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 MongeAmpere 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 SeibergWitten equations. A slightly more complicated case is the self dual YangMills 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
Main article: Numerical partial differential equationsNumerical 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
then it usually has an infinite number of first integrals, which help to study it.
EulerLagrange equations
Systems of PDEs often arise as the EulerLagrange equations for a variational problem. Systems of this form can sometimes be solved by finding an extremum of the original variational problem.
Hamilton equations
Further information: Hamiltonian mechanicsIntegrable systems
Main article: integrable systemsPDEs 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 YangMills 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 Fluid mechanics BenjaminOno 1+1 internal waves in deep water Boomeron 1+1 Solitons BornInfeld 1+1 Boussinesq 1+1 Fluid mechanics Buckmaster 1+1 Thin viscous fluid sheet flow Burgers 1+1 Fluid mechanics CahnHilliard equation Any Phase separation Calabi flow Any CalabiYau manifolds Camassa–Holm 1+1 Peakons Carleman 1+1 Cauchy momentum any Momentum transport CaudreyDoddGibbon SawadaKotera 1+1 Same as (rescaled) SawadaKotera Chiral field 1+1 Clairaut equation any Differential geometry Complex MongeAmpère Any lower order terms Calabi conjecture Davey–Stewartson 1+2 Finite depth waves DegasperisProcesi 1+1 Peakons Dispersive long wave 1+1 , w_{t} = (2uw + w_{x})_{x} Drinfel'd Sokolov Wilson 1+1 Dym equation 1+1 Solitons Eckhaus equation 1+1 Integrable systems Eikonal equation any optics Einstein field equations Any General relativity Ernst equation 2 Euler equations 1+3 nonviscous fluids Fisher's equation 1+1 Gene propagation FitzhughNagumo 1+1 G–K
Name Dim Equation Applications Gardner equation 1+1 Garnier equation isomonodromic deformations GaussCodazzi surfaces GinzburgLandau 1+3 Superconductivity GrossNeveu 1+1 Gross –Pitaevskii 1+n Bose–Einstein condensate Hartree equation Any where .
HasegawaMima 1+3 Turbulence in plasma Heisenberg ferromagnet 1+1 Magnetism Hirota equation 1+1 Hirota Satsuma 1+1 , Hunter–Saxton 1+1 Liquid crystals Ishimori equation 1+2 Integrable systems Kadomtsev –Petviashvili 1+2 Shallow water waves von Karman 2 , Kaup 1+1 Kaup –Kupershmidt 1+1 Integrable systems Klein Gordon Maxwell any , Klein Gordon (nonlinear) any Klein Gordon Zakharov Khokhlov Zabolotskaya 1+2 Korteweg–de Vries (KdV) 1+1 Shallow waves, Integrable systems KdV (generalized) 1+1 KdV (modified) 1+1 KdV (super) 1+1 , There are more minor variations listed in the article on KdV equations. Kuramoto Sivashinsky 1+n L–R
Name Dim Equation Applications Landau–Lifshitz model 1+n Magnetic field in solids LinTsien equation 1+2 Liouville any Minimal surface 3 minimal surfaces Molenbroeck 2 Monge–Ampère any lower order terms Navier–Stokes
(and its derivation)1+3
+ mass conservation:
+ an equation of state to relate p and ρ, e.g. for an incompressible flow:Fluid flow Nonlinear Schrödinger (cubic) 1+1 optics, water waves Nonlinear Schrödinger (derivative) 1+1 optics, water waves Novikov–Veselov equation 1+2 see Veselov–Novikov equation below Omega equation 1+3 atmospheric physics Plateau 2 Pohlmeyer Lund Regge 2 Porous medium 1+n diffusion Prandtl 1+2 , boundary layer Primitive equations 1+3 Atmospheric models S–Z, α–ω
Name Dim Equation Applications Rayleigh 2 Ricci flow Any Poincaré conjecture Richards equation 1+3 Variablysaturated flow in porous media SawadaKotera 1+1 Schlesinger Any isomonodromic deformations SeibergWitten 1+3 SeibergWitten invariants, QFT Shallow water 1+2 shallow water waves SineGordon 1+1 Solitons, QFT SinhGordon 1+1 Solitons, QFT Sinhpoisson 1+n SwiftHohenberg any pattern forming Threewave equation 1+n Integrable systems Thomas equation 2 Thirring model 1+1 , Dirac field, QFT Toda lattice any Veselov–Novikov equation 1+2 , , shallow water waves Wadati Konno Ichikawa Schimizu 1+1 WDVV equations Any Topological field theory, QFT WZW model 1+1 QFT Witham equation phase averaging Yamabe n Differential geometry YangMills equation (sourcefree) Any Gauge theory, QFT YangMills (selfdual/antiselfdual) 4 Instantons, Donaldson theory, QFT Yukawa equation 1+n Mesonnucleon interactions, QFT Zakharov system 1+3 Langmuir waves Zakharov–Schulman 1+3 Acoustic waves Zoomeron 1+1 Solitons φ^{4} equation 1+1 QFT σmodel 1+1 Harmonic maps, integrable systems, QFT See also
 EulerLagrange equation
 Nonlinear system
 Integrable system
 Inverse scattering transform
 Dispersive partial differential equation
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, AmsterdamNew York: NorthHolland Publishing Co., ISBN 0444863680, MR0680040
 Pokhozhaev, S.I. (2001), "Nonlinear partial differential equation", in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Springer, ISBN 9781556080104, 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 1584883553, MR2042347
 Scott, Alwyn, ed. (2004), Encyclopedia of Nonlinear Science, Routledge, ISBN 9781579583859. For errata, see this
 Zwillinger, Daniel (1998), Handbook of differential equations (3rd ed.), Boston, MA: Academic Press, Inc., ISBN 9780127843964, MR0977062
External links
Categories: Partial differential equations
 Solitons
 Differential geometry
 Exactly solvable models
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