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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
$\ \displaystyle \frac{\partial^2 w}{\partial
t^2}\,\frac{\partial^2 w}{\partial x^2}-\left(\frac{\partial^2
w}{\partial t\partial x}\right)^2+\frac{\partial^2 w}{\partial
t\partial y}+\frac{\partial^2 w}{\partial x\partial z}=0$.
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second Heavenly equation.}
1. Suppose $w(t,x,y,z)$ is a solution of the equation in question.

1.1. The function

$\displaystyle w_1(t,x,y,z)=w(t,x,y,z)+(x-c t)H(y+c z)+F(y,z)$,

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

1.2. The function

$\displaystyle w_2(t,x,y,z)=w(t,x,y,z)+x\,\frac{\partial
H(y,z)}{\partial y}-t\,\frac{\partial H(y,z)}{\partial z}+F(y,z)$,

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

1.3. The function $w_3(t,x,y,z)=w(t',x',y',z')$ with
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$\displaystyle t'=a_1 t+a_2 x-\frac{a_2 a_3}{a_1}y+a_3 z$,
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$\displaystyle x'=-\frac{a_1 c a_2^2-h^2}{a_2^2 h}t-\frac{c
a_2}{h}x+\frac{c a_2 a_3}{a_1 h}y-\frac{(a_1 c a_2^2-h^2)a_3}{a_1
a_2^2 h}z$,
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$\displaystyle y'=c y-\frac{a_1 c a_2^2-h^2}{a_2^3}z$,
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$\displaystyle z'=h y-\frac{a_1 h}{a_2}z$,
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where $a_1$, $a_2$, $a_3$, $c$, and $h$ are arbitrary constants, is also a
solution of the equation.

2. Exact solutions:
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$\displaystyle w(t,x,y,z) =G(x+ct,y-cz)$,
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$\displaystyle w(t,x,y,z) =\frac{cx^2}{2}+a x+G(t-cy,z)$,
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$\displaystyle w(t,x,y,z) =\frac{\partial K(t,z)}{\partial
t}\,x+\left[\int\left(\frac{\partial^2 K(t,z)}{\partial
t^2}\right)^2\,dt-\frac{\partial K(t,z)}{\partial
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where $G$ and $K$ are arbitrary functions, $a$ and $c$ are arbitrary constants.
Remarks:See also exact solutions in References [2].
Novelty:New solution(s) & transformation(s)
References:[1] J. F. Plebanski, Some solutions of complex Einstein equations, J. Math. Phys., 1975, Vol. 16, pp. 2395--2402.
[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:Wed 25 Jun 2008 10:07
Edits by author:1
Last edit by author:Mon 30 Jun 2008 09:39

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