| Name |
Dim |
Equation |
Applications |
| Rayleigh |
1+1 |
 |
|
| Ricci flow |
Any |
 |
Poincaré conjecture |
| Richards equation |
1+3 |
![{\displaystyle \displaystyle \theta _{t}=\left[K(\theta )\left(\psi _{z}+1\right)\right]_{z}}](//wikimedia.org/api/rest_v1/media/math/render/svg/e1a88a581feb802680f23c628c9287f81639a7bd) |
Variably saturated flow in porous media |
| Rosenau–Hyman |
1+1 |
 |
compacton solutions |
| Sawada–Kotera |
1+1 |
 |
|
| 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}](//wikimedia.org/api/rest_v1/media/math/render/svg/37b933b68cbc812a2c58a39d93497fe33d13f157) |
plasmas |
| Schamel equation |
1+1 |
 |
plasmas, solitons, optics |
| 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}](//wikimedia.org/api/rest_v1/media/math/render/svg/76907bf442734c0aff03f2145ca2be80c5665df9) |
isomonodromic deformations |
| Seiberg–Witten |
1+3 |
 |
Seiberg–Witten invariants, QFT |
| Shallow water |
1+2 |
 |
shallow water waves |
| Sine–Gordon |
1+1 |
 |
Solitons, QFT |
| Sinh–Gordon |
1+1 |
 |
Solitons, QFT |
| Sinh–Poisson |
1+n |
 |
Fluid Mechanics |
|
| Swift–Hohenberg |
any |
 |
pattern forming |
| Thomas |
2 |
 |
|
| Thirring |
1+1 |
 ,  |
Dirac field, QFT |
| Toda lattice |
any |
 |
|
| Veselov–Novikov |
1+2 |
 ,  ,  |
shallow water waves |
| Vorticity equation |
|
 |
Fluid Mechanics |
| Wadati–Konno–Ichikawa–Schimizu |
1+1 |
 |
|
| WDVV equations |
Any |
  |
Topological field theory, QFT |
| WZW model |
1+1 |
![{\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)}](//wikimedia.org/api/rest_v1/media/math/render/svg/7ab8e448239ea98f4d6d03629262e1b317f0fb65)
|
QFT |
| Whitham equation |
1+1 |
 |
water waves |
| Williams spray equation |
|
 |
Combustion |
| Yamabe |
n |
 |
Differential geometry |
| Yang–Mills (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 }]}](//wikimedia.org/api/rest_v1/media/math/render/svg/eb7faaabf521c2d147d286ac8a1cb9f74548b2ad) |
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 }]}](//wikimedia.org/api/rest_v1/media/math/render/svg/f0da8bc8bfc9f5376e8c15acddc714dee8863006) |
Instantons, Donaldson theory, QFT |
| Yukawa |
1+n |
 |
Meson-nucleon interactions, QFT |
| Zakharov system |
1+3 |
 |
Langmuir waves |
| Zakharov–Schulman |
1+3 |
 |
Acoustic waves |
| Zeldovich–Frank-Kamenetskii equation |
1+3 |
 |
Combustion |
| Zoomeron |
1+1 |
 |
Solitons |
| φ4 equation |
1+1 |
 |
QFT |
| σ-model |
1+1 |
 |
Harmonic maps, integrable systems, QFT |
, 
...
, 
...
![{\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'} }](http://wikimedia.org/api/rest_v1/media/math/render/svg/81897aaeb638554eb93a78924a0794aafc8e4489)
, 
...
, 
...
![{\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 }}](http://wikimedia.org/api/rest_v1/media/math/render/svg/470886b4973dcb3812c3f7ae69fea1da2fbdfeb0)
![{\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]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/5a0ed42f78205aa810e0dc9d1305c72999a63e23)
, 
...
, 
...
lower order terms![{\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}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/3dcc6555162e4c7f246a33e18c378e2f5eeeaefe)
, 
, 
...![{\displaystyle \displaystyle \theta _{t}=\left[K(\theta )\left(\psi _{z}+1\right)\right]_{z}}](http://wikimedia.org/api/rest_v1/media/math/render/svg/e1a88a581feb802680f23c628c9287f81639a7bd)
![{\displaystyle {\ddot {V}}+\partial _{\eta }\left[{\frac {1}{1-{\ddot {V}}}}\partial _{\eta }\left({\frac {1-{\ddot {V}}}{V}}\right)\right]=0}](http://wikimedia.org/api/rest_v1/media/math/render/svg/37b933b68cbc812a2c58a39d93497fe33d13f157)
![{\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}](http://wikimedia.org/api/rest_v1/media/math/render/svg/76907bf442734c0aff03f2145ca2be80c5665df9)
![{\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)}](http://wikimedia.org/api/rest_v1/media/math/render/svg/7ab8e448239ea98f4d6d03629262e1b317f0fb65)
![{\displaystyle \displaystyle D_{\mu }F^{\mu \nu }=0,\quad F_{\mu \nu }=A_{\mu ,\nu }-A_{\nu ,\mu }+[A_{\mu },\,A_{\nu }]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/eb7faaabf521c2d147d286ac8a1cb9f74548b2ad)
![{\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 }]}](http://wikimedia.org/api/rest_v1/media/math/render/svg/f0da8bc8bfc9f5376e8c15acddc714dee8863006)
