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

G–K

Name	Dim	Equation	Applications
G equation	1+3	$G_t+\mathbf{𝐯}\cdot \mathrm{\nabla }G=S_L(G)|\mathrm{\nabla }G|$	turbulent combustion
Generic scalar transport	1+3	${\phi }_t+\mathrm{\nabla }\cdot f(t,x,\phi ,\mathrm{\nabla }\phi )=g(t,x,\phi )$	transport
Ginzburg–Landau	1+3	$\alpha \psi +\beta |\psi {|}^2\psi +\frac{1}{2m}{(-i\hslash \mathrm{\nabla }-2e\mathbf{𝐀})}^2\psi =0$	Superconductivity
Gross–Pitaevskii	1 + n	$i{\partial }_t\psi =(-\frac{1}{2}{\mathrm{\nabla }}^2+V(x)+g|\psi {|}^2)\psi$	Bose–Einstein condensate
Gyrokinetics equation	1 + 5	$\frac{\partial h_s}{\partial t}+(v_{||}\hat{b}+{\overrightarrow{V}}_{ds}+{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi })\cdot {\overrightarrow{\mathrm{\nabla }}}_{\overrightarrow{R}}{h}_s-\sum _{s^{\prime }}{⟨C[h_s,h_{s^{\prime }}]⟩}_{\phi }=\frac{Z_{s}e{f}_{s0}}{T_s}\frac{\partial {⟨\varphi ⟩}_{\phi }}{\partial t}-\frac{\partial f_{s0}}{\partial \psi }{⟨{\overrightarrow{V}}_{\varphi }⟩}_{\phi }\cdot \overrightarrow{\mathrm{\nabla }}\psi$	Microturbulence in plasma
Guzmán	1 + n	$J_t+g{J}_x+1/2{\sigma }^2{J}_{xx}-\lambda {\sigma }^2(J_x{)}^2+f=0$	Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation	Any	$i{\partial }_{t}u+\mathrm{\Delta }u=(\pm |x{|}^{-n}|u{|}^2)u$
Hasegawa–Mima	1+3	$0=\frac{\partial }{\partial t}({\mathrm{\nabla }}^2\phi -\phi )-[(\mathrm{\nabla }\phi \times \widehat{\mathbf{𝐳}})\cdot \mathrm{\nabla }][{\mathrm{\nabla }}^2\phi -\ln \left(\frac{n_0}{{\omega }_{ci}}\right)]$	Turbulence in plasma
Heisenberg ferromagnet	1+1	${\mathbf{𝐒}}_t=\mathbf{𝐒}\wedge {\mathbf{𝐒}}_{xx}.$	Magnetism
Hicks	1+1	${\psi }_{rr}-{\psi }_r/r+{\psi }_{zz}=r^2\mathrm{d}H/\mathrm{d}\psi -\mathrm{\Gamma }\mathrm{d}\mathrm{\Gamma }/\mathrm{d}\psi$	Fluid dynamics
Hunter–Saxton	1+1	${(u_t+u{u}_x)}_x=\frac{1}{2}{u}_x^2$	Liquid crystals
Ishimori equation	1+2	${\mathbf{𝐒}}_t=\mathbf{𝐒}\wedge ({\mathbf{𝐒}}_{xx}+{\mathbf{𝐒}}_{yy})+u_x{\mathbf{𝐒}}_y+u_y{\mathbf{𝐒}}_x,u_{xx}-{\alpha }^2{u}_{yy}=-2{\alpha }^2\mathbf{𝐒}\cdot ({\mathbf{𝐒}}_x\wedge {\mathbf{𝐒}}_y)$	Integrable systems
Kadomtsev –Petviashvili	1+2	${\partial }_x({\partial }_{t}u+u{\partial }_{x}u+{\epsilon }^2{\partial }_{xxx}u)+\lambda {\partial }_{yy}u=0$	Shallow water waves
Kardar–Parisi–Zhang equation	1+3	$h_t=\nu {\mathrm{\nabla }}^{2}h+\lambda (\mathrm{\nabla }h{)}^2/2+\eta$	Stochastics
von Karman	2	${\mathrm{\nabla }}^{4}u=E(w_{xy}^2-w_{xx}{w}_{yy}),{\mathrm{\nabla }}^{4}w=a+b(u_{yy}{w}_{xx}+u_{xx}{w}_{yy}-2u_{xy}{w}_{xy})$
Kaup	1+1	$f_x=2fgc(x-t)=g_t$
Kaup–Kupershmidt	1+1	$u_t=u_{xxxxx}+10u_{xxx}u+25u_{xx}{u}_x+20u^2{u}_x$	Integrable systems
Klein–Gordon–Maxwell	any	${\mathrm{\nabla }}^{2}s=(|\mathbf{𝐚}{|}^2+1)s,{\mathrm{\nabla }}^2\mathbf{𝐚}=\mathrm{\nabla }(\mathrm{\nabla }\cdot \mathbf{𝐚})+s^2\mathbf{𝐚}$
Klein–Gordon (nonlinear)	any	${\mathrm{\nabla }}^{2}u+\lambda u^p=0$	Relativistic quantum mechanics
Khokhlov–Zabolotskaya	1+2	$u_{xt}-(u{u}_x{)}_x=u_{yy}$
Korteweg–de Vries (KdV)	1+1	$u_t+u_{xxx}-6u{u}_x=0$	Shallow waves, Integrable systems
KdV (super)	1+1	$u_t=6u{u}_x-u_{xxx}+3w{w}_{xx},w_t=3u_{x}w+6u{w}_x-4w_{xxx}$
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation	1 + n	$u_t+{\mathrm{\nabla }}^{4}u+{\mathrm{\nabla }}^{2}u+\frac{1}{2}|\mathrm{\nabla }u{|}^2=0$	Combustion

L–Q

Name	Dim	Equation	Applications
Landau–Lifshitz model	1+n	$\frac{\partial \mathbf{𝐒}}{\partial t}=\mathbf{𝐒}\wedge \sum _i\frac{{\partial }^2\mathbf{𝐒}}{\partial x_i^2}+\mathbf{𝐒}\wedge J\mathbf{𝐒}$	Magnetic field in solids
Lin–Tsien equation	1+2	$2u_{tx}+u_x{u}_{xx}-u_{yy}=0$
Liouville equation	any	${\mathrm{\nabla }}^{2}u+e^{\lambda u}=0$
Liouville–Bratu–Gelfand equation	any	${\mathrm{\nabla }}^2\psi +\lambda e^{\psi }=0$	combustion, astrophysics
Logarithmic Schrödinger equation	any	$i\frac{\partial \psi }{\partial t}+\mathrm{\Delta }\psi +\psi \ln |\psi {|}^2=0.$	Superfluids, Quantum gravity
Minimal surface	3	$\mathrm{div}(Du/\sqrt{1+|Du{|}^2})=0$	minimal surfaces
Monge–Ampère	any	$\det ({\partial }_{ij}\phi )=$ lower order terms
Navier–Stokes (and its derivation)	1+3	$\rho (\frac{\partial v_i}{\partial t}+v_j\frac{\partial v_i}{\partial x_j})=-\frac{\partial p}{\partial x_i}+\frac{\partial }{\partial x_j}[\mu (\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i})+\lambda \frac{\partial v_k}{\partial x_k}]+\rho f_i$ + mass conservation: $\frac{\partial \rho }{\partial t}+\frac{\partial \left(\rho v_i\right)}{\partial x_i}=0$ + an equation of state to relate p and ρ, e.g. for an incompressible flow: $\frac{\partial v_i}{\partial x_i}=0$	Fluid flow, gas flow
Nonlinear Schrödinger (cubic)	1+1	$i{\partial }_t\psi =-\frac{1}{2}\underset{x}{\overset{2}{\partial }}\psi +\kappa |\psi {|}^2\psi$	optics, water waves
Nonlinear Schrödinger (derivative)	1+1	$i{\partial }_t\psi =-\frac{1}{2}\underset{x}{\overset{2}{\partial }}\psi +{\partial }_x(i\kappa |\psi {|}^2\psi )$	optics, water waves
Omega equation	1+3	${\mathrm{\nabla }}^2\omega +\frac{f^2}{\sigma }\frac{{\partial }^2\omega }{\partial p^2}$ $=\frac{f}{\sigma }\frac{\partial }{\partial p}{\mathbf{𝐕}}_g\cdot {\mathrm{\nabla }}_p({\zeta }_g+f)+\frac{R}{\sigma p}\underset{p}{\overset{2}{\mathrm{\nabla }}}({\mathbf{𝐕}}_g\cdot {\mathrm{\nabla }}_{p}T)$	atmospheric physics
Plateau	2	$(1+u_y^2)u_{xx}-2u_x{u}_y{u}_{xy}+(1+u_x^2)u_{yy}=0$	minimal surfaces
Pohlmeyer–Lund–Regge	2	$u_{xx}-u_{yy}\pm \sin u\cos u+\frac{\cos u}{{\sin }^{3}u}(v_x^2-v_y^2)=0,(v_x{\cot }^{2}u{)}_x=(v_y{\cot }^{2}u{)}_y$
Porous medium	1+n	$u_t=\mathrm{\Delta }(u^{\gamma })$	diffusion
Prandtl	1+2	$u_t+u{u}_x+v{u}_y=U_t+U{U}_x+\frac{\mu }{\rho }{u}_{yy}$, $u_x+v_y=0$	boundary layer

R–Z, α–ω

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