Nonlinear partial differential equation

In mathematics and physics, non-linear partial differential equations are (as their name suggests) partial differential equations with non-linear 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 Poincare 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.

List of equations

Name	Dim	Equation	Applications
Benjamin-Bona-Mahony	1+1	$u_{t}-u_{xxx}+uu_{x}=0$	Fluid mechanics
Benjamin-Ono	1+1	$u_{t}+Hu_{xx}+uu_{x}$	internal waves in deep water
Boussinesq	1+1	$u_{tt}-u_{xx}-u_{xxxx}+3(u^{2})_{xx}=0$	Fluids
Burgers	1+1	$u_{t}+uu_{x}=\nu u_{xx}$	Fluid mechanics
Cahn-Hilliard equation	Any	${\frac {\partial c}{\partial t}}=D\nabla ^{2}\left(c^{3}-c-\gamma \nabla ^{2}c\right)$	Phase separation
Calabi flow	Any		Calabi-Yau manifolds
Cauchy momentum equation	any	$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=\nabla \cdot \sigma +\mathbf {f}$	Momentum transport
Chaplygin's equation	1+3	$\Psi _{\theta \theta }+{\frac {v^{2}}{1-{\frac {v^{2}}{c^{2}}}}}\Psi _{vv}+v\Psi _{v}=0.$	transonic flow
Complex Monge-Ampère equation	Any		Calabi conjecture
Davey–Stewartson equation	1+2	$iu_{t}+c_{0}u_{xx}+u_{yy}=c_{1}|u|^{2}u+c_{2}u\phi _{x}$ $\phi _{xx}+c_{3}\phi _{yy}=(|u|^{2})_{x}$	Finite depth waves
Dym equation	1+1	$u_{t}=u^{3}u_{xxx}.\,$	Solitons
Eikonal equation	1+3	$|\nabla u(x)|=F(x),\ x\in \Omega$	optics
Einstein field equations	1+3, Any	$R_{ab}-{\textstyle 1 \over 2}R\,g_{ab}=\kappa T_{ab}$	General relativity
Fisher's equation	1+1	${\frac {\partial u}{\partial t}}=u(1-u)+{\frac {\partial ^{2}u}{\partial x^{2}}}.\,$	Gene propagation
Generalized Korteweg-de Vries equation
Generic scalar transport equation	1+3	${\frac {\partial \phi }{\partial t}}+\nabla \cdot f(t,x,\phi ,\nabla \phi )=g(t,x,\phi )$	transport
Ginzburg-Landau-Schrodinger equation
Hartree equation
Hasegawa-Mima equation	1+3	$0={\frac {\partial }{\partial t}}\left(\nabla ^{2}\phi -\phi \right)$ $-\left[\left(\nabla \phi \times \mathbf {\hat {z}} \right)\cdot \nabla \right]\left[\nabla ^{2}\phi -ln\left({\frac {n_{0}}{\omega _{ci}}}\right)\right]$	Turbulence in plasma
Ishimori equation	1+2	${\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \left({\frac {\partial ^{2}\mathbf {S} }{\partial x^{2}}}+{\frac {\partial ^{2}\mathbf {S} }{\partial y^{2}}}\right)+{\frac {\partial u}{\partial x}}{\frac {\partial \mathbf {S} }{\partial y}}+{\frac {\partial u}{\partial y}}{\frac {\partial \mathbf {S} }{\partial x}}$ ${\frac {\partial ^{2}u}{\partial x^{2}}}-\alpha ^{2}{\frac {\partial ^{2}u}{\partial y^{2}}}=-2\alpha ^{2}\mathbf {S} \cdot \left({\frac {\partial \mathbf {S} }{\partial x}}\wedge {\frac {\partial \mathbf {S} }{\partial y}}\right)$
Kadomtsev–Petviashvili equation	1+2	$\partial _{x}(\partial _{t}u+u\partial _{x}u+\epsilon ^{2}\partial _{xxx}u)+\lambda \partial _{yy}u=0$	Shallow water waves
Kaup–Kupershmidt equation	1+1	$u_{t}=u_{xxxxx}+10u_{xxx}u+25u_{xx}u_{x}+20u^{2}u_{x}$ $={\frac {1}{6}}(6u_{xxxx}+60uu_{xx}+45u_{x}^{2}+40u^{3})_{x}$	Integrable systems
Klein-Gordon-Zakharov system
Korteweg–de Vries equation, integrable	1+1	$\partial _{t}\phi +\partial _{x}^{3}\phi +6\phi \partial _{x}\phi =0$	Shallow water waves
Landau–Lifshitz model	1+any	${\frac {\partial \mathbf {S} }{\partial t}}=\mathbf {S} \wedge \sum _{i}{\frac {\partial ^{2}\mathbf {S} }{\partial x_{i}^{2}}}+\mathbf {S} \wedge J\mathbf {S}$	Magnetic field in solids
Lin-Tsien equation	1+2	$2u_{tx}+u_{x}u_{xx}-u_{yy}$
Modified Korteweg-de Vries equation
Monge–Ampère equation	any
Myrzakulov equations
Navier-Stokes equations	1+3	$\rho \left({\frac {\partial \mathbf {v} }{\partial t}}+\mathbf {v} \cdot \nabla \mathbf {v} \right)=-\nabla p+\nabla \cdot \mathbb {T} +\mathbf {f}$	Fluid flow
Nonlinear Schrödinger equation	1+1	$i\partial _{t}\psi =-{1 \over 2}\partial _{x}^{2}\psi +\kappa |\psi |^{2}\psi$	optics, water waves
Omega equation	1+3	$\nabla ^{2}\omega +{\frac {f^{2}}{\sigma }}{\frac {\partial ^{2}\omega }{\partial p^{2}}}$ $={\frac {f}{\sigma }}{\frac {\partial }{\partial p}}\mathbf {V} _{g}\cdot \nabla _{p}(\zeta _{g}+f)+{\frac {R}{\sigma p}}\nabla _{p}^{2}(\mathbf {V} _{g}\cdot \nabla _{p}T),$	atmospheric physics
Primitive equations	1+3		Atmospheric models
Ricci flow	Any	$\partial _{t}g_{ij}=-2R_{ij}$	Poincare conjecture
Richards equation	1+3	${\frac {\partial \theta }{\partial t}}={\frac {\partial }{\partial z}}\left[K(\psi )\left({\frac {\partial \psi }{\partial z}}+1\right)\right]\$	Water flow
Seiberg-Witten equations	1+3	$D^{A}\phi =0$, $F_{A}^{+}=\sigma (\phi )$	Seiberg-Witten invariants
Self dual Yang-Mills equation	1+3, 4		Donaldson theory
Shallow water equations	1+3	${\begin{aligned}{\frac {\partial \eta }{\partial t}}+{\frac {\partial (\eta u)}{\partial x}}+{\frac {\partial (\eta v)}{\partial y}}=0\\[3pt]{\frac {\partial (\eta u)}{\partial t}}+{\frac {\partial }{\partial x}}\left(\eta u^{2}+{\frac {1}{2}}g\eta ^{2}\right)+{\frac {\partial (\eta uv)}{\partial y}}=0\\[3pt]{\frac {\partial (\eta v)}{\partial t}}+{\frac {\partial (\eta uv)}{\partial x}}+{\frac {\partial }{\partial y}}\left(\eta v^{2}+{\frac {1}{2}}g\eta ^{2}\right)=0\end{aligned}}$	shallow waves
Sine-Gordon equation	1+1	$\,\phi _{tt}-\phi _{xx}+\sin \phi =0$	Solitons
Swift-Hohenberg equation	any	${\frac {\partial u}{\partial t}}=ru-(1+\nabla ^{2})^{2}u+N(u)$	pattern forming
Thirring model	1+n		Dirac field
Yang-Mills equation	Any		gauge theory
Zakharov system	1+3	$i\partial _{t}^{}u+\Delta u=un$ $\Box n=-\Delta (|u|_{}^{2})$	Langmuir waves
Zakharov-Schulman system	1+3	$iut+L_{1}u=\phi u$ $L_{2}\phi =L_{3}(|u|^{2})$	Acoustic waves

See also

• Euler-Lagrange equation
• Nonlinear system
• soliton
• Integrable system
• Inverse scattering transform
• Dispersive partial differential equation

References

• 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
• Zwillinger, Daniel (1998), Handbook of differential equations (3rd ed.), Boston, MA: Academic Press, Inc., ISBN 978-0127843964, MR0977062

External links

• EqWorld
• dispersive PDE wiki