List of nonlinear partial differential equations

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In mathematics and physics, nonlinear partial differential equations are (as their name suggests) partial differential equations with nonlinear terms. They describe many different physical systems, ranging from gravitation to fluid dynamics, and have been used in mathematics to solve problems such as the Poincare conjecture and the Calabi conjecture. They are difficult to study: there are almost no general techniques that work for all such equations, and usually each individual equation has to be studied as a separate problem.

[edit] Methods for studying nonlinear partial differential equations

[edit] Existence and uniqueness of solutions

A fundamental question for any PDE is the existence and uniqueness of a solution for given boundary conditions. For nonlinear equations these questions are in general very hard: for example, the hardest part of Yau's solution of the Calabi conjecture was the proof of existence for a Monge-Ampere equation .

[edit] Singularities

The basic questions about singularities (their formation, propagation, and removal, and regularity of solutions) are the same as for linear PDE, but as usual much harder to study. In the linear case one can just use spaces of distributions, but nonlinear PDEs are not usually defined on arbitrary distributions, so one replaces spaces of distributions by refinements such as Sobolev spaces.

An example of singularity formation is given by the Ricci flow: Hamilton showed that while short time solutions exist, singularities will usually form after a finite time. Perelman's solution of the Poincare conjecture depended on a deep study of these singularities, where he showed how to continue the solution past the singularities.

[edit] Linear approximation

The solutions in a neighborhood of a known solution can sometimes be studied by linearizing the PDE around the solution. This corresponds to studying the tangent space of a point of the moduli space of all solutions.

[edit] Moduli space of solutions

Ideally one would like to describe the (moduli) space of all solutions explicitly, and for some very special PDEs this is possible. (In general this is a hopeless problem: it is unlikely that there is any useful description of all solutions of the Navier–Stokes equation for example, as this would involve describing all possible fluid motions.) If the equation has a very large symmetry group, then one is usually only interested in the moduli space of solutions modulo the symmetry group, and this is sometimes a finite dimensional compact manifold, possibly with singularities; for example, this happens in the case of the Seiberg-Witten equations. A slightly more complicated case is the self dual Yang-Mills equations, when the moduli space is finite dimensional but not necessarily compact, though it can often be compactified explicitly. Another case when one can sometimes hope to describe all solutions is the case of completely integrable models, when solutions are sometimes a sort of superposition of solitons; for example, this happens for the KdV equation.

[edit] Exact solutions

It is often possible to write down some special solutions explicitly in terms of elementary functions (though it is rarely possible to describe all solutions like this). One way of finding such explicit solutions is to reduce the equations to equations of lower dimension, preferably ordinary differential equations, which can often be solved exactly. This can sometimes be done using separation of variables, or by looking for highly symmetric solutions.

Some equations have several different exact solutions; there are a particularly large number of known exact solutions of Einstein's field equations.

[edit] Numerical solutions

Main article: Numerical partial differential equations

Numerical solution on a computer is almost the only method that can be used for getting information about arbitrary systems of PDEs. There has been a lot of work done on solving certain systems numerically, especially for the Navier–Stokes and other equations related to weather prediction.

[edit] Lax pair

If a system of PDEs can be put into Lax pair form

$\frac{dL}{dt}=LA-AL$

then it usually has an infinite number of first integrals, which help to study it.

[edit] Euler-Lagrange equations

Systems of PDEs often arise as the Euler-Lagrange equations for a variational problem. Systems of this form can sometimes be solved by finding an extremum of the original variational problem.

[edit] Hamilton equations

[edit] Integrable systems

Main article: integrable systems

PDEs that arise from integrable systems are often the easiest to study, and can sometimes be completely solved. A well known example is the KdV equation.

[edit] Symmetry

Some systems of PDEs have large symmetry groups. For example, the Yang-Mills equations are invariant under and infinite dimensional gauge group, and many systems of equations (such as the Einstein field equations) are invariant under diffeomorphisms of the underlying manifold.

Sometimes equations are parabolic or hyperbolic "modulo the action of some group": for example, the Ricci flow equation is not quite parabolic, but is "parabolic modulo the action of the diffeomorphism group", which implies that it has most of the good properties of parabolic equations.

[edit] Look it up

There are several tables of previously studied PDEs such as (Polyanin & Zaitsev 2004) and (Zwillinger 1998) and the tables below.

[edit] List of equations

[edit] A–F

Name	Dim	Equation	Applications
Benjamin-Bona-Mahony	1+1	$\displaystyle u_t+u_x+uu_x-u_{xxt}=0$	Fluid mechanics
Benjamin-Ono	1+1	$\displaystyle u_t+Hu_{xx}+uu_x$	internal waves in deep water
Boomeron	1+1	$\displaystyle u_t=\mathbf{b}\cdot\mathbf{v}_x$ $\displaystyle \mathbf{v}_{xt}=u_{xx}\mathbf{b}+\mathbf{a}\times\mathbf{v}_x- 2\mathbf{v}\times(\mathbf{v}\times\mathbf{b})$	Solitons
Born-Infeld	1+1	$\displaystyle (1-u_t^2)u_{xx} +2u_xu_tu_{xt}-(1+u_x^2)u_{tt}=0$
Boussinesq	1+1	$\displaystyle u_{tt} - u_{xx} - u_{xxxx} - 3(u^2)_{xx} = 0$	Fluid mechanics
Buckmaster	1+1	$\displaystyle u_t=(u^4)_{xx}+(u^3)_x$	Thin viscous fluid sheet flow
Burgers	1+1	$\displaystyle u_t+uu_x=\nu u_{xx}$	Fluid mechanics
Cahn-Hilliard equation	Any	$\displaystyle \frac{\partial c}{\partial t} = D\nabla^2\left(c^3-c-\gamma\nabla^2 c\right)$	Phase separation
Calabi flow	Any		Calabi-Yau manifolds
Caudrey-Dodd-Gibbon- Sawada-Kotera	1+1	Same as (rescaled) Sawada-Kotera
Carleman	1+1	$\displaystyle u_t+u_x=v^2-u^2=v_x-v_t$
Camassa-Holm	1+1	$\displaystyle u_t=2f_x+fu_x$ $\displaystyle u=f_{xx}/2-2f$	Peakons
Cauchy momentum	any	$\displaystyle \rho \left(\frac{\partial \mathbf{v}}{\partial t} + \mathbf{v} \cdot \nabla \mathbf{v}\right) = \nabla \cdot \sigma + \mathbf{f}$	Momentum transport
Chiral field	1+1
Clairaut equation	any	$x\cdot Du+f(Du)=u$	Differential geometry
Complex Monge-Ampère	Any	$\displaystyle \det(\partial_{i\bar j}\phi) =$ lower order terms	Calabi conjecture
Davey–Stewartson	1+2	$\displaystyle i u_t + c_0 u_{xx} + u_{yy} = c_1 \|u\|^2 u + c_2 u \phi_x$ $\displaystyle \phi_{xx} + c_3 \phi_{yy} = ( \|u\|^2 )_x$	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	$\displaystyle u_t=(u^2-u_x+2w)_x$ , $w t = (2 u w + w x) x$
Drinfel'd -Sokolov -Wilson	1+1	$\displaystyle u_t=3ww_x$ $\displaystyle w_t=2w_{xxx}+2uw_x+u_xw$
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	$\displaystyle R_{ab} - {\textstyle 1 \over 2}R\,g_{ab} = \kappa T_{ab}$	General relativity
Ernst equation	2	$\displaystyle \Re(u)(u_{rr}+u_r/r+u_{zz}) = (u_r)^2+(u_z)^2$
Euler equations	1+3	$\begin{align} &{\partial\rho\over\partial t}+ \nabla\cdot(\rho\bold u)=0\\[1.2ex] &{\partial\rho{\bold u}\over\partial t}+ \nabla\cdot(\bold u\otimes(\rho \bold \bold u))+\nabla p=0\\[1.2ex] &{\partial E\over\partial t}+ \nabla\cdot(\bold u(E+p))=0, \end{align}$	non-viscous fluids
Fisher's equation	1+1	$\displaystyle \frac{\partial u}{\partial t}=u(1-u)+\frac{\partial^2 u}{\partial x^2}.\,$	Gene propagation
Fitzhugh-Nagumo	1+1	$\displaystyle u_t=u_{xx}+u(u-a)(1-u)+w$ $\displaystyle w_t=\epsilon u$

[edit] G–K

Name	Dim	Equation	Applications
Gardner equation	1+1	$\displaystyle u_t=6(u+\epsilon^2u^2)u_x+u_{xxx}$
Garnier equation			isomonodromic deformations
Gauss-Codazzi			surfaces
Ginzburg-Landau	1+3	$\displaystyle \alpha \psi + \beta \|\psi\|^2 \psi + \frac{1}{2m} \left(-i\hbar\nabla - 2e\mathbf{A} \right)^2 \psi = 0$	Superconductivity
Gross-Neveu	1+1
Gross –Pitaevskii	1+n	$\displaystyle i\partial_t\psi = (-\tfrac12\Delta^2 + V(x) + g\|\psi\|^2) \psi$	Bose–Einstein condensate
Hartree equation	Any	$\displaystyle i\partial_tu + \Delta u= V(u)u$ where $\displaystyle V(u)= \pm \|x\|^{-n} * \|u\|^2$ .
Hasegawa -Mima	1+3	$\displaystyle 0=\frac{\partial}{\partial t}\left(\nabla^2\phi-\phi\right)$ $\displaystyle -\left[\left(\nabla\phi\times \mathbf{\hat z}\right)\cdot\nabla\right]\left[\nabla^2\phi-ln\left(\frac{n_0}{\omega_{ci}}\right)\right]$	Turbulence in plasma
Heisenberg ferromagnet	1+1	$\displaystyle \mathbf{S}_t=\mathbf{S}\wedge \mathbf{S}_{xx}.$	Magnetism
Hirota equation	1+1
Hirota -Satsuma	1+1	$\displaystyle u_t=u_{xxx}/2 +3uu_x-6ww_x$ , $\displaystyle w_t+w_{xxx}+3uw_x=0$
Hunter–Saxton	1+1	$\displaystyle (u_t + u u_x)_x = \frac{1}{2} \, u_x^2$	Liquid crystals
Ishimori equation	1+2	$\displaystyle \frac{\partial \mathbf{S}}{\partial t} = \mathbf{S}\wedge \left(\frac{\partial^2 \mathbf{S}}{\partial x^{2}} + \frac{\partial^2 \mathbf{S}}{\partial y^{2}}\right)+ \frac{\partial u}{\partial x}\frac{\partial \mathbf{S}}{\partial y} + \frac{\partial u}{\partial y}\frac{\partial \mathbf{S}}{\partial x}$ $\displaystyle \frac{\partial^2 u}{\partial x^2}-\alpha^2 \frac{\partial^2 u}{\partial y^2}=-2\alpha^2 \mathbf{S}\cdot\left(\frac{\partial \mathbf{S}}{\partial x}\wedge \frac{\partial \mathbf{S}}{\partial y}\right)$	Integrable systems
Kadomtsev –Petviashvili	1+2	$\displaystyle \partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0$	Shallow water waves
von Karman	2	$\displaystyle \nabla^4 u = E(w_{xy}^2-w_{xx}w_{yy})$ , $\displaystyle \nabla^4w = a+b(u_{yy}w_{xx}+u_{xx}w_{yy}-2u_{xy}w_{xy}$
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	$\displaystyle \nabla^2s=(\|\mathbf a\|^2+1)s$ , $\displaystyle \nabla^2\mathbf a =\nabla(\nabla\cdot\mathbf a)+s^2\mathbf a$
Klein -Gordon (nonlinear)	any	$\nabla^2u+\lambda u^p=0$
Klein -Gordon -Zakharov
Khokhlov -Zabolotskaya	1+2	$\displaystyle u_{xt} -(uu_x)_x =u_{yy}$
Korteweg–de Vries (KdV)	1+1	$\displaystyle \partial_tu+\partial^3_x u+6u\partial_x u=0$	Shallow waves, Integrable systems
KdV (generalized)	1+1	$\displaystyle \partial_t u + \partial_x^3 u + \partial_x f(u)) = 0$
KdV (modified)	1+1	$\displaystyle \partial_t u + \partial_x^3 u \pm 6u^2\partial_x u = 0$
KdV (super)	1+1	$\displaystyle u_t=6uu_x-u_{xxx}+3ww_{xx}$ , $\displaystyle w_t=3u_xw+6uw_x-4w_{xxx}$
There are more minor variations listed in the article on KdV equations.
Kuramoto -Sivashinsky	1+n	$\displaystyle u_t+\nabla^4u+\nabla^2u+\|\nabla^2u\|^2/2=0$

[edit] L–R

Name	Dim	Equation	Applications
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	$\displaystyle 2u_{tx}+u_xu_{xx}-u_{yy}$
Liouville	any	$\displaystyle \nabla^2u+e^{\lambda u}=0$
Minimal surface	3	$\displaystyle div(Du/\sqrt{1+\|Du\|^2})=0$	minimal surfaces
Molenbroeck	2
Monge–Ampère	any	$\displaystyle \det(\partial_{ij}\phi) =$ lower order terms
Navier–Stokes (and its derivation)	1+3	$\displaystyle \rho \left( \frac{\partial v_i}{\partial t} + v_j \frac{\partial v_i}{\partial x_j} \right) = - \frac{\partial p}{\partial x_i} + \frac{\partial}{\partial x_j} \left[ \mu \left( \frac{\partial v_i}{\partial x_j} + \frac{\partial v_j}{\partial x_i} \right) + \lambda \frac{\partial v_k}{\partial x_k} \right] + f_i$ + 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
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	$\displaystyle (1+u_x^2)u_{xx} -2u_xu_yu_{xy} +(1+u_y^2)u_{yy}=0$
Pohlmeyer -Lund -Regge	2	$\displaystyle u_{xx}-u_{yy}\pm \sin u \cos u +\frac{\cos u}{\sin^3 u}(v_x^2-v_y^2)=0$ $\displaystyle (v_x\cot^2u)_x = (v_y\cot^2 u)_y$
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
Primitive equations	1+3		Atmospheric models

[edit] S–Z, α–ω

Name	Dim	Equation	Applications
Rayleigh	2	$\displaystyle u_{tt}-u_{xx} = \epsilon(u_t-u_t^3)$
Ricci flow	Any	$\displaystyle \partial_t g_{ij}=-2 R_{ij}$	Poincare conjecture
Richards equation	1+3	$\displaystyle \frac{\partial \theta}{\partial t}= \frac{\partial}{\partial z} \left[ K(\psi) \left (\frac{\partial \psi}{\partial z} + 1 \right) \right]\$	Variably-saturated flow in porous media
Sawada-Kotera	1+1	$\displaystyle u_t+45u^2u_x+15u_xu_{xx}+15uu_{xxx}+u_{xxxx}=0$
Schlesinger	Any	$\displaystyle \begin{align} {\partial A_i \over \partial t_j} &= {\left[ A_i, \ A_j \right] \over t_i - t_j}, \quad i\neq j \\ {\partial A_i \over \partial t_i} &=- \sum_{j=1 \atop j\neq i}^n {\left[ A_i, \ A_j \right] \over t_i - t_j}, \quad 1\leq i, j \leq n \end{align}$	isomonodromic deformations
Seiberg-Witten	1+3	$\displaystyle D^A\phi=0, \qquad F^+_A=\sigma(\phi)$	Seiberg-Witten invariants , QFT
Shallow water	1+2	$\displaystyle \begin{align} \frac{\partial \eta }{\partial t} + \frac{\partial (\eta u)}{\partial x} + \frac{\partial (\eta v)}{\partial y} = 0\\[3pt] \frac{\partial (\eta u)}{\partial t}+ \frac{\partial}{\partial x}\left( \eta u^2 + \frac{1}{2}g \eta^2 \right) + \frac{\partial (\eta u v)}{\partial y} = 0\\[3pt] \frac{\partial (\eta v)}{\partial t} + \frac{\partial (\eta uv)}{\partial x} + \frac{\partial}{\partial y}\left(\eta v^2 + \frac{1}{2}g \eta ^2\right) = 0 \end{align}$	shallow water waves
Sine-Gordon	1+1	$\displaystyle \, \phi_{tt}- \phi_{xx} + \sin\phi = 0$	Solitons, QFT
Sinh-Gordon	1+1	$\displaystyle u_{xt}= \sinh u$	Solitons, QFT
Sinh-poisson	1+n	$\displaystyle \nabla^2u+\sinh u=0$
Swift-Hohenberg	any	$\displaystyle \frac{\partial u}{\partial t} = r u - (1+\nabla^2)^2u + N(u)$	pattern forming
Three-wave equation	1+n		Integrable systems
Thomas equation	2	$\displaystyle u_{xy}+\alpha u_x+\beta u_y+\gamma u_xu_y=0$
Thirring model	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	$\displaystyle (\partial_t+\partial_z^3+\partial_{\bar z}^3)v+\partial_z(uv)+\partial_{\bar z}(uw) =0$ , $\displaystyle \partial_{\bar z}u=3\partial_zv$ , $\displaystyle \partial_zw=3\partial_{\bar z} v$
Wadati- Konno- Ichikawa- Schimizu	1+1	$\displaystyle iu_t+((1+\|u\|^2)^{-1/2}u)_{xx}=0$
WDVV equations	Any	$\displaystyle \sum_{\sigma, \tau = 1}^n\left({\partial^3 F \over \partial t^\alpha t^\beta t^\sigma} \eta^{\sigma \tau} {\partial^3 F \over \partial t^\mu t^\nu t^\tau} \right)$ $\displaystyle = \sum_{\sigma, \tau = 1}^n\left({\partial^3 F \over \partial t^\alpha t^\nu t^\sigma} \eta^{\sigma \tau} {\partial^3 F \over \partial t^\mu t^\beta t^\tau} \right)$	Topological field theory, QFT
WZW model	1+1		QFT
Witham equation			phase averaging
Yamabe	n	$\displaystyle\Delta \phi+h(x)\phi = \lambda f(x)\phi^{(n+2)/(n-2)}$	Differential geometry
Yang-Mills equation (source-free)	Any	$\displaystyle D_\mu F^{\mu\nu}=0, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]$	Gauge theory, QFT
Yang-Mills (self-dual/anti-self-dual)	4	$F_{\alpha \beta} = \pm \epsilon_{\alpha \beta \mu \nu} F^{\mu \nu}, \quad F_{\mu \nu} = A_{\mu, \nu} - A_{\nu, \mu }+ [A_\mu, \, A_\nu]$	Instantons, Donaldson theory, QFT
Yukawa equation	1+n	$\displaystyle i \partial_t^{}u + \Delta u = -A u$ $\displaystyle\Box A = m^2_{} A + \|u\|^2$	Meson-nucleon interactions, QFT
Zakharov system	1+3	$\displaystyle i \partial_t^{} u + \Delta u = un$ $\displaystyle \Box n = -\Delta (\|u\|^2_{})$	Langmuir waves
Zakharov-Schulman	1+3	$\displaystyle iut + L_1u = \phi u$ $\displaystyle L_2 \phi = L_3( \| u \|^2)$	Acoustic waves
Zoomeron	1+1	$\displaystyle (u_{xt}/u)_{tt}-(u_{xt}/u)_{xx} +2(u^2)_{xt}=0$	Solitons
φ⁴ equation	1+1	$\displaystyle \phi_{tt}-\phi_{xx}-\phi+\phi^3=0$	QFT
σ-model	1+1	$\displaystyle {\mathbf v}_{xt}+({\mathbf v}_x{\mathbf v}_t){\mathbf v}=0$	Harmonic maps, integrable systems, QFT

[edit] See also

[edit] References

Calogero, Francesco & Degasperis, Antonio (1982), Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations, vol. 13, Studies in Mathematics and its Applications, Amsterdam-New York: North-Holland Publishing Co., MR 0680040, ISBN 0-444-86368-0
Pokhozhaev, S.I. (2001), “Non-linear partial differential equation”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
Polyanin, Andrei D. & Zaitsev, Valentin F. (2004), Handbook of nonlinear partial differential equations, Boca Raton, FL: Chapman & Hall/CRC, pp. xx+814, MR 2042347, ISBN 1-58488-355-3
Scott, Alwyn, ed. (2004), Encyclopedia of Nonlinear Science, Routledge, ISBN 978-1579583859 . For errata, see this
Zwillinger, Daniel (1998), Handbook of differential equations (3rd ed.), Boston, MA: Academic Press, Inc., MR 0977062, ISBN 978-0127843964

[edit] External links