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View Equation

The database contains 327 equations (8 equations are awaiting activation).

Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):\noindent
$\ \displaystyle \frac{\partial^2 w}{\partial
t\partial z}\,\frac{\partial^2 w}{\partial x\partial
y}-\frac{\partial^2 w}{\partial t\partial x}\,\frac{\partial^2
w}{\partial y\partial z}-1=0.$
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\emph{Plebanski
first Heavenly equation.} (See References [1])
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
Suppose $w(t,x,y,z)$ is a solution of the equation in question.

1. The function
 
$w_1(t,x,y,z)=\pm w(t,x,y,z)+H(t,y)+F(x,z),$

where $F$
and $H$ are arbitrary functions, is also a solution of the equation.


2. The function $w_2(t,x,y,z)=w(t',x',y',z')$ with

$\displaystyle t'=a_1 t+a_2y$

$\displaystyle x'=b_1 x+b_2 z$

$\displaystyle y'=c_1t+(\frac{c_1 a_2}{a_1}+\frac{1}{a_1(b_1
h_2-b_2h_1)})y$

$\displaystyle z'=h_1x+h_2z$ 

where $a_1,a_2,b_1,b_2,c_1,h_1,h_2$ are
arbitrary constants, is also a solution of the equation.

3. The function $w_3(t,x,y,z)=a w(t',x',y',z')$ with

$\displaystyle t'=\tau(t)$

$\displaystyle x'=\frac{1}{a}\int_s^x \frac{d\xi}{D_2
K\left[\xi,W(x,z,\xi)\right]}+G\left[K(x,z)\right]$

$\displaystyle y'=\frac{c\,y}{\tau'(t)}+\theta (t)$

$\displaystyle z'=K(x,z)$

where $D_2K(x_1,x_2)=\frac{\partial K}{\partial x_2}$ and
$W(x,z,\xi)=W$ is any solution of the following transcendental
equation $K(\xi,W)=K(x,z)$, $K$,$\tau$, $\theta$ and  $G$ are
arbitrary functions, $a,c,s$ are arbitrary constants, is also a
solution of the equation.
\medskip

Exact solutions:

See also solutions with arbitrary constants in References [2].
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$\displaystyle w(t,x,y,z)
=\sqrt{z}\left(\frac{xy}{\phi\,'(t)}+2\phi(t)\right)\,,$

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$\displaystyle w(t,x,y,z)
=2[\phi(t)+\psi(x)]\sqrt{\frac{yz}{\phi\,'(t)\,\psi\,'(x)}}\,,$
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$\displaystyle w(t,x,y,z)
=K(x,y)+\frac{t\phi(x)}{\psi\,'(y)}-\frac{z\psi(y)}{\phi\,'(x)}\,,$
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$\displaystyle w(t,x,y,z)
=K(x,y)+G(x,y)t-z\int\frac{dy}{\frac{\partial G(x,y)}{\partial
x}}\,,$
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$\displaystyle w(t,x,y,z)
=\left[y\,\theta(t)+\eta(t)\right]\int\frac{dx}{\zeta(x)}+\left[z\,\zeta(x)+\phi(x)\right]\int\frac{dt}{\theta(t)}+K\left[y\,\theta(t)+\eta(t),z\,\zeta(x)+\phi(x)\right]\,,
\notag$

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\begin{align}
\displaystyle w(t,x,y,z) =&
\sqrt{2[t\phi(y)+\psi(y)]}\left(\zeta(y)+\eta(x)+\theta(x)\int\frac{dy}{\phi(y)}+\frac{z}{\theta\,'(x)}\right)+\notag\\\notag\\&\int
G\left(y,\zeta(y)+\eta(x)+\theta(x)\int\frac{dy}{\phi(y)}+\frac{z}{\theta\,'(x)}\right)\,dy\,,
\notag
\end{align}


where $G,K$ and $\phi,\psi,\eta,\theta,\zeta$ are arbitrary
functions.
Novelty:New solution(s) & transformation(s)
References:[1] J. F. Plebanski, Some solutions of complex Einstein equations, J. Math. Phys. 16, 2395-2402 (1975).
[2] V. Dryuma, On solutions of a Heavenly equations and their generalizations, arXiv:gr-qc/0611001v1
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Fri 18 Apr 2008 10:18
Edits by author:7
Last edit by author:Wed 09 Jul 2008 12:17

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