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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
$\ \displaystyle \frac{\partial^2 w}{\partial
t\partial z}\,\frac{\partial^2 w}{\partial x\partial
y}-\frac{\partial^2 w}{\partial t\partial x}\,\frac{\partial^2
w}{\partial y\partial z}-1=0.$
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first Heavenly equation.} (See References [1])
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

$\displaystyle z'=h_1x+h_2z$ 

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

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

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

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.

Exact solutions:

See also solutions with arbitrary constants in References [2].
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$\displaystyle w(t,x,y,z)

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$\displaystyle w(t,x,y,z)
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$\displaystyle w(t,x,y,z)
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$\displaystyle w(t,x,y,z)
=K(x,y)+G(x,y)t-z\int\frac{dy}{\frac{\partial G(x,y)}{\partial
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$\displaystyle w(t,x,y,z)

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\displaystyle w(t,x,y,z) =&

where $G,K$ and $\phi,\psi,\eta,\theta,\zeta$ are arbitrary
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
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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