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 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 1+1 Cauchy momentum any Momentum transport Chafee–Infante equation Clairaut equation any Differential geometry Clarke's equation 1+1 Combustion Any lower order terms Calabi conjecture Davey–Stewartson 1+2 Finite depth waves Degasperis–Procesi 1+1 Peakons 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 2 1+1 Kaup–Kupershmidt 1+1 Integrable systems any Klein–Gordon (nonlinear) any Relativistic quantum mechanics 1+2 Korteweg–de Vries (KdV) 1+1 Shallow waves, Integrable systems 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 2 minimal surfaces 2 1+n diffusion 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 1+1 compacton solutions 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 1+n Fluid Mechanics Swift–Hohenberg any pattern forming 2 Thirring 1+1 , Dirac field, QFT Toda lattice any Veselov–Novikov 1+2 , , shallow water waves Vorticity equation Fluid Mechanics 1+1 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 (source-free) Any Gauge theory, QFT Yang–Mills (self-dual/anti-self-dual) 4 Instantons, Donaldson theory, QFT 1+n Meson-nucleon interactions, QFT Zakharov system 1+3 Langmuir waves Zakharov–Schulman 1+3 Acoustic waves 1+1 Solitons φ4 equation 1+1 QFT σ-model 1+1 Harmonic maps, integrable systems, QFT
References[]
Categories:
- Partial differential equations