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
Name Dim Equation Applications Bateman-Burgers equation 1+1 Fluid mechanics Benjamin–Bona–Mahony 1+1 Fluid mechanics Benjamin–Ono 1+1 internal waves in deep water Boomeron 1+1 Solitons Boltzmann equation 1+6 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 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 Chafee–Infante equation Clairaut equation any Differential geometry Clarke's equation 1+1 Combustion 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
G–K
Name Dim Equation Applications G equation 1+3 turbulent combustion Generic scalar transport 1+3 transport Ginzburg–Landau 1+3 Superconductivity Gross–Pitaevskii 1 + n Bose–Einstein condensate Gyrokinetics equation 1 + 5 Microturbulence in plasma 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 Hicks 1+1 Fluid dynamics Hunter–Saxton 1+1 Liquid crystals Ishimori equation 1+2 Integrable systems Kadomtsev –Petviashvili 1+2 Shallow water waves Kardar–Parisi–Zhang equation 1+3 Stochastics 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
L–Q
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 Logarithmic Schrödinger equation any Superfluids, Quantum gravity Minimal surface 3 minimal surfaces 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, gas flow Nonlinear Schrödinger (cubic) 1+1 optics, water waves Nonlinear Schrödinger (derivative) 1+1 optics, water waves Omega equation 1+3 atmospheric physics Plateau 2 Pohlmeyer–Lund–Regge 2 Porous medium 1+n diffusion Prandtl 1+2 , boundary layer
R–Z, α–ω
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 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 QFT Whitham equation 1+1 water waves Williams spray equation Combustion 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
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References
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