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

 * {|class="wikitable" style="background: white; color: black; text-align: left"

!Name !Dim !Equation !Applications \displaystyle \mathbf{v}_{xt}=u_{xx}\mathbf{b}+\mathbf{a}\times\mathbf{v}_x- 2\mathbf{v}\times(\mathbf{v}\times\mathbf{b})$$ \displaystyle \varphi_{xx} + c_3 \varphi_{yy} = ( |u|^2 )_x$$ \displaystyle w_t=2w_{xxx}+2uw_x+u_xw$$ \displaystyle w_t=\varepsilon u$$
 * -style="background: #eee"
 * Bateman-Burgers equation
 * 1+1
 * $$\displaystyle u_t+uu_x=\nu u_{xx}$$
 * Fluid mechanics
 * Benjamin–Bona–Mahony
 * 1+1
 * $$\displaystyle u_t+u_x+uu_x-u_{xxt}=0$$
 * Fluid mechanics
 * Benjamin–Ono
 * 1+1
 * $$\displaystyle u_t+Hu_{xx}+uu_x=0$$
 * internal waves in deep water
 * Boomeron
 * 1+1
 * $$\displaystyle u_t=\mathbf{b}\cdot\mathbf{v}_x, \quad
 * Boomeron
 * 1+1
 * $$\displaystyle u_t=\mathbf{b}\cdot\mathbf{v}_x, \quad
 * $$\displaystyle u_t=\mathbf{b}\cdot\mathbf{v}_x, \quad
 * Solitons
 * Boltzmann equation
 * 1+6
 * $$\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_if_j] \,d\Omega\,d^3\mathbf{p'}$$
 * Statistical mechanics
 * Born–Infeld
 * 1+1
 * $$\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0$$
 * Electrodynamics
 * Boussinesq
 * 1+1
 * $$\displaystyle u_{tt} - u_{xx} - u_{xxxx} - 3(u^2)_{xx} = 0$$
 * Fluid mechanics
 * Boussinesq type equation
 * 1+1
 * $$\displaystyle u_{tt}-u_{xx}-2 \alpha (u u_x)_{x}-\beta u_{xxtt}=0$$
 * Fluid mechanics
 * Buckmaster
 * 1+1
 * $$\displaystyle u_t=(u^4)_{xx}+(u^3)_x$$
 * Thin viscous fluid sheet flow
 * Cahn–Hilliard equation
 * Any
 * $$\displaystyle c_t = D\nabla^2\left(c^3-c-\gamma\nabla^2 c\right)$$
 * Phase separation
 * Calabi flow
 * Any
 * $$\frac{\partial g_{ij}}{\partial t}=(\Delta R)g_{ij}$$
 * Calabi–Yau manifolds
 * Camassa–Holm
 * 1+1
 * $$u_t + 2\kappa u_x - u_{xxt} + 3 u u_x = 2 u_x u_{xx} + u u_{xxx}\,$$
 * Peakons
 * Carleman
 * 1+1
 * $$\displaystyle u_t+u_x=v^2-u^2=v_x-v_t$$
 * Cauchy momentum
 * any
 * $$\displaystyle \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \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 \delta e^{\theta})_{tt}=\nabla^2(\theta_t-\delta e^\theta)$$
 * Combustion
 * Complex Monge–Ampère
 * Any
 * $$\displaystyle \det(\partial_{i\bar j}\varphi) = $$ lower order terms
 * Calabi conjecture
 * Constant astigmatism
 * 1+1
 * $$z_{yy} + \left(\frac{1}{z}\right)_{xx} + 2 = 0$$
 * Differential geometry
 * Davey–Stewartson
 * 1+2
 * $$\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \varphi_x, \quad
 * Clarke's equation
 * 1+1
 * $$(\theta_t-\gamma \delta e^{\theta})_{tt}=\nabla^2(\theta_t-\delta e^\theta)$$
 * Combustion
 * Complex Monge–Ampère
 * Any
 * $$\displaystyle \det(\partial_{i\bar j}\varphi) = $$ 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
 * $$\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \varphi_x, \quad
 * Davey–Stewartson
 * 1+2
 * $$\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \varphi_x, \quad
 * $$\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 |u|^2 u + c_2 u \varphi_x, \quad
 * Finite depth waves
 * Degasperis–Procesi
 * 1+1
 * $$\displaystyle u_t - u_{xxt} + 4u u_x = 3 u_x u_{xx} + u u_{xxx}$$
 * Peakons
 * Dispersive long wave
 * 1+1
 * $$\displaystyle u_t=(u^2-u_x+2w)_x$$, $$w_t=(2uw+w_x)_x$$
 * Drinfeld–Sokolov–Wilson
 * 1+1
 * $$\displaystyle u_t=3ww_x, \quad
 * Drinfeld–Sokolov–Wilson
 * 1+1
 * $$\displaystyle u_t=3ww_x, \quad
 * $$\displaystyle u_t=3ww_x, \quad
 * Dym equation
 * 1+1
 * $$\displaystyle u_t = u^3u_{xxx}.\,$$
 * Solitons
 * Eckhaus equation
 * 1+1
 * $$iu_t+u_{xx}+2|u|^2_xu+|u|^4u=0$$
 * Integrable systems
 * Eikonal equation
 * any
 * $$\displaystyle |\nabla u(x)|=F(x), \ x\in \Omega$$
 * optics
 * Einstein field equations
 * Any
 * $$\displaystyle R_{\mu\nu} - {\textstyle 1 \over 2}R\,g_{\mu\nu}+\Lambda g_{\mu\nu} = \frac{8\pi G}{c^{4}} T_{\mu\nu} $$
 * General relativity
 * Ernst equation
 * 2
 * $$\displaystyle \Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2$$
 * 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
 * $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0,\quad \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=-\nabla p + \rho\mathbf{f},\quad \frac{\partial s}{\partial t}+\mathbf{v}\cdot\nabla s=0$$
 * non-viscous fluids
 * Fisher's equation
 * 1+1
 * $$\displaystyle u_t=u(1-u)+u_{xx} $$
 * Gene propagation
 * FitzHugh–Nagumo model
 * 1+1
 * $$\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w, \quad
 * Euler equations
 * 1+3
 * $$\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0,\quad \rho\left(\frac{\partial\mathbf{u}}{\partial t}+\mathbf{v}\cdot\nabla\mathbf{v}\right)=-\nabla p + \rho\mathbf{f},\quad \frac{\partial s}{\partial t}+\mathbf{v}\cdot\nabla s=0$$
 * non-viscous fluids
 * Fisher's equation
 * 1+1
 * $$\displaystyle u_t=u(1-u)+u_{xx} $$
 * Gene propagation
 * FitzHugh–Nagumo model
 * 1+1
 * $$\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w, \quad
 * 1+1
 * $$\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w, \quad
 * Biological neuron model
 * Föppl–von Kármán equations
 * $$\frac{Eh^3}{12(1-\nu^2)}\nabla^4 w-h\frac{\partial}{\partial x_\beta}\left(\sigma_{\alpha\beta}\frac{\partial w}{\partial x_\alpha}\right)=P, \quad \frac{\partial\sigma_{\alpha\beta}}{\partial x_\beta}=0$$
 * Solid Mechanics
 * Fujita–Storm equation
 * $$ u_{t}=a (u^{-2} u_x)_x $$
 * }
 * Fujita–Storm equation
 * $$ u_{t}=a (u^{-2} u_x)_x $$
 * }
 * }
 * }

G–K

 * {|class="wikitable" style="background: white; color: black; text-align: left"

!Name !Dim !Equation !Applications
 * -style="background: #eee"
 * G equation
 * 1+3
 * $$G_t + \mathbf{v}\cdot\nabla G = S_L(G) |\nabla G|$$
 * turbulent combustion
 * Generic scalar transport
 * 1+3
 * $$\displaystyle \varphi_t + \nabla \cdot f(t,x,\varphi,\nabla\varphi) = g(t,x,\varphi) $$
 * transport
 * Ginzburg–Landau
 * 1+3
 * $$\displaystyle \alpha \psi + \beta |\psi|^2 \psi + \tfrac{1}{2m} \left(-i\hbar\nabla - 2e\mathbf{A} \right)^2 \psi = 0 $$
 * Superconductivity
 * Gross–Pitaevskii
 * $$\displaystyle i\partial_t\psi = \left (-\tfrac12\nabla^2 + V(x) + g|\psi|^2 \right ) \psi $$
 * Bose–Einstein condensate
 * Gyrokinetics equation
 * $${\displaystyle {\frac {\partial h_{s}}{\partial t}}+\left(v_{||}{\hat {b}}+{\vec {V}}_{ds}+\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\right)\cdot {\vec {\nabla }}_{\vec {R}}h_{s}-\sum _{s'}\left\langle C\left[h_{s},h_{s'}\right]\right\rangle _{\varphi }={\frac {Z_{s}ef_{s0}}{T_{s}}}{\frac {\partial \left\langle \phi \right\rangle _{\varphi }}{\partial t}}-{\frac {\partial f_{s0}}{\partial \psi }}\left\langle {\vec {V}}_{\phi }\right\rangle _{\varphi }\cdot {\vec {\nabla }}\psi } $$
 * Microturbulence in plasma
 * Guzmán
 * $$\displaystyle J_t+gJ_x+1/2\sigma^2J_{xx}-\lambda\sigma^2(J_x)^2+f=0 $$
 * Hamilton–Jacobi–Bellman equation for risk aversion
 * Hartree equation
 * Any
 * $$\displaystyle i\partial_tu + \Delta u= \left (\pm |x|^{-n} |u|^2 \right) u$$
 * Hasegawa–Mima
 * 1+3
 * $$\displaystyle 0 = \frac{\partial}{\partial t} \left( \nabla^2 \varphi - \varphi \right) - \left[ \left( \nabla\varphi \times \hat{\mathbf{z}} \right)\cdot \nabla \right] \left[ \nabla^2 \varphi - \ln \left(\frac{n_0}{\omega_{ci}}\right)\right]$$
 * Turbulence in plasma
 * Heisenberg ferromagnet
 * 1+1
 * $$\displaystyle \mathbf{S}_t=\mathbf{S}\wedge \mathbf{S}_{xx}. $$
 * Magnetism
 * Hicks
 * 1+1
 * $$\psi_{rr} - \psi_r/r + \psi_{zz} = r^2 \mathrm{d}H/\mathrm{d} \psi - \Gamma \mathrm{d} \Gamma/\mathrm{d}\psi$$
 * Fluid dynamics
 * Hunter–Saxton
 * 1+1
 * $$\displaystyle \left (u_t + u u_x \right )_x = \tfrac{1}{2} u_x^2 $$
 * Liquid crystals
 * Ishimori equation
 * 1+2
 * $$\displaystyle \mathbf{S}_t = \mathbf{S}\wedge \left(\mathbf{S}_{xx} + \mathbf{S}_{yy}\right)+  u_x\mathbf{S}_y + u_y\mathbf{S}_x,\quad \displaystyle  u_{xx}-\alpha^2 u_{yy}=-2\alpha^2  \mathbf{S}\cdot\left(\mathbf{S}_x\wedge \mathbf{S}_y\right)$$
 * Integrable systems
 * Kadomtsev –Petviashvili
 * 1+2
 * $$\displaystyle \partial_x \left (\partial_t u+u \partial_x u+\varepsilon^2\partial_{xxx}u \right )+\lambda\partial_{yy}u=0$$
 * Shallow water waves
 * Kardar–Parisi–Zhang equation
 * 1+3
 * $$\displaystyle h_t=\nu \nabla^2 h + \lambda (\nabla h)^2 /2+ \eta$$
 * Stochastics
 * von Karman
 * 2
 * $$\displaystyle \nabla^4 u = E \left (w_{xy}^2-w_{xx}w_{yy} \right ), \quad \nabla^4 w = a+b \left (u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy} \right)$$
 * Kaup
 * 1+1
 * $$\displaystyle f_x=2fgc(x-t)=g_t$$
 * Kaup–Kupershmidt
 * 1+1
 * $$\displaystyle u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x $$
 * Integrable systems
 * Klein–Gordon–Maxwell
 * any
 * $$\displaystyle \nabla^2s= \left (|\mathbf a|^2+1 \right )s, \quad \nabla^2\mathbf a =\nabla(\nabla\cdot\mathbf a)+s^2\mathbf a$$
 * Klein–Gordon (nonlinear)
 * any
 * $$\nabla^2u+\lambda u^p=0$$
 * Relativistic quantum mechanics
 * Khokhlov–Zabolotskaya
 * 1+2
 * $$\displaystyle u_{xt} -(uu_x)_x =u_{yy}$$
 * Kompaneyets
 * 1+1
 * $$\displaystyle n_{t} =x^{-2}[x^4(n_x+n^2+n)]_x$$
 * Physical kinetics
 * Korteweg–de Vries (KdV)
 * 1+1
 * $$\displaystyle u_{t}+u_{xxx}-6u u_{x}=0$$
 * Shallow waves, Integrable systems
 * Kaup–Kupershmidt
 * 1+1
 * $$\displaystyle u_t = u_{xxxxx}+10u_{xxx}u+25u_{xx}u_x+20u^2u_x $$
 * Integrable systems
 * Klein–Gordon–Maxwell
 * any
 * $$\displaystyle \nabla^2s= \left (|\mathbf a|^2+1 \right )s, \quad \nabla^2\mathbf a =\nabla(\nabla\cdot\mathbf a)+s^2\mathbf a$$
 * Klein–Gordon (nonlinear)
 * any
 * $$\nabla^2u+\lambda u^p=0$$
 * Relativistic quantum mechanics
 * Khokhlov–Zabolotskaya
 * 1+2
 * $$\displaystyle u_{xt} -(uu_x)_x =u_{yy}$$
 * Kompaneyets
 * 1+1
 * $$\displaystyle n_{t} =x^{-2}[x^4(n_x+n^2+n)]_x$$
 * Physical kinetics
 * Korteweg–de Vries (KdV)
 * 1+1
 * $$\displaystyle u_{t}+u_{xxx}-6u u_{x}=0$$
 * Shallow waves, Integrable systems
 * 1+1
 * $$\displaystyle n_{t} =x^{-2}[x^4(n_x+n^2+n)]_x$$
 * Physical kinetics
 * Korteweg–de Vries (KdV)
 * 1+1
 * $$\displaystyle u_{t}+u_{xxx}-6u u_{x}=0$$
 * Shallow waves, Integrable systems
 * Shallow waves, Integrable systems


 * KdV (super)
 * 1+1
 * $$\displaystyle u_t=6uu_x-u_{xxx}+3ww_{xx}, \quad w_t=3u_xw+6uw_x-4w_{xxx}$$
 * $$\displaystyle u_t=6uu_x-u_{xxx}+3ww_{xx}, \quad w_t=3u_xw+6uw_x-4w_{xxx}$$


 * colspan="4" |There are more minor variations listed in the article on KdV equations.
 * Kuramoto–Sivashinsky equation
 * $$\displaystyle u_t+\nabla^4u+\nabla^2u+ \tfrac{1}{2}|\nabla u|^2=0$$
 * Combustion
 * }
 * $$\displaystyle u_t+\nabla^4u+\nabla^2u+ \tfrac{1}{2}|\nabla u|^2=0$$
 * Combustion
 * }

L–Q

 * {|class="wikitable" style="background: white; color: black; text-align: left"

!Name !Dim !Equation !Applications \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 $$ + mass conservation: $$\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 $$
 * -style="background: #eee"
 * Landau–Lifshitz model
 * 1+n
 * $$\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 2u_{tx}+u_xu_{xx}-u_{yy}=0$$
 * Liouville equation
 * any
 * $$\displaystyle \nabla^2u+e^{\lambda u}=0$$
 * Liouville–Bratu–Gelfand equation
 * any
 * $$\nabla^2 \psi + \lambda e^\psi=0$$
 * combustion, astrophysics
 * Logarithmic Schrödinger equation
 * any
 * $$ i \frac{\partial \psi}{\partial t} + \Delta \psi + \psi \ln |\psi|^2 = 0. $$
 * Superfluids, Quantum gravity
 * Minimal surface
 * 3
 * $$\displaystyle \operatorname{div}(Du/\sqrt{1+|Du|^2})=0$$
 * minimal surfaces
 * Monge–Ampère
 * any
 * $$\displaystyle \det(\partial_{ij}\varphi) = $$ lower order terms
 * Navier–Stokes (and its derivation)
 * 1+3
 * $$ \displaystyle
 * $$\displaystyle \operatorname{div}(Du/\sqrt{1+|Du|^2})=0$$
 * minimal surfaces
 * Monge–Ampère
 * any
 * $$\displaystyle \det(\partial_{ij}\varphi) = $$ lower order terms
 * Navier–Stokes (and its derivation)
 * 1+3
 * $$ \displaystyle
 * Navier–Stokes (and its derivation)
 * 1+3
 * $$ \displaystyle
 * Fluid flow, gas flow
 * Nonlinear Schrödinger (cubic)
 * 1+1
 * $$\displaystyle i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\kappa|\psi|^2 \psi$$
 * optics, water waves
 * Nonlinear Schrödinger (derivative)
 * 1+1
 * $$\displaystyle i\partial_t\psi=-{1\over 2}\partial^2_x\psi+\partial_x(i\kappa|\psi|^2 \psi)$$
 * optics, water waves
 * Omega equation
 * 1+3
 * $$\displaystyle \nabla^2\omega + \frac{f^2}{\sigma}\frac{\partial^2\omega}{\partial p^2} $$ $$\displaystyle = \frac{f}{\sigma}\frac{\partial}{\partial p}\mathbf{V}_g\cdot\nabla_p (\zeta_g + f) + \frac{R}{\sigma p}\nabla^2_p(\mathbf{V}_g\cdot\nabla_p T)$$
 * atmospheric physics
 * Plateau
 * 2
 * $$\displaystyle (1+u_y^2)u_{xx} -2u_xu_yu_{xy} +(1+u_x^2)u_{yy}=0$$
 * minimal surfaces
 * Pohlmeyer–Lund–Regge
 * 2
 * $$\displaystyle u_{xx}-u_{yy}\pm \sin u \cos u +\frac{\cos u}{\sin^3 u}(v_x^2-v_y^2)=0,\quad \displaystyle (v_x\cot^2u)_x = (v_y\cot^2 u)_y$$
 * Porous medium
 * 1+n
 * $$\displaystyle u_t=\Delta(u^\gamma)$$
 * diffusion
 * Prandtl
 * 1+2
 * $$\displaystyle u_t+uu_x+vu_y=U_t+UU_x+\frac{\mu}{\rho}u_{yy}$$, $$\displaystyle u_x+v_y=0$$
 * boundary layer
 * }
 * $$\displaystyle u_t=\Delta(u^\gamma)$$
 * diffusion
 * Prandtl
 * 1+2
 * $$\displaystyle u_t+uu_x+vu_y=U_t+UU_x+\frac{\mu}{\rho}u_{yy}$$, $$\displaystyle u_x+v_y=0$$
 * boundary layer
 * }
 * }

R–Z, &alpha;–&omega;

 * {|class="wikitable" style="background: white; color: black; text-align: left"

!Name !Dim !Equation !Applications \ddot V + \partial_\eta \left[\frac{1}{1-\ddot V} \partial_\eta \left(\frac{1-\ddot V}{V}\right) \right] =0 $$ u_t = r u - (1+\nabla^2)^2u + N(u) $$ \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) $$ $$\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) $$ \mathcal{K} (\gamma^{-1} \partial^\mu \gamma \,, \,   \gamma^{-1} \partial_\mu \gamma) + 2\pi k\, S^{\mathrm WZ}(\gamma)$$ $$S^{\mathrm WZ}(\gamma) = - \, \frac{1}{48\pi^2} \int_{B^3} d^3y\, \varepsilon^{ijk} \mathcal{K} \left( \gamma^{-1} \, \frac {\partial \gamma} {\partial y^i} \,, \, \left[ \gamma^{-1} \, \frac {\partial \gamma} {\partial y^j} \, , \, \gamma^{-1} \, \frac {\partial \gamma} {\partial y^k} \right] \right)$$ $$ \quad  F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu] $$
 * -style="background: #eee"
 * Rayleigh
 * 1+1
 * $$\displaystyle u_{tt}-u_{xx} = \varepsilon(u_t-u_t^3)$$
 * Ricci flow
 * Any
 * $$\displaystyle \partial_t g_{ij}=-2 R_{ij}$$
 * Poincaré conjecture
 * Richards equation
 * 1+3
 * $$\displaystyle \theta_t=\left[ K(\theta) \left (\psi_z + 1 \right) \right]_z$$
 * 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
 * $$\displaystyle u_t+45u^2u_x+15u_xu_{xx}+15uu_{xxx}+u_{xxxxx}=0$$
 * Sack–Schamel equation
 * 1+1
 * Sawada–Kotera
 * 1+1
 * $$\displaystyle u_t+45u^2u_x+15u_xu_{xx}+15uu_{xxx}+u_{xxxxx}=0$$
 * Sack–Schamel equation
 * 1+1
 * Sack–Schamel equation
 * 1+1
 * 1+1
 * plasmas
 * Schamel equation
 * 1+1
 * $$\phi_t + (1 + b \sqrt \phi ) \phi_x + \phi_{xxx} = 0$$
 * plasmas, solitons, optics
 * Schlesinger
 * Any
 * $$\displaystyle  {\partial A_i \over \partial t_j}  {\left[ A_i, \ A_j \right] \over t_i - t_j}, \quad i\neq j,  \quad {\partial A_i \over \partial t_i} =- \sum_{j=1 \atop j\neq i}^n {\left[ A_i, \ A_j \right] \over t_i - t_j}, \quad 1\leq i, j \leq n $$
 * isomonodromic deformations
 * Seiberg–Witten
 * 1+3
 * $$\displaystyle D^A\varphi=0, \qquad F^+_A=\sigma(\varphi)$$
 * Seiberg–Witten invariants, QFT
 * Shallow water
 * 1+2
 * $$\displaystyle \eta_t + (\eta u)_x + (\eta v)_y = 0,\ (\eta u)_t+ \left( \eta u^2 + \frac{1}{2}g \eta^2 \right)_x + (\eta uv)_y = 0,\ (\eta v)_t + (\eta uv)_x + \left(\eta v^2 + \frac{1}{2}g \eta ^2\right)_y = 0$$
 * shallow water waves
 * Sine–Gordon
 * 1+1
 * $$\displaystyle \, \varphi_{tt}- \varphi_{xx} + \sin\varphi = 0$$
 * Solitons, QFT
 * Sinh–Gordon
 * 1+1
 * $$\displaystyle u_{xt}= \sinh u $$
 * Solitons, QFT
 * Sinh–Poisson
 * 1+n
 * $$\displaystyle \nabla^2u+\sinh u=0$$
 * Fluid Mechanics
 * Swift–Hohenberg
 * any
 * $$\displaystyle
 * Sinh–Poisson
 * 1+n
 * $$\displaystyle \nabla^2u+\sinh u=0$$
 * Fluid Mechanics
 * Swift–Hohenberg
 * any
 * $$\displaystyle
 * any
 * $$\displaystyle
 * pattern forming
 * Thomas
 * 2
 * $$\displaystyle u_{xy}+\alpha u_x+\beta u_y+\gamma u_xu_y=0$$
 * Thirring
 * 1+1
 * $$\displaystyle iu_x+v+u|v|^2=0$$, $$\displaystyle iv_t+u+v|u|^2=0$$
 * Dirac field, QFT
 * Toda lattice
 * any
 * $$\displaystyle \nabla^2\log u_n = u_{n+1}-2u_n+u_{n-1}$$
 * Veselov–Novikov
 * 1+2
 * $$\displaystyle (\partial_t+\partial_z^3+\partial_{\bar z}^3)v+\partial_z(uv)+\partial_{\bar z}(uw) =0$$, $$\displaystyle \partial_{\bar z}u=3\partial_zv$$, $$\displaystyle \partial_zw=3\partial_{\bar z} v$$
 * shallow water waves
 * Vorticity equation
 * $$\frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega = (\boldsymbol \omega \cdot \nabla) \mathbf u - \boldsymbol \omega (\nabla \cdot \mathbf u) + \frac{1}{\rho^2}\nabla \rho \times \nabla p + \nabla \times \left( \frac{\nabla \cdot \tau}{\rho} \right) + \nabla \times \left( \frac{\mathbf{f}}{\rho} \right), \ \boldsymbol{\omega}=\nabla\times\mathbf{u}$$
 * Fluid Mechanics
 * Wadati–Konno–Ichikawa–Schimizu
 * 1+1
 * $$\displaystyle iu_t+((1+|u|^2)^{-1/2}u)_{xx}=0$$
 * WDVV equations
 * Any
 * $$\displaystyle
 * $$\frac{\partial \boldsymbol \omega}{\partial t} + (\mathbf u \cdot \nabla) \boldsymbol \omega = (\boldsymbol \omega \cdot \nabla) \mathbf u - \boldsymbol \omega (\nabla \cdot \mathbf u) + \frac{1}{\rho^2}\nabla \rho \times \nabla p + \nabla \times \left( \frac{\nabla \cdot \tau}{\rho} \right) + \nabla \times \left( \frac{\mathbf{f}}{\rho} \right), \ \boldsymbol{\omega}=\nabla\times\mathbf{u}$$
 * Fluid Mechanics
 * Wadati–Konno–Ichikawa–Schimizu
 * 1+1
 * $$\displaystyle iu_t+((1+|u|^2)^{-1/2}u)_{xx}=0$$
 * WDVV equations
 * Any
 * $$\displaystyle
 * WDVV equations
 * Any
 * $$\displaystyle
 * Topological field theory, QFT
 * WZW model
 * 1+1
 * $$S_k(\gamma)= - \, \frac {k}{8\pi} \int_{S^2} d^2x\,
 * $$S_k(\gamma)= - \, \frac {k}{8\pi} \int_{S^2} d^2x\,
 * QFT
 * Whitham equation
 * 1+1
 * $$\displaystyle \eta_t + \alpha \eta \eta_x + \int_{-\infty}^{+\infty} K(x-\xi)\, \eta_\xi(\xi,t)\, \text{d}\xi = 0$$
 * water waves
 * Williams spray equation
 * $$\frac{\partial f_j}{\partial t} + \nabla_x\cdot(\mathbf{v}f_j) + \nabla_v\cdot(F_jf_j) =- \frac{\partial }{\partial r}(R_jf_j) - \frac{\partial }{\partial T}(E_jf_j) + Q_j + \Gamma_j,\ F_j = \dot{\mathbf{v}},\ R_j = \dot{r},\ E_j = \dot{T},\ j = 1,2,...,M$$
 * Combustion
 * Yamabe
 * n
 * $$\displaystyle\Delta \varphi+h(x)\varphi = \lambda f(x)\varphi^{(n+2)/(n-2)}$$
 * Differential geometry
 * Yang–Mills (source-free)
 * Any
 * $$\displaystyle D_\mu F^{\mu\nu}=0, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]
 * Differential geometry
 * Yang–Mills (source-free)
 * Any
 * $$\displaystyle D_\mu F^{\mu\nu}=0, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]
 * $$\displaystyle D_\mu F^{\mu\nu}=0, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]
 * Gauge theory, QFT
 * Yang–Mills (self-dual/anti-self-dual)
 * 4
 * $$ F_{\alpha \beta} = \pm \varepsilon_{\alpha \beta \mu \nu} F^{\mu \nu},
 * $$ F_{\alpha \beta} = \pm \varepsilon_{\alpha \beta \mu \nu} F^{\mu \nu},
 * Instantons, Donaldson theory, QFT
 * Yukawa
 * 1+n
 * $$\displaystyle i \partial_t^{}u + \Delta u = -A u,\quad \displaystyle\Box A = m^2_{} A + |u|^2 $$
 * Meson-nucleon interactions, QFT
 * Zakharov system
 * 1+3
 * $$\displaystyle i \partial_t^{} u + \Delta u = un,\quad \displaystyle \Box n = -\Delta (|u|^2_{})$$
 * Langmuir waves
 * Zakharov–Schulman
 * 1+3
 * $$\displaystyle iu_t + L_1u = \varphi u,\quad \displaystyle L_2 \varphi = L_3( | u |^2)$$
 * Acoustic waves
 * Zeldovich–Frank-Kamenetskii equation
 * 1+3
 * $$\displaystyle u_t = D\nabla^2 u + \frac{\beta^2}{2}u(1-u) e^{-\beta(1-u)}$$
 * Combustion
 * Zoomeron
 * 1+1
 * $$\displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx} +2(u^2)_{xt}=0$$
 * Solitons
 * &phi;4 equation
 * 1+1
 * $$\displaystyle \varphi_{tt}-\varphi_{xx}-\varphi+\varphi^3=0$$
 * QFT
 * &sigma;-model
 * 1+1
 * $$\displaystyle {\mathbf v}_{xt}+({\mathbf v}_x{\mathbf v}_t){\mathbf v}=0$$
 * Harmonic maps, integrable systems, QFT
 * }
 * QFT
 * &sigma;-model
 * 1+1
 * $$\displaystyle {\mathbf v}_{xt}+({\mathbf v}_x{\mathbf v}_t){\mathbf v}=0$$
 * Harmonic maps, integrable systems, QFT
 * }
 * }