Any	${\displaystyle {\displaystyle c_t=D{\mathrm{\nabla }}^2(c^3-c-\gamma {\mathrm{\nabla }}^{2}c)}}$	Phase separation
Calabi flow	Any	${\displaystyle \frac{\partial g_{ij}}{\partial t}=(\mathrm{\Delta }R)g_{ij}}$	Calabi–Yau manifolds
Camassa–Holm	1+1	${\displaystyle u_t+2\kappa u_x-u_{xxt}+3u{u}_x=2u_x{u}_{xx}+u{u}_{xxx}}$	Peakons
Carleman	1+1	${\displaystyle {\displaystyle u_t+u_x=v^2-u^2=v_x-v_t}}$
Cauchy momentum	any	${\displaystyle {\displaystyle \rho (\frac{\partial \mathbf{𝐯}}{\partial t}+\mathbf{𝐯}\cdot \mathrm{\nabla }\mathbf{𝐯})=\mathrm{\nabla }\cdot \sigma +\rho \mathbf{𝐟}}}$	Momentum transport
Chafee–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	${\displaystyle {\displaystyle \det ({\partial }_{i\overline{j}}\phi )=}}$ lower order terms	Calabi conjecture
Constant astigmatism	1+1	${\displaystyle z_{yy}+{\left(\frac{1}{z}\right)}_{xx}+2=0}$	Differential geometry
Davey–Stewartson	1+2	${\displaystyle {\displaystyle i{u}_t+c_0{u}_{xx}+u_{yy}=c_1|u{|}^{2}u+c_{2}u{\phi }_x,{\displaystyle {\phi }_{xx}+c_3{\phi }_{yy}=(|u{|}^2{)}_x}}}$	Finite depth waves
Degasperis–Procesi	1+1	${\displaystyle {\displaystyle u_t-u_{xxt}+4u{u}_x=3u_x{u}_{xx}+u{u}_{xxx}}}$	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	${\displaystyle {\displaystyle u_t=3w{w}_x,{\displaystyle w_t=2w_{xxx}+2u{w}_x+u_{x}w}}}$
Dym equation	1+1	${\displaystyle {\displaystyle u_t=u^3{u}_{xxx}.}}$	Solitons
Eckhaus equation	1+1	${\displaystyle i{u}_t+u_{xx}+2|u{|}_x^{2}u+|u{|}^{4}u=0}$	Integrable systems
Eikonal equation	any	${\displaystyle {\displaystyle |\mathrm{\nabla }u(x)|=F(x),x\in \mathrm{\Omega }}}$	optics
Einstein field equations	Any	${\displaystyle {\displaystyle R_{\mu \nu }-\frac{1}{2}R{g}_{\mu \nu }+\mathrm{\Lambda }{g}_{\mu \nu }=\frac{8\pi G}{c^4}{T}_{\mu \nu }}}$	General relativity
Ernst equation	2	${\displaystyle {\displaystyle \mathrm{\Re }(u)(u_{rr}+u_r/r+u_{zz})=(u_r{)}^2+(u_z{)}^2}}$
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	${\displaystyle \frac{\partial \rho }{\partial t}+\mathrm{\nabla }\cdot (\rho \mathbf{𝐮})=0,\rho (\frac{\partial \mathbf{𝐮}}{\partial t}+\mathbf{𝐯}\cdot \mathrm{\nabla }\mathbf{𝐯})=-\mathrm{\nabla }p+\rho \mathbf{𝐟},\frac{\partial s}{\partial t}+\mathbf{𝐯}\cdot \mathrm{\nabla }s=0}$	non-viscous fluids
Fisher's equation	1+1	${\displaystyle {\displaystyle u_t=u(1-u)+u_{xx}}}$	Gene propagation
FitzHugh–Nagumo model	1+1	${\displaystyle {\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w,{\displaystyle w_t=\epsilon u}}}$	Biological neuron model
Föppl–von Kármán equations		${\displaystyle \frac{E{h}^3}{12(1-{\nu }^2)}{\mathrm{\nabla }}^{4}w-h\frac{\partial }{\partial x_{\beta }}\left({\sigma }_{\alpha \beta }\frac{\partial w}{\partial x_{\alpha }}\right)=P,\frac{\partial {\sigma }_{\alpha \beta }}{\partial x_{\beta }}=0}$	Solid Mechanics
Fujita–Storm equation		$${\displaystyle u_t=a(u^{-2}{u}_x{)}_x}$$

Name	Dim	Equation	Applications
G equation	1+3	${\displaystyle G_t+\mathbf{𝐯}\cdot \mathrm{\nabla }G=S_L(G)|\mathrm{\nabla }G|}$	turbulent combustion
Generic scalar transport	1+3	${\displaystyle {\displaystyle {\phi }_t+\mathrm{\nabla }\cdot f(t,x,\phi ,\mathrm{\nabla }\phi )=g(t,x,\phi )}}$	transport
Ginzburg–Landau	1+3	${\displaystyle {\displaystyle \alpha \psi +\beta |\psi {|}^2\psi +\frac{1}{2m}{(-i\hslash \mathrm{\nabla }-2e\mathbf{𝐀})}^2\psi =0}}$	Superconductivity
Gross–Pitaevskii	1 + n	${\displaystyle {\displaystyle i{\partial }_t\psi =(-\frac{1}{2}{\mathrm{\nabla }}^2+V(x)+g|\psi {|}^2)\psi }}$	Bose–Einstein condensate
Gyrokinetics equation	1 + 5	${\displaystyle {\displaystyle \frac{\partial h_s}{\partial t}}+(v_{||}\hat{b}+{\overrightarrow{V}}_{ds}+{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi })\cdot {\overrightarrow{\mathrm{\nabla }}}_{\overrightarrow{R}}{h}_s-\sum _{s^{\prime }}{⟨C[h_s,h_{s^{\prime }}]⟩}_{\phi }=\frac{Z_{s}e{f}_{s0}}{T_s}\frac{\partial {⟨\varphi ⟩}_{\phi }}{\partial t}-\frac{\partial f_{s0}}{\partial \psi }{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi }\cdot \overrightarrow{\mathrm{\nabla }}\psi }$	Microturbulence in plasma
Guzmán	1 + n	${\displaystyle {\displaystyle J_t+g{J}_x+1/2{\sigma }^2{J}_{xx}-\lambda {\sigma }^2(J_x{)}^2+f=0}}$	Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation	Any	${\displaystyle {\displaystyle i{\partial }_{t}u+\mathrm{\Delta }u=(\pm |x{|}^{-n}|u{|}^2)u}}$
Hasegawa–Mima	1+3	${\displaystyle {\displaystyle 0=\frac{\partial }{\partial t}({\mathrm{\nabla }}^2\phi -\phi )-[(\mathrm{\nabla }\phi \times \widehat{\mathbf{𝐳}})\cdot \mathrm{\nabla }][{\mathrm{\nabla }}^2\phi -\ln \left(\frac{n_0}{{\omega }_{ci}}\right)]}}$	Turbulence in plasma
Heisenberg ferromagnet	1+1	${\displaystyle {\displaystyle {\mathbf{𝐒}}_t=\mathbf{𝐒}\wedge {\mathbf{𝐒}}_{xx}.}}$	Magnetism
Hicks	1+1	${\displaystyle {\psi }_{rr}-{\psi }_r/r+{\psi }_{zz}=r^2\mathrm{d}H/\mathrm{d}\psi -\mathrm{\Gamma }\mathrm{d}\mathrm{\Gamma }/\mathrm{d}\psi }$	Fluid dynamics
Hunter–Saxton	1+1	${\displaystyle {\displaystyle {(u_t+u{u}_x)}_x=\frac{1}{2}{u}_x^2}}$	Liquid crystals
Ishimori equation	1+2	${\displaystyle {\displaystyle {\mathbf{𝐒}}_t=\mathbf{𝐒}\wedge ({\mathbf{𝐒}}_{xx}+{\mathbf{𝐒}}_{yy})+u_x{\mathbf{𝐒}}_y+u_y{\mathbf{𝐒}}_x,{\displaystyle u_{xx}-{\alpha }^2{u}_{yy}=-2{\alpha }^2\mathbf{𝐒}\cdot ({\mathbf{𝐒}}_x\wedge {\mathbf{𝐒}}_y)}}}$	Integrable systems
Kadomtsev –Petviashvili	1+2	${\displaystyle {\displaystyle {\partial }_x({\partial }_{t}u+u{\partial }_{x}u+{\epsilon }^2{\partial }_{xxx}u)+\lambda {\partial }_{yy}u=0}}$	Shallow water waves
Kardar–Parisi–Zhang equation	1+3	${\displaystyle {\displaystyle h_t=\nu {\mathrm{\nabla }}^{2}h+\lambda (\mathrm{\nabla }h{)}^2/2+\eta }}$	Stochastics
von Karman	2	${\displaystyle {\displaystyle {\mathrm{\nabla }}^{4}u=E(w_{xy}^2-w_{xx}{w}_{yy}),{\mathrm{\nabla }}^{4}w=a+b(u_{yy}{w}_{xx}+u_{xx}{w}_{yy}-2u_{xy}{w}_{xy})}}$
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	${\displaystyle {\displaystyle {\mathrm{\nabla }}^{2}s=(|\mathbf{𝐚}{|}^2+1)s,{\mathrm{\nabla }}^2\mathbf{𝐚}=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐚})+s^2\mathbf{𝐚}}}$
Klein–Gordon (nonlinear)	any	${\displaystyle {\mathrm{\nabla }}^{2}u+\lambda u^p=0}$	Relativistic quantum mechanics
Khokhlov–Zabolotskaya	1+2	${\displaystyle {\displaystyle u_{xt}-(u{u}_x{)}_x=u_{yy}}}$
Korteweg–de Vries (KdV)	1+1	${\displaystyle {\displaystyle u_t+u_{xxx}-6u{u}_x=0}}$	Shallow waves, Integrable systems
KdV (super)	1+1	${\displaystyle {\displaystyle u_t=6u{u}_x-u_{xxx}+3w{w}_{xx},w_t=3u_{x}w+6u{w}_x-4w_{xxx}}}$
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation	1 + n	${\displaystyle {\displaystyle u_t+{\mathrm{\nabla }}^{4}u+{\mathrm{\nabla }}^{2}u+\frac{1}{2}|\mathrm{\nabla }u{|}^2=0}}$	Combustion

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	${\displaystyle {\displaystyle 2u_{tx}+u_x{u}_{xx}-u_{yy}=0}}$
Liouville equation	any	${\displaystyle {\displaystyle {\mathrm{\nabla }}^{2}u+e^{\lambda u}=0}}$
Liouville–Bratu–Gelfand equation	any	${\displaystyle {\mathrm{\nabla }}^2\psi +\lambda e^{\psi }=0}$	combustion, astrophysics
Logarithmic Schrödinger equation	any	${\displaystyle i\frac{\partial \psi }{\partial t}+\mathrm{\Delta }\psi +\psi \ln |\psi {|}^2=0.}$	Superfluids, Quantum gravity
Minimal surface	3	${\displaystyle {\displaystyle \mathrm{div}(Du/\sqrt{1+|Du{|}^2})=0}}$	minimal surfaces
Monge–Ampère	any	${\displaystyle {\displaystyle \det ({\partial }_{ij}\phi )=}}$ lower order terms
Navier–Stokes (and its derivation)	1+3	${\displaystyle {\displaystyle \rho (\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j})=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}[\mu (\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})+\lambda \frac{\partial v_k}{\partial x_k}]+\rho f_i}}$ + 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	${\displaystyle {\displaystyle i{\partial }_t\psi =-\frac{1}{2}{\displaystyle \underset{x}{\overset{2}{\partial }}}\psi +\kappa |\psi {|}^2\psi }}$	optics, water waves
Nonlinear Schrödinger (derivative)	1+1	${\displaystyle {\displaystyle i{\partial }_t\psi =-\frac{1}{2}{\displaystyle \underset{x}{\overset{2}{\partial }}}\psi +{\partial }_x(i\kappa |\psi {|}^2\psi )}}$	optics, water waves
Omega equation	1+3	${\displaystyle {\displaystyle {\mathrm{\nabla }}^2\omega +\frac{f^2}{\sigma }\frac{{\partial }^2\omega }{\partial p^2}}}$ ${\displaystyle {\displaystyle =\frac{f}{\sigma }\frac{\partial }{\partial p}{\mathbf{𝐕}}_g\cdot {\mathrm{\nabla }}_p({\zeta }_g+f)+\frac{R}{\sigma p}{\displaystyle \underset{p}{\overset{2}{\mathrm{\nabla }}}}({\mathbf{𝐕}}_g\cdot {\mathrm{\nabla }}_{p}T)}}$	atmospheric physics
Plateau	2	${\displaystyle {\displaystyle (1+u_y^2)u_{xx}-2u_x{u}_y{u}_{xy}+(1+u_x^2)u_{yy}=0}}$	minimal surfaces
Pohlmeyer–Lund–Regge	2	${\displaystyle {\displaystyle u_{xx}-u_{yy}\pm \sin u\cos u+\frac{\cos u}{{\sin }^{3}u}(v_x^2-v_y^2)=0,{\displaystyle (v_x{\cot }^{2}u{)}_x=(v_y{\cot }^{2}u{)}_y}}}$
Porous medium	1+n	${\displaystyle {\displaystyle u_t=\mathrm{\Delta }(u^{\gamma })}}$	diffusion
Prandtl	1+2	${\displaystyle {\displaystyle u_t+u{u}_x+v{u}_y=U_t+U{U}_x+\frac{\mu }{\rho }{u}_{yy}}}$, ${\displaystyle {\displaystyle u_x+v_y=0}}$	boundary layer

Name	Dim	Equation	Applications
Rayleigh	1+1	${\displaystyle {\displaystyle u_{tt}-u_{xx}=\epsilon (u_t-u_t^3)}}$
Ricci flow	Any	${\displaystyle {\displaystyle {\partial }_t{g}_{ij}=-2R_{ij}}}$	Poincaré conjecture
Richards equation	1+3	${\displaystyle {\displaystyle {\theta }_t={[K(\theta )({\psi }_z+1)]}_z}}$	Variably saturated flow in porous media
Rosenau–Hyman	1+1	${\displaystyle u_t+a{\left(u^n\right)}_x+{\left(u^n\right)}_{xxx}=0}$	compacton solutions
Sawada–Kotera	1+1	${\displaystyle {\displaystyle u_t+45u^2{u}_x+15u_x{u}_{xx}+15u{u}_{xxx}+u_{xxxxx}=0}}$
Sack–Schamel equation	1+1	${\displaystyle \ddot{V}+{\partial }_{\eta }\left[\frac{1}{1-\ddot{V}}{\partial }_{\eta }\left(\frac{1-\ddot{V}}{V}\right)\right]=0}$	plasmas
Schamel equation	1+1	${\displaystyle {\varphi }_t+(1+b\sqrt{\varphi }){\varphi }_x+{\varphi }_{xxx}=0}$	plasmas, solitons, optics
Schlesinger	Any	${\displaystyle {\displaystyle \frac{\partial A_i}{\partial t_j}\frac{[A_i,A_j]}{t_i-t_j},i\ne j,\frac{\partial A_i}{\partial t_i}=-{\displaystyle \sum _{\genfrac{}{}{0ex}{}{j=1}{j\ne i}}^n}\frac{[A_i,A_j]}{t_i-t_j},1\le i,j\le n}}$	isomonodromic deformations
Seiberg–Witten	1+3	${\displaystyle {\displaystyle D^A\phi =0,F_A^{+}=\sigma (\phi )}}$	Seiberg–Witten invariants, QFT
Shallow water	1+2	${\displaystyle {\displaystyle {\eta }_t+(\eta u{)}_x+(\eta v{)}_y=0,(\eta u{)}_t+{(\eta u^2+\frac{1}{2}g{\eta }^2)}_x+(\eta uv{)}_y=0,(\eta v{)}_t+(\eta uv{)}_x+{(\eta v^2+\frac{1}{2}g{\eta }^2)}_y=0}}$	shallow water waves
Sine–Gordon	1+1	${\displaystyle {\displaystyle {\phi }_{tt}-{\phi }_{xx}+\sin \phi =0}}$	Solitons, QFT
Sinh–Gordon	1+1	${\displaystyle {\displaystyle u_{xt}=\sinh u}}$	Solitons, QFT
Sinh–Poisson	1+n	${\displaystyle {\displaystyle {\mathrm{\nabla }}^{2}u+\sinh u=0}}$	Fluid Mechanics
Swift–Hohenberg	any	${\displaystyle {\displaystyle u_t=ru-(1+{\mathrm{\nabla }}^2{)}^{2}u+N(u)}}$	pattern forming
Thomas	2	${\displaystyle {\displaystyle u_{xy}+\alpha u_x+\beta u_y+\gamma u_x{u}_y=0}}$
Thirring	1+1	${\displaystyle {\displaystyle i{u}_x+v+u|v{|}^2=0}}$, ${\displaystyle {\displaystyle i{v}_t+u+v|u{|}^2=0}}$	Dirac field, QFT
Toda lattice	any	${\displaystyle {\displaystyle {\mathrm{\nabla }}^2\log u_n=u_{n+1}-2u_n+u_{n-1}}}$
Veselov–Novikov	1+2	${\displaystyle {\displaystyle ({\partial }_t+{\displaystyle \underset{z}{\overset{3}{\partial }}}+{\displaystyle \underset{\overline{z}}{\overset{3}{\partial }}})v+{\partial }_z(uv)+{\partial }_{\overline{z}}(uw)=0}}$, ${\displaystyle {\displaystyle {\partial }_{\overline{z}}u=3{\partial }_{z}v}}$, ${\displaystyle {\displaystyle {\partial }_{z}w=3{\partial }_{\overline{z}}v}}$	shallow water waves
Vorticity equation		${\displaystyle \frac{\partial \mathit{\omega }}{\partial t}+(\mathbf{𝐮}\cdot \mathrm{\nabla })\mathit{\omega }=(\mathit{\omega }\cdot \mathrm{\nabla })\mathbf{𝐮}-\mathit{\omega }(\mathrm{\nabla }\cdot \mathbf{𝐮})+\frac{1}{{\rho }^2}\mathrm{\nabla }\rho \times \mathrm{\nabla }p+\mathrm{\nabla }\times \left(\frac{\mathrm{\nabla }\cdot \tau }{\rho }\right)+\mathrm{\nabla }\times \left(\frac{\mathbf{𝐟}}{\rho }\right),\mathit{\omega }=\mathrm{\nabla }\times \mathbf{𝐮}}$	Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu	1+1	${\displaystyle {\displaystyle i{u}_t+((1+|u{|}^2{)}^{-1/2}u{)}_{xx}=0}}$
WDVV equations	Any	${\displaystyle {\displaystyle {\displaystyle \sum _{\sigma ,\tau =1}^n}\left(\frac{{\partial }^{3}F}{\partial t^{\alpha }{t}^{\beta }{t}^{\sigma }}{\eta }^{\sigma \tau }\frac{{\partial }^{3}F}{\partial t^{\mu }{t}^{\nu }{t}^{\tau }}\right)}}$ ${\displaystyle {\displaystyle ={\displaystyle \sum _{\sigma ,\tau =1}^n}\left(\frac{{\partial }^{3}F}{\partial t^{\alpha }{t}^{\nu }{t}^{\sigma }}{\eta }^{\sigma \tau }\frac{{\partial }^{3}F}{\partial t^{\mu }{t}^{\beta }{t}^{\tau }}\right)}}$	Topological field theory, QFT
WZW model	1+1	${\displaystyle S_k(\gamma )=-\frac{k}{8\pi }\underset{S^2}{\int }{d}^{2}x{𝒦}({\gamma }^{-1}{\partial }^{\mu }\gamma ,{\gamma }^{-1}{\partial }_{\mu }\gamma )+2\pi k{S}^{\mathrm{W}Z}(\gamma )}$ ${\displaystyle S^{\mathrm{W}Z}(\gamma )=-\frac{1}{48{\pi }^2}\underset{B^3}{\int }{d}^{3}y{\epsilon }^{ijk}{𝒦}({\gamma }^{-1}\frac{\partial \gamma }{\partial y^i},[{\gamma }^{-1}\frac{\partial \gamma }{\partial y^j},{\gamma }^{-1}\frac{\partial \gamma }{\partial y^k}])}$	QFT
Whitham equation	1+1	${\displaystyle {\displaystyle {\eta }_t+\alpha \eta {\eta }_x+{\displaystyle \underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}}K(x-\xi ){\eta }_{\xi }(\xi ,t)\text{d}\xi =0}}$	water waves
Williams spray equation		${\displaystyle \frac{\partial f_j}{\partial t}+{\mathrm{\nabla }}_x\cdot (\mathbf{𝐯}{f}_j)+{\mathrm{\nabla }}_v\cdot (F_j{f}_j)=-\frac{\partial }{\partial r}(R_j{f}_j)-\frac{\partial }{\partial T}(E_j{f}_j)+Q_j+{\mathrm{\Gamma }}_j,F_j=\dot{\mathbf{𝐯}},R_j=\dot{r},E_j=\dot{T},j=1,2,...,M}$	Combustion
Yamabe	n	${\displaystyle {\displaystyle \mathrm{\Delta }\phi +h(x)\phi =\lambda f(x){\phi }^{(n+2)/(n-2)}}}$	Differential geometry
Yang–Mills (source-free)	Any	${\displaystyle {\displaystyle D_{\mu }{F}^{\mu \nu }=0,F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },A_{\nu }]}}$	Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual)	4	${\displaystyle F_{\alpha \beta }=\pm {\epsilon }_{\alpha \beta \mu \nu }{F}^{\mu \nu },F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },A_{\nu }]}$	Instantons, Donaldson theory, QFT
Yukawa	1+n	${\displaystyle {\displaystyle i{\displaystyle \underset{t}{\overset{}{\partial }}}u+\mathrm{\Delta }u=-Au,{\displaystyle ◻A=m_{}^{2}A+|u{|}^2}}}$	Meson-nucleon interactions, QFT
Zakharov system	1+3	${\displaystyle {\displaystyle i{\displaystyle \underset{t}{\overset{}{\partial }}}u+\mathrm{\Delta }u=un,{\displaystyle ◻n=-\mathrm{\Delta }(|u{|}_{}^2)}}}$	Langmuir waves
Zakharov–Schulman	1+3	${\displaystyle {\displaystyle i{u}_t+L_{1}u=\phi u,{\displaystyle L_2\phi =L_3(|u{|}^2)}}}$	Acoustic waves
Zeldovich–Frank-Kamenetskii equation	1+3	${\displaystyle {\displaystyle u_t=D{\mathrm{\nabla }}^{2}u+\frac{{\beta }^2}{2}u(1-u)e^{-\beta (1-u)}}}$	Combustion
Zoomeron	1+1	${\displaystyle {\displaystyle (u_{xt}/u{)}_{tt}-(u_{xt}/u{)}_{xx}+2(u^2{)}_{xt}=0}}$	Solitons
φ4 equation	1+1	${\displaystyle {\displaystyle {\phi }_{tt}-{\phi }_{xx}-\phi +{\phi }^3=0}}$	QFT
σ-model	1+1	${\displaystyle {\displaystyle {\mathbf{𝐯}}_{xt}+({\mathbf{𝐯}}_x{\mathbf{𝐯}}_t)\mathbf{𝐯}=0}}$	Harmonic maps, integrable systems, QFT

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