List of nonlinear partial differential equations

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 Poincaré 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.

Contents

Methods for studying nonlinear partial differential equations

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.

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 Poincaré conjecture depended on a deep study of these singularities, where he showed how to continue the solution past the singularities.

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.

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 Korteweg–de Vries equation.

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.

Numerical solutions

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.

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.

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.

Hamilton equations

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 Korteweg–de Vries equation.

Symmetry

Some systems of PDEs have large symmetry groups. For example, the Yang-Mills equations are invariant under an infinite dimensional gauge group, and many systems of equations (such as the Einstein field equations) are invariant under diffeomorphisms of the underlying manifold. Any such symmetry groups can usually be used to help study the equations; in particular if one solution is known one can trivially generate more by acting with the symmetry group.

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.

Look it up

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

List of equations

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=0 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
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 + \mathbf{f} Momentum transport
Caudrey-Dodd-Gibbon- Sawada-Kotera 1+1 Same as (rescaled) Sawada-Kotera
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, wt = (2uw + wx)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

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 u|^2/2=0

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 \operatorname{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
Novikov–Veselov equation 1+2 see Veselov–Novikov equation below
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

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} Poincaré 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 equation 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
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 iu_t + 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

See also

References

  • Calogero, Francesco; Degasperis, Antonio (1982), Spectral transform and solitons. Vol. I. Tools to solve and investigate nonlinear evolution equations, Studies in Mathematics and its Applications, 13, Amsterdam-New York: North-Holland Publishing Co., ISBN 0-444-86368-0, MR0680040 
  • Pokhozhaev, S.I. (2001), "Non-linear partial differential equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 978-1556080104, http://eom.springer.de/N/n067170.htm 
  • Polyanin, Andrei D.; Zaitsev, Valentin F. (2004), Handbook of nonlinear partial differential equations, Boca Raton, FL: Chapman & Hall/CRC, pp. xx+814, ISBN 1-58488-355-3, MR2042347 
  • 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., ISBN 978-0127843964, MR0977062 

External links


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