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 |