List of nonlinear partial differential equations

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See also Nonlinear partial differential equation and List of partial differential equation topics.


Name Dim Equation Applications
Benjamin–Bona–Mahony 1+1 Fluid mechanics
Benjamin–Ono 1+1 internal waves in deep water
Boomeron 1+1

Boltzmann equation Statistical mechanics
Born–Infeld 1+1 Electrodynamics
Boussinesq 1+1 Fluid mechanics
Boussinesq type equation 1+1 Fluid mechanics
Buckmaster 1+1 Thin viscous fluid sheet flow
Burgers 1+1 Fluid mechanics
Cahn–Hilliard equation Any Phase separation
Calabi flow Any Calabi–Yau manifolds
Camassa–Holm 1+1 Peakons
Carleman 1+1
Cauchy momentum any Momentum transport
Caudrey–Dodd–Gibbon–Sawada–Kotera 1+1 Same as (rescaled) Sawada–Kotera
Chafee–Infante equation
Chiral field 1+1
Clairaut equation any Differential geometry
Complex Monge–Ampère Any lower order terms Calabi conjecture
Davey–Stewartson 1+2

Finite depth waves
Degasperis–Procesi 1+1 Peakons
Dispersive long wave 1+1 ,
Drinfeld–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
Estevez–Mansfield–Clarkson equation
Euler equations 1+3 non-viscous fluids
Fisher's equation 1+1 Gene propagation
Fitzhugh–Nagumo 1+1

Biological neuron model
Föppl–von Kármán equations Solid Mechanics


Name Dim Equation Applications
G equation

turbulent combustion
Gardner equation 1+1
Garnier equation isomonodromic deformations
Gauss–Codazzi surfaces
Ginzburg–Landau 1+3 Superconductivity
Gross–Neveu 1+1
Gross–Pitaevskii 1 + n Bose–Einstein condensate
Guzmán 1 + n Hamilton–Jacobi–Bellman equation for risk aversion
Hartree equation Any
Hasegawa–Mima 1+3 Turbulence in plasma
Heisenberg ferromagnet 1+1 Magnetism
Hirota equation[1] 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 Relativistic quantum mechanics
Khokhlov–Zabolotskaya 1+2
Korteweg–de Vries (KdV) 1+1 Shallow waves, Integrable systems
KdV (super) 1+1
There are more minor variations listed in the article on KdV equations.
Kuramoto–Sivashinsky equation 1 + n Combustion


Name Dim Equation Applications
Landau–Lifshitz model 1+n Magnetic field in solids
Lin–Tsien equation 1+2
Liouville equation any
Liouville–Bratu–Gelfand equation any combustion, astrophysics
Minimal surface 3 minimal surfaces
Molenbroeck 2
Monge–Ampère any lower order terms
(and its derivation)

+ mass conservation:
+ an equation of state to relate p and ρ, e.g. for an incompressible flow:

Fluid flow, gas 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

R–Z, α–ω[edit]

Name Dim Equation Applications
Rayleigh 1+1
Ricci flow Any Poincaré conjecture
Richards equation 1+3 Variably saturated flow in porous media
Rosenau–Hyman equation 1+1 compacton solutions
Sawada–Kotera 1+1
Schlesinger Any 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
Swift–Hohenberg any pattern forming
Three-wave 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
Vorticity equation Fluid Mechanics
Wadati–Konno–Ichikawa–Schimizu 1+1
WDVV equations Any Topological field theory, QFT
WZW model 1+1

Whitham equation 1+1 water waves
Williams spray equation

Yamabe n Differential geometry
Yang–Mills equation (source-free) Any Gauge theory, QFT
Yang–Mills (self-dual/anti-self-dual) 4 Instantons, Donaldson theory, QFT
Yukawa equation 1+n

Meson-nucleon interactions, QFT
Zakharov system 1+3

Langmuir waves
Zakharov–Schulman 1+3

Acoustic waves
Zoomeron 1+1 Solitons
φ4 equation 1+1 QFT
σ-model 1+1 Harmonic maps, integrable systems, QFT


  1. ^ Shu, ±Jian-Jun (2003). "Exact N-envelope-soliton solutions of the Hirota equation". Optica Applicata. 33 (2-3): 539–546.