List of nonlinear partial differential equations

!Name

!Dim

!Equation

!Applications

|1+1

| $\backslash displaystyle\; u\_t+uu\_x=\backslash nu\; u\_\{xx\}$

|Fluid mechanics

|1+1

|$\backslash displaystyle\; u\_t+u\_x+uu\_x-u\_\{xxt\}=0$

|Fluid mechanics

|1+1

| $\backslash displaystyle\; u\_t+Hu\_\{xx\}+uu\_x=0$

| internal waves in deep water

|1+1

|$\backslash displaystyle\; u\_t=\backslash mathbf\{b\}\backslash cdot\backslash mathbf\{v\}\_x,\; \backslash quad$

\displaystyle \mathbf{v}_{xt}=u_{xx}\mathbf{b}+\mathbf{a}\times\mathbf{v}_x-

2\mathbf{v}\times(\mathbf{v}\times\mathbf{b})

|Solitons

|1+6

|$\backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash mathbf\{p\}\_i\}\{m\_i\}\backslash cdot\backslash nabla\; f\_i\; +\; \backslash mathbf\{F\}\backslash cdot\backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; \backslash mathbf\{p\}\_i\}\; =\; \backslash left(\backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; t\}\; \backslash right)\_\backslash mathrm\{coll\},$

$\backslash left(\backslash frac\{\backslash partial\; f\_i\}\{\backslash partial\; t\}\; \backslash right)\_\{\backslash mathrm\{coll\}\}\; =\; \backslash sum\_\{j=1\}^n\; \backslash iint\; g\_\{ij\}\; I\_\{ij\}(g\_\{ij\},\; \backslash Omega)[f\text{'}\_i\; f\text{'}\_j\; -\; f\_if\_j]\; \backslash ,d\backslash Omega\backslash ,d^3\backslash mathbf\{p\text{'}\}$

|Statistical mechanics

|1+1

|$\backslash displaystyle\; (1-u\_t^2)u\_\{xx\}\; +2u\_xu\_tu\_\{xt\}-(1+u\_x^2)u\_\{tt\}=0$

| Electrodynamics

| 1+1

| $\backslash displaystyle\; u\_\{tt\}\; -\; u\_\{xx\}\; -\; u\_\{xxxx\}\; -\; 3(u^2)\_\{xx\}\; =\; 0$

|Fluid mechanics

| 1+1

| $\backslash displaystyle\; u\_\{tt\}-u\_\{xx\}-2\; \backslash alpha\; (u\; u\_x)\_\{x\}-\backslash beta\; u\_\{xxtt\}=0$

|Fluid mechanics

|1+1

|$\backslash displaystyle\; u\_t=(u^4)\_\{xx\}+(u^3)\_x$

|Thin viscous fluid sheet flow

|Any

|$\backslash displaystyle\; c\_t\; =\; D\backslash nabla^2\backslash left(c^3-c-\backslash gamma\backslash nabla^2\; c\backslash right)$

|Phase separation

|Any

|$\backslash frac\{\backslash partial\; g\_\{ij\}\}\{\backslash partial\; t\}=(\backslash Delta\; R)g\_\{ij\}$

|Calabi–Yau manifolds

|1+1

|$u\_t\; +\; 2\backslash kappa\; u\_x\; -\; u\_\{xxt\}\; +\; 3\; u\; u\_x\; =\; 2\; u\_x\; u\_\{xx\}\; +\; u\; u\_\{xxx\}\backslash ,$

|Peakons

|1+1

|$\backslash displaystyle\; u\_t+u\_x=v^2-u^2=v\_x-v\_t$

|

|any

|$\backslash displaystyle\; \backslash rho\; \backslash left(\backslash frac\{\backslash partial\; \backslash mathbf\{v\}\}\{\backslash partial\; t\}\; +\; \backslash mathbf\{v\}\; \backslash cdot\; \backslash nabla\; \backslash mathbf\{v\}\backslash right)\; =\; \backslash nabla\; \backslash cdot\; \backslash sigma\; +\; \backslash rho\backslash mathbf\{f\}$

|Momentum transport

|

| $u\_t-u\_\{xx\}+\backslash lambda(u^3-u)=0$

|

|any

|$x\backslash cdot\; Du+f(Du)=u$

|Differential geometry

|1+1

|$(\backslash theta\_t-\backslash gamma\; \backslash delta\; e^\{\backslash theta\})\_\{tt\}=\backslash nabla^2(\backslash theta\_t-\backslash delta\; e^\backslash theta)$

|Combustion

|Any

|$\backslash displaystyle\; \backslash det(\backslash partial\_\{i\backslash bar\; j\}\backslash varphi)\; =$ lower order terms

|Calabi conjecture

|1+1

|$z\_\{yy\}\; +\; \backslash left(\backslash frac\{1\}\{z\}\backslash right)\_\{xx\}\; +\; 2\; =\; 0$

|Differential geometry

|1+2

|$\backslash displaystyle\; i\; u\_t\; +\; c\_0\; u\_\{xx\}\; +\; u\_\{yy\}\; =\; c\_1\; |u|^2\; u\; +\; c\_2\; u\; \backslash varphi\_x,\; \backslash quad$

\displaystyle \varphi_{xx} + c_3 \varphi_{yy} = ( |u|^2 )_x

|Finite depth waves

|1+1

|$\backslash displaystyle\; u\_t\; -\; u\_\{xxt\}\; +\; 4u\; u\_x\; =\; 3\; u\_x\; u\_\{xx\}\; +\; u\; u\_\{xxx\}$

|Peakons

|1+1

|$\backslash displaystyle\; u\_t=(u^2-u\_x+2w)\_x$, $w\_t=(2uw+w\_x)\_x$

|

|1+1

|$\backslash displaystyle\; u\_t=3ww\_x,\; \backslash quad$

\displaystyle w_t=2w_{xxx}+2uw_x+u_xw

|

|1+1

|$\backslash displaystyle\; u\_t\; =\; u^3u\_\{xxx\}.\backslash ,$

|Solitons

|1+1

|$iu\_t+u\_\{xx\}+2|u|^2\_xu+|u|^4u=0$

|Integrable systems

|any

|$\backslash displaystyle\; |\backslash nabla\; u(x)|=F(x),\; \backslash \; x\backslash in\; \backslash Omega$

|optics

| Any

|$\backslash displaystyle\; R\_\{\backslash mu\backslash nu\}\; -\; \{\backslash textstyle\; 1\; \backslash over\; 2\}R\backslash ,g\_\{\backslash mu\backslash nu\}+\backslash Lambda\; g\_\{\backslash mu\backslash nu\}\; =\; \backslash frac\{8\backslash pi\; G\}\{c^\{4\}\}\; T\_\{\backslash mu\backslash nu\}$

|General relativity

|1+1

|$\backslash varphi\_t\; =\; (\backslash varphi\_x\; +\; a\backslash varphi\_x^3)\_x$

|Fluid dynamics

|2

|$\backslash displaystyle\; \backslash Re(u)(u\_\{rr\}+u\_r/r+u\_\{zz\})\; =\; (u\_r)^2+(u\_z)^2$

|

|

| $U\_\{tyyy\}+\backslash beta\; U\_y\; U\_\{yt\}+\backslash beta\; U\_\{yy\}\; U\_t+U\_\{tt\}=0\; \backslash text\{\; in\; which\; \}\; U=u(x,y,t)$

|

|1+3

|$\backslash frac\{\backslash partial\backslash rho\}\{\backslash partial\; t\}+\backslash nabla\backslash cdot(\backslash rho\backslash mathbf\{u\})=0,\backslash quad\; \backslash rho\backslash left(\backslash frac\{\backslash partial\backslash mathbf\{u\}\}\{\backslash partial\; t\}+\backslash mathbf\{v\}\backslash cdot\backslash nabla\backslash mathbf\{v\}\backslash right)=-\backslash nabla\; p\; +\; \backslash rho\backslash mathbf\{f\},\backslash quad\; \backslash frac\{\backslash partial\; s\}\{\backslash partial\; t\}+\backslash mathbf\{v\}\backslash cdot\backslash nabla\; s=0$

|non-viscous fluids

|1+1

|$\backslash displaystyle\; u\_t=u(1-u)+u\_\{xx\}$

|Gene propagation

|1+1

|$\backslash displaystyle\; u\_t=u\_\{xx\}+u(u-a)(1-u)+w,\; \backslash quad$

\displaystyle w_t=\varepsilon u

|Biological neuron model

|

| $\backslash frac\{Eh^3\}\{12(1-\backslash nu^2)\}\backslash nabla^4\; w-h\backslash frac\{\backslash partial\}\{\backslash partial\; x\_\backslash beta\}\backslash left(\backslash sigma\_\{\backslash alpha\backslash beta\}\backslash frac\{\backslash partial\; w\}\{\backslash partial\; x\_\backslash alpha\}\backslash right)=P,\; \backslash quad\; \backslash frac\{\backslash partial\backslash sigma\_\{\backslash alpha\backslash beta\}\}\{\backslash partial\; x\_\backslash beta\}=0$

|Solid Mechanics

|

| $$u\_\{t\}=a\; (u^\{-2\}\; u\_x)\_x$$

|

!Name

!Dim

!Equation

!Applications

|1+3

|$G\_t\; +\; \backslash mathbf\{v\}\backslash cdot\backslash nabla\; G\; =\; S\_L(G)\; |\backslash nabla\; G|$

| turbulent combustion

|1+3

|$\backslash displaystyle\; \backslash varphi\_t\; +\; \backslash nabla\; \backslash cdot\; f(t,x,\backslash varphi,\backslash nabla\backslash varphi)\; =\; g(t,x,\backslash varphi)$

|transport

|1+3

|$\backslash displaystyle\; \backslash alpha\; \backslash psi\; +\; \backslash beta\; |\backslash psi|^2\; \backslash psi\; +\; \backslash tfrac\{1\}\{2m\}\; \backslash left(-i\backslash hbar\backslash nabla\; -\; 2e\backslash mathbf\{A\}\; \backslash right)^2\; \backslash psi\; =\; 0$

|Superconductivity

|{{math|1 + n}}

|$\backslash displaystyle\; i\backslash partial\_t\backslash psi\; =\; \backslash left\; (-\backslash tfrac12\backslash nabla^2\; +\; V(x)\; +\; g|\backslash psi|^2\; \backslash right\; )\; \backslash psi$

| Bose–Einstein condensate

|{{math|1 + 5}}

|$\{\backslash displaystyle\; \{\backslash frac\; \{\backslash partial\; h\_\{s\}\}\{\backslash partial\; t\}\}+\backslash left(v\_\{|$

!Name

!Dim

!Equation

!Applications

|1+n

|$\backslash displaystyle\; \backslash frac\{\backslash partial\; \backslash mathbf\{S\}\}\{\backslash partial\; t\}\; =\; \backslash mathbf\{S\}\backslash wedge\; \backslash sum\_i\backslash frac\{\backslash partial^2\; \backslash mathbf\{S\}\}\{\backslash partial\; x\_i^\{2\}\}\; +\; \backslash mathbf\{S\}\backslash wedge\; J\backslash mathbf\{S\}$

|Magnetic field in solids

|1+2

| $\backslash displaystyle\; 2u\_\{tx\}+u\_xu\_\{xx\}-u\_\{yy\}=0$

|

|any

|$\backslash displaystyle\; \backslash nabla^2u+e^\{\backslash lambda\; u\}=0$

|

|any

|$\backslash nabla^2\; \backslash psi\; +\; \backslash lambda\; e^\backslash psi=0$

|combustion, astrophysics

|any

|$i\; \backslash frac\{\backslash partial\; \backslash psi\}\{\backslash partial\; t\}\; +\; \backslash Delta\; \backslash psi\; +\; \backslash psi\; \backslash ln\; |\backslash psi|^2\; =\; 0.$

|Superfluids, Quantum gravity

|3

|$\backslash displaystyle\; \backslash operatorname\{div\}(Du/\backslash sqrt\{1+|Du|^2\})=0$

|minimal surfaces

|any

|$\backslash displaystyle\; \backslash det(\backslash partial\_\{ij\}\backslash varphi)\; =$ lower order terms

|

|1+3

|$\backslash displaystyle$

\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: $\backslash frac\{\backslash partial\; \backslash rho\}\{\backslash partial\; t\}\; +\; \backslash frac\{\backslash partial\; \backslash left(\; \backslash rho\backslash ,\; v\_i\; \backslash right)\}\{\backslash partial\; x\_i\}\; =\; 0$

+ an equation of state to relate p and ρ, e.g. for an incompressible flow: $\backslash frac\{\backslash partial\; v\_i\}\{\backslash partial\; x\_i\}\; =\; 0$

|Fluid flow, gas flow

|1+1

|$\backslash displaystyle\; i\backslash partial\_t\backslash psi=-\{1\backslash over\; 2\}\backslash partial^2\_x\backslash psi+\backslash kappa|\backslash psi|^2\; \backslash psi$

|optics, water waves

|1+1

|$\backslash displaystyle\; i\backslash partial\_t\backslash psi=-\{1\backslash over\; 2\}\backslash partial^2\_x\backslash psi+\backslash partial\_x(i\backslash kappa|\backslash psi|^2\; \backslash psi)$

|optics, water waves

|1+3

|$\backslash displaystyle\; \backslash nabla^2\backslash omega\; +\; \backslash frac\{f^2\}\{\backslash sigma\}\backslash frac\{\backslash partial^2\backslash omega\}\{\backslash partial\; p^2\}$ $\backslash displaystyle\; =\; \backslash frac\{f\}\{\backslash sigma\}\backslash frac\{\backslash partial\}\{\backslash partial\; p\}\backslash mathbf\{V\}\_g\backslash cdot\backslash nabla\_p\; (\backslash zeta\_g\; +\; f)\; +\; \backslash frac\{R\}\{\backslash sigma\; p\}\backslash nabla^2\_p(\backslash mathbf\{V\}\_g\backslash cdot\backslash nabla\_p\; T)$

|atmospheric physics

|2

|$\backslash displaystyle\; (1+u\_y^2)u\_\{xx\}\; -2u\_xu\_yu\_\{xy\}\; +(1+u\_x^2)u\_\{yy\}=0$

|minimal surfaces

|2

|$\backslash displaystyle\; u\_\{xx\}-u\_\{yy\}\backslash pm\; \backslash sin\; u\; \backslash cos\; u\; +\backslash frac\{\backslash cos\; u\}\{\backslash sin^3\; u\}(v\_x^2-v\_y^2)=0,\backslash quad\; \backslash displaystyle\; (v\_x\backslash cot^2u)\_x\; =\; (v\_y\backslash cot^2\; u)\_y$

|

|1+n

|$\backslash displaystyle\; u\_t=\backslash Delta(u^\backslash gamma)$

|diffusion

|1+2

|$\backslash displaystyle\; u\_t+uu\_x+vu\_y=U\_t+UU\_x+\backslash frac\{\backslash mu\}\{\backslash rho\}u\_\{yy\}$, $\backslash displaystyle\; u\_x+v\_y=0$

|boundary layer

!Name

!Dim

!Equation

!Applications

|1+1

|$\backslash displaystyle\; u\_\{tt\}-u\_\{xx\}\; =\; \backslash varepsilon(u\_t-u\_t^3)$

|

|Any

|$\backslash displaystyle\; \backslash partial\_t\; g\_\{ij\}=-2\; R\_\{ij\}$

|Poincaré conjecture

|1+3

|$\backslash displaystyle\; \backslash theta\_t=\backslash left[\; K(\backslash theta)\; \backslash left\; (\backslash psi\_z\; +\; 1\; \backslash right)\; \backslash right]\_z$

|Variably saturated flow in porous media

|1+1

|$u\_t\; +\; a\; \backslash left(u^n\backslash right)\_x\; +\; \backslash left(u^n\backslash right)\_\{xxx\}\; =\; 0$

| compacton solutions

|1+1

|$\backslash displaystyle\; u\_t+45u^2u\_x+15u\_xu\_\{xx\}+15uu\_\{xxx\}+u\_\{xxxxx\}=0$

|

|1+1

|$$

\ddot V + \partial_\eta \left[\frac{1}{1-\ddot V} \partial_\eta \left(\frac{1-\ddot V}{V}\right) \right] =0

|plasmas

|1+1

|$\backslash phi\_t\; +\; (1\; +\; b\; \backslash sqrt\; \backslash phi\; )\; \backslash phi\_x\; +\; \backslash phi\_\{xxx\}\; =\; 0$

|plasmas, solitons, optics

| Any

|$\backslash displaystyle\; \{\backslash partial\; A\_i\; \backslash over\; \backslash partial\; t\_j\}\; \{\backslash left[\; A\_i,\; \backslash \; A\_j\; \backslash right]\; \backslash over\; t\_i\; -\; t\_j\},\; \backslash quad\; i\backslash neq\; j,\; \backslash quad\; \{\backslash partial\; A\_i\; \backslash over\; \backslash partial\; t\_i\}\; =-\; \backslash sum\_\{j=1\; \backslash atop\; j\backslash neq\; i\}^n\; \{\backslash left[\; A\_i,\; \backslash \; A\_j\; \backslash right]\; \backslash over\; t\_i\; -\; t\_j\},\; \backslash quad\; 1\backslash leq\; i,\; j\; \backslash leq\; n$

|isomonodromic deformations

|1+3

|$\backslash displaystyle\; D^A\backslash varphi=0,\; \backslash qquad\; F^+\_A=\backslash sigma(\backslash varphi)$

|Seiberg–Witten invariants, QFT

|1+2

|$\backslash displaystyle\; \backslash eta\_t\; +\; (\backslash eta\; u)\_x\; +\; (\backslash eta\; v)\_y\; =\; 0,\backslash \; (\backslash eta\; u)\_t+\; \backslash left(\; \backslash eta\; u^2\; +\; \backslash frac\{1\}\{2\}g\; \backslash eta^2\; \backslash right)\_x\; +\; (\backslash eta\; uv)\_y\; =\; 0,\backslash \; (\backslash eta\; v)\_t\; +\; (\backslash eta\; uv)\_x\; +\; \backslash left(\backslash eta\; v^2\; +\; \backslash frac\{1\}\{2\}g\; \backslash eta\; ^2\backslash right)\_y\; =\; 0$

|shallow water waves

|1+1

|$\backslash displaystyle\; \backslash ,\; \backslash varphi\_\{tt\}-\; \backslash varphi\_\{xx\}\; +\; \backslash sin\backslash varphi\; =\; 0$

|Solitons, QFT

|1+1

|$\backslash displaystyle\; u\_\{xt\}=\; \backslash sinh\; u$

|Solitons, QFT

|1+n

|$\backslash displaystyle\; \backslash nabla^2u+\backslash sinh\; u=0$

|Fluid Mechanics

|any

|$\backslash displaystyle$

u_t = r u - (1+\nabla^2)^2u + N(u)

|pattern forming

|2

|$\backslash displaystyle\; u\_\{xy\}+\backslash alpha\; u\_x+\backslash beta\; u\_y+\backslash gamma\; u\_xu\_y=0$

|

|1+1

|$\backslash displaystyle\; iu\_x+v+u|v|^2=0$, $\backslash displaystyle\; iv\_t+u+v|u|^2=0$

|Dirac field, QFT

|any

|$\backslash displaystyle\; \backslash nabla^2\backslash log\; u\_n\; =\; u\_\{n+1\}-2u\_n+u\_\{n-1\}$

|

|1+2

|$\backslash displaystyle\; (\backslash partial\_t+\backslash partial\_z^3+\backslash partial\_\{\backslash bar\; z\}^3)v+\backslash partial\_z(uv)+\backslash partial\_\{\backslash bar\; z\}(uw)\; =0$, $\backslash displaystyle\; \backslash partial\_\{\backslash bar\; z\}u=3\backslash partial\_zv$, $\backslash displaystyle\; \backslash partial\_zw=3\backslash partial\_\{\backslash bar\; z\}\; v$

| shallow water waves

|

|$\backslash frac\{\backslash partial\; \backslash boldsymbol\; \backslash omega\}\{\backslash partial\; t\}\; +\; (\backslash mathbf\; u\; \backslash cdot\; \backslash nabla)\; \backslash boldsymbol\; \backslash omega\; =\; (\backslash boldsymbol\; \backslash omega\; \backslash cdot\; \backslash nabla)\; \backslash mathbf\; u\; -\; \backslash boldsymbol\; \backslash omega\; (\backslash nabla\; \backslash cdot\; \backslash mathbf\; u)\; +\; \backslash frac\{1\}\{\backslash rho^2\}\backslash nabla\; \backslash rho\; \backslash times\; \backslash nabla\; p\; +\; \backslash nabla\; \backslash times\; \backslash left(\; \backslash frac\{\backslash nabla\; \backslash cdot\; \backslash tau\}\{\backslash rho\}\; \backslash right)\; +\; \backslash nabla\; \backslash times\; \backslash left(\; \backslash frac\{\backslash mathbf\{f\}\}\{\backslash rho\}\; \backslash right),\; \backslash \; \backslash boldsymbol\{\backslash omega\}=\backslash nabla\backslash times\backslash mathbf\{u\}$

|Fluid Mechanics

|1+1

|$\backslash displaystyle\; iu\_t+((1+|u|^2)^\{-1/2\}u)\_\{xx\}=0$

|

|Any

|$\backslash displaystyle$

\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) $\backslash displaystyle$

= \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

|1+1

|$S\_k(\backslash gamma)=\; -\; \backslash ,\; \backslash frac\; \{k\}\{8\backslash pi\}\; \backslash int\_\{S^2\}\; d^2x\backslash ,$

\mathcal{K} (\gamma^{-1} \partial^\mu \gamma \, , \,

\gamma^{-1} \partial_\mu \gamma) + 2\pi k\, S^{\mathrm WZ}(\gamma)

$S^\{\backslash mathrm\; WZ\}(\backslash gamma)\; =\; -\; \backslash ,\; \backslash frac\{1\}\{48\backslash pi^2\}\; \backslash int\_\{B^3\}\; d^3y\backslash ,$

\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

|1+1

|$\backslash displaystyle\; \backslash eta\_t\; +\; \backslash alpha\; \backslash eta\; \backslash eta\_x\; +\; \backslash int\_\{-\backslash infty\}^\{+\backslash infty\}\; K(x-\backslash xi)\backslash ,\; \backslash eta\_\backslash xi(\backslash xi,t)\backslash ,\; \backslash text\{d\}\backslash xi\; =\; 0$

|water waves

|

|$\backslash frac\{\backslash partial\; f\_j\}\{\backslash partial\; t\}\; +\; \backslash nabla\_x\backslash cdot(\backslash mathbf\{v\}f\_j)\; +\; \backslash nabla\_v\backslash cdot(F\_jf\_j)\; =-\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; r\}(R\_jf\_j)\; -\; \backslash frac\{\backslash partial\; \}\{\backslash partial\; T\}(E\_jf\_j)\; +\; Q\_j\; +\; \backslash Gamma\_j,\backslash \; F\_j\; =\; \backslash dot\{\backslash mathbf\{v\}\},\backslash \; R\_j\; =\; \backslash dot\{r\},\backslash \; E\_j\; =\; \backslash dot\{T\},\backslash \; j\; =\; 1,2,...,M$

|Combustion

|n

|$\backslash displaystyle\backslash Delta\; \backslash varphi+h(x)\backslash varphi\; =\; \backslash lambda\; f(x)\backslash varphi^\{(n+2)/(n-2)\}$

|Differential geometry

|Any

|$\backslash displaystyle\; D\_\backslash mu\; F^\{\backslash mu\backslash nu\}=0,\; \backslash quad\; F\_\{\backslash mu\; \backslash nu\}\; =\; A\_\{\backslash mu,\; \backslash nu\}\; -\; A\_\{\backslash nu,\; \backslash mu\; \}+\; [A\_\backslash mu,\; \backslash ,\; A\_\backslash nu]$

|Gauge theory, QFT

| 4

| $F\_\{\backslash alpha\; \backslash beta\}\; =\; \backslash pm\; \backslash varepsilon\_\{\backslash alpha\; \backslash beta\; \backslash mu\; \backslash nu\}\; F^\{\backslash mu\; \backslash nu\},$

\quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]

| Instantons, Donaldson theory, QFT

|1+n

|$\backslash displaystyle\; i\; \backslash partial\_t^\{\}u\; +\; \backslash Delta\; u\; =\; -A\; u,\backslash quad\; \backslash displaystyle\backslash Box\; A\; =\; m^2\_\{\}\; A\; +\; |u|^2$

|Meson-nucleon interactions, QFT

|1+3

|$\backslash displaystyle\; i\; \backslash partial\_t^\{\}\; u\; +\; \backslash Delta\; u\; =\; un,\backslash quad\; \backslash displaystyle\; \backslash Box\; n\; =\; -\backslash Delta\; (|u|^2\_\{\})$

|Langmuir waves

|1+3

|$\backslash displaystyle\; iu\_t\; +\; L\_1u\; =\; \backslash varphi\; u,\backslash quad\; \backslash displaystyle\; L\_2\; \backslash varphi\; =\; L\_3(\; |\; u\; |^2)$

|Acoustic waves

|1+3

|$\backslash displaystyle\; u\_t\; =\; D\backslash nabla^2\; u\; +\; \backslash frac\{\backslash beta^2\}\{2\}u(1-u)\; e^\{-\backslash beta(1-u)\}$

|Combustion

|1+1

|$\backslash displaystyle\; (u\_\{xt\}/u)\_\{tt\}-(u\_\{xt\}/u)\_\{xx\}\; +2(u^2)\_\{xt\}=0$

|Solitons

|1+1

|$\backslash displaystyle\; \backslash varphi\_\{tt\}-\backslash varphi\_\{xx\}-\backslash varphi+\backslash varphi^3=0$

| QFT

|1+1

|$\backslash displaystyle\; \{\backslash mathbf\; v\}\_\{xt\}+(\{\backslash mathbf\; v\}\_x\{\backslash mathbf\; v\}\_t)\{\backslash mathbf\; v\}=0$

|Harmonic maps, integrable systems, QFT