List of nonlinear partial differential equations explained

\displaystyleu_t+uux=\nuu_xx

\displaystyleu_{t+ux+uux-uxxt}=0

\displaystyleu_t+Huxx+uu_x=0

\displaystyleu_{t=b ⋅ vx,}   \displaystylev_xt=u_xxb+a x v_{x- 2v x (v x b)}

	\partialfi
	\partialt

+

	pi
	mi

⋅ \nablaf_i+F ⋅

	\partialfi
	\partialpi

=\left(

	\partialfi
	\partialt

\right)_coll,   \left(

	\partialfi
	\partialt

\right)_coll=

	n
\sum
	j=1

\iintg_ijI_ij(g_ij,\Omega)[f'_if'_j-f_ifj]d\Omegad^3p'

\displaystyle

	2)u
(1-u
	xx

+2u_xutuxt

	2)u
-(1+u
	tt

=0

\displaystyleu_tt-u_xx-u_xxxx-

	2)
3(u
	xx

=0

\displaystyleu_tt-u_xx-2\alpha(uu_x)x-\betau_xxtt=0

\displaystyle

	4)
u
	xx

	3)
+(u
	x

\displaystylec_t=D\nabla^{2\left(c3-c-\gamma\nabla2}c\right)

	\partialgij
	\partialt

=(\DeltaR)g_ij

u_t+2\kappau_x-u_xxt+3uu_x=2u_xu_xx+uu_xxx

\displaystyleu_t+u

	2-u
	x=v

	2=v
	x-v

_t

\displaystyle\rho\left(

	\partialv
	\partialt

+v ⋅ \nablav\right)=\nabla ⋅ \sigma+\rhof

u_t-uxx+λ(u^3-u)=0

x ⋅ Du+f(Du)=u

(\theta_t-\gamma\deltae^\theta)_tt

	2(\theta
=\nabla
	t-\delta

e^\theta)

\displaystyle\det(\partial_i\bar\varphi)=

z_yy+\left(

	1
	z

\right)_xx+2=0

\displaystyleiu_t+c₀u_xx+u_yy=c₁

\displaystyleu_t-u_xxt+4uu_x=3u_xu_xx+uu_xxx

\displaystyle

	2-u
u
	x+2w)

_x

w_t=(2uw+wx)x

\displaystyleu_t=3wwx,   \displaystylew_t=2wxxx+2uw_x+uxw

\displaystyleu_t=

	3u
u
	xxx

.

iu_t+uxx+2

\displaystyle

\displaystyleR_\mu\nu-{style1\over2}Rg_\mu\nu+Λg_\mu\nu=

	8\piG
	c4

T_\mu\nu

\displaystyle\Re(u)(u_rr+u_r/r+uzz)=

	2
(u
	z)

U_tyyy+\betaU_yU_yt+\betaU_yyU_t+Utt=0inwhichU=u(x,y,t)

	\partial\rho
	\partialt

+\nabla ⋅ (\rhou)=0,   \rho\left(

	\partialu
	\partialt

+v ⋅ \nablav\right)=-\nablap+\rhof,  

	\partials
	\partialt

+v ⋅ \nablas=0

\displaystyleu_t=u(1-u)+uxx

\displaystyleu_t=uxx+u(u-a)(1-u)+w,   \displaystylew_{t=\varepsilon}u

	Eh3
	12(1-\nu2)

\nabla⁴w-h

	\partial
	\partialx\beta

\left(\sigma_\alpha\beta

	\partialw
	\partialx\alpha

\right)=P,  

	\partial\sigma\alpha\beta
	\partialx\beta

=0

G_t+v ⋅ \nablaG=S_L(G)

\displaystyle\varphi_t+\nabla ⋅ f(t,x,\varphi,\nabla\varphi)=g(t,x,\varphi)

\displaystyle\alpha\psi+\beta

\displaystylei\partial_t\psi=\left(-\tfrac12\nabla²+V(x)+g

{\displaystyle{

	\partialhs
	\partialt

\displaystyleJ_t+gJ

	2J
	xx

	2+f=0
-λ\sigma
	x)

\displaystylei\partial_tu+\Deltau=\left(\pm

\displaystyle0=

	\partial
	\partialt

\left(\nabla²\varphi-\varphi\right)-\left[\left(\nabla\varphi x \hat{z

\displaystyleS_t=S\wedgeS_xx.

\psi_rr-\psi_r/r+\psi_zz=r²dH/d\psi-\Gammad\Gamma/d\psi

\displaystyle\left(u_t+uu_x\right)_x=\tfrac{1}{2}

	2
u
	x

\displaystyleS_t=S\wedge\left(S_xx+S_yy\right)+u_xSy+u_ySx, \displaystyleu_xx-\alpha²u_yy=-2\alpha²S ⋅ \left(S_x\wedgeS_y\right)

\displaystyle\partial_x\left(\partial_tu+u\partial_x

	2\partial
u+\varepsilon
	xxx

u\right)+λ\partial_yyu=0

\displaystyleh_t=\nu\nabla²h+λ(\nablah)²/2+η

\displaystyle\nabla⁴u=E\left

	2-w
(w
	xx

w_yy\right),   \nabla⁴w=a+b\left(u_yyw_xx+u_xxw_yy-2u_xyw_xy\right)

\displaystylef_{x=2fgc(x-t)=gt}

\displaystyleu_t=u_xxxxx+10u_xxxu+25u_xx

	2u
u
	x

\displaystyle\nabla^2s=\left(

\nabla^2u+λu^p=0

\displaystyleu_xt-(uu_x)x=u_yy

\displaystylen_t=x^-2

	2+n)]
[x
	x

\displaystyleu_t+u_xxx-6uu_x=0

\displaystyle\partial_tu+

	3
\partial
	x

u-6u\partial_xu+u/2t=0

\displaystyle\partial_tu+\left(u_xx-2η

	2/2
u
	x

\left(η+u²\right)\right)_x=0

u_t

	n)
+a(u
	x

	n)
+(u
	xxx

=0

\displaystyle\partial_tu+

	3
\partial
	x

u+\partial_xf(u)=0

\displaystyle\partial_tu+

	3
\partial
	x

u\pm

	2\partial
6u
	x

u=0

\displaystyleu_t+u_xxx-\tfrac{1}{8}

	3
u
	x

+u_x\left(Ae^au+B+Ce^-au\right)=0

\displaystyle\partial_tu+

	3
\partial
	x

u-6\partial_xu+u/t=0

\displaystyleu_t=6uux-uxxx+3ww_xx,   w_{t=3uxw+6uwx-4wxxx}

\displaystyle\partial_tu+

	3
\partial
	x

u-6f(t)u\partial_xu=0

\displaystyle\partial_tu+\beta

	3
t
	x

u+\alpha

	nu\partial
t
	x

u=0

\displaystyle\partial_tu+

	3
\mu\partial
	x

u+2u\partial_xu-\nuu_xx=0

\displaystyle

	4u+\nabla
u
	t+\nabla

^2u+\tfrac{1}{2}

\displaystyle

	\partialS
	\partialt

=S\wedge

\sum
	i \partial2S \partial 2 x i

+S\wedgeJS

\displaystyle2u_tx+u_xuxx-u_yy=0

\displaystyle\nabla^2u+eλ=0

\nabla²\psi+λe^\psi=0

i

	\partial\psi
	\partialt

+\Delta\psi+\psiln

\displaystyle\operatorname{div}(Du/\sqrt{1+|Du|^2})=0

\displaystyle\det(\partial_ij\varphi)=

\displaystyle\rho\left(

	\partialvi
	\partialt

+v_j

	\partialvi
	\partialxj

\right)=-

	\partialp
	\partialxi

+

	\partial
	\partialxj

\left[\mu\left(

	\partialvi
	\partialxj

+

	\partialvj
	\partialxi

\right)+λ

	\partialvk
	\partialxk

\right]+\rhof_i

	\partial\rho
	\partialt

+

	\partial\left(\rhovi\right)
	\partialxi

=0

	\partialvi
	\partialxi

=0

\displaystylei\partial_{t\psi=-{1\over}

	2
2}\partial
	x\psi+\kappa

\displaystylei\partial_{t\psi=-{1\over}

	2
2}\partial
	x\psi+\partial

_x(i\kappa

\displaystyle\nabla^2\omega+

	f2
	\sigma

	\partial2\omega
	\partialp2

\displaystyle=

	f
	\sigma

	\partial
	\partialp

V_{g ⋅ \nablap}(\zeta_g+f)+

	R
	\sigmap

	2
\nabla
	p(V

_{g ⋅ \nablap}T)

\displaystyle

	2)u
(1+u
	xx

-2u_xuyuxy

	2)u
+(1+u
	yy

=0

\displaystyleu_xx-u_yy\pm\sinu\cosu+

	\cosu
	\sin3u

	2)=0,
(v
	y

\displaystyle

	2u)
(v
	x

=

	2
(v
	y\cot

u)_y

\displaystyle

	\gamma)
u
	t=\Delta(u

\displaystyleu_{t+uux+vuy=Ut+UU}

	x+ \mu \rho

u_yy

\displaystyleu_x+vy=0

\displaystyleu_tt-u_xx=\varepsilon(u_t-u

	3)
	t

\displaystyle\partial_tg_ij=-2R_ij

\displaystyle\theta_t=\left[K(\theta)\left(\psi_z+1\right)\right]_z

u_t+a

	n\right)
\left(u
	x

+

	n\right)
\left(u
	xxx

=0

\displaystyle

	2u
u
	x+15u

_xuxx+15uu_xxx+u_xxxxx=0

\ddotV+\partial_η\left[

	1
	1-\ddotV

\partial_η\left(

	1-\ddotV
	V

\right)\right]=0

\phi_t+(1+b\sqrt\phi)\phi_x+\phi_xxx=0

\displaystyle{\partialA_i\over\partialt_j}{\left[A_i, A_j\right]\overt_i-t_j},   i ≠ j,   {\partialA_i\over\partialt_i}=-

	n
\sum
	j=1\atopj ≠ i

{\left[A_i, A_j\right]\overt_i-t_j},   1\leqi,j\leqn

\displaystyleD^A\varphi=0,   

	+
F
	A=\sigma(\varphi)

\displaystyleη_t+(ηu)_x+(ηv)_y=0, (ηu)_t+\left(ηu²+

	1
	2

gη²\right)_x+(ηuv)_y=0, (ηv)_t+(ηuv)_x+\left(ηv²+

	1
	2

gη

	2\right)
	y

=0

\displaystyle\varphi_tt-\varphi_xx+\sin\varphi=0

\displaystyleu_xt=\sinhu

\displaystyle\nabla^2u+\sinhu=0

\displaystyleu_t=ru-(1+\nabla^2)2u+N(u)

\displaystyleu_xy+\alphau_x+\betau_y+\gammau_xuy=0

\displaystyleiu_x+v+u

\displaystyleiv_t+u+v

\displaystyle\nabla^2logu_n=u_n+1-2u_n+un-1

\displaystyle(\partial_t+\partial

	3)v+\partial
	z(uv)+\partial

_\bar(uw)=0

\displaystyle\partial_\baru=3\partial_zv

\displaystyle\partial_{zw=3\partial\bar}v

	\partial\boldsymbol\omega
	\partialt

+(u ⋅ \nabla)\boldsymbol\omega=(\boldsymbol\omega ⋅ \nabla)u-\boldsymbol\omega(\nabla ⋅ u)+

	1
	\rho2

\nabla\rho x \nablap+\nabla x \left(

	\nabla ⋅ \tau
	\rho

\right)+\nabla x \left(

	f
	\rho

\right), \boldsymbol{\omega}=\nabla x u

\displaystyleiu_t+((1+

\displaystyle

	n\left({\partial
\sum
	\sigma,\tau=1

³F\over\partialt^\alphat^\betat^\sigma}η^\sigma{\partial³F\over\partialt^\mut^\nut^\tau}\right)

\displaystyle=

	n\left({\partial
\sum
	\sigma,\tau=1

³F\over\partialt^\alphat^\nut^\sigma}η^\sigma{\partial³F\over\partialt^\mut^\betat^\tau}\right)

S_k(\gamma)=-

	k
	8\pi

\int
	S2

d^2xl{K}(\gamma^-1\partial^\mu\gamma,\gamma^-1\partial_\mu\gamma)+2\pikS^WZ(\gamma)

S^WZ(\gamma)=-

	1
	48\pi2

\int
	B3

d^3y\varepsilon^ijkl{K}\left(\gamma^-1

	\partial\gamma
	\partialyi

,\left[ \gamma^-1

	\partial\gamma
	\partialyj

, \gamma^-1

	\partial\gamma
	\partialyk

\right] \right)

\displaystyleη_t+\alphaηη_x+

	+infty
\int
	-infty

K(x-\xi)η_\xi(\xi,t)d\xi=0

	\partialfj
	\partialt

+\nabla_{x ⋅ (vfj)}+\nabla_{v ⋅ (Fjfj)}=-

	\partial
	\partialr

(R_jfj)-

	\partial
	\partialT

(E_jfj)+Q_j+\Gamma_j, Fj=

•

v

, R_j=

•

r

, E_j=

•

T

, j=1,2,...,M

\displaystyle\Delta\varphi+h(x)\varphi=λf(x)\varphi^(n+2)/(n-2)

\displaystyleD_\muF^\mu\nu=0,   F_\mu=A_\mu,-A_\nu,+[A_\mu,A_\nu]

F_\alpha=\pm\varepsilon_\alphaF^\mu,   F_\mu=A_\mu,-A_\nu,+[A_\mu,A_\nu]

\displaystylei

\partial
	t

u+\Deltau=-Au,   \displaystyle\BoxA=

	2
m

A+

\displaystylei

\partial
	t

u+\Deltau=un,   \displaystyle\Boxn=-\Delta(

\displaystyleiu_t+L_1u=\varphiu,   \displaystyleL₂\varphi=L₃₍

\displaystyleu_t=D\nabla²u+

	\beta2
	2

u(1-u)e^-\beta(1-u)

\displaystyle(u_xt/u)_tt-(u_xt/u)_xx

	2)
+2(u
	xt

=0

\displaystyle\varphi_tt-\varphi_xx-\varphi+\varphi³⁼⁰

\displaystyle{v}_xt+({v}_x{v}t){v}=0