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Equation data
Category:4. Nonlinear Partial Differential Equations
Subcategory:4.2. Second-Order Quasilinear Hyperbolic Equations
Equation(s):\noindent
$\displaystyle 
\frac{\partial^2 w}{\partial t \, \partial x}  = 
w \frac{\partial^2 w}{\partial x^2 }+a\left(\frac{\partial w}{\partial x }\right)^2$.
Solution(s),
Transformation(s),
Integral(s)
:
\noindent
The general solution in parametric form
($a\neq 0$):
$$
w=f'_t(t)+\int [g(z)-at]^{\frac{1-a}a}dz,
$$
$$
x=-f(t)+\int [g(z)-at]^{\frac{1}a}dz,
$$
where $f(t)$ and $g(z)$ are arbitrary functions, and $z$ is the parameter.
Remarks:It is a special case of the Calogero equation.
Novelty:Material has been fully published elsewhere
References:1. Calogero, F., A solvable nonlinear wave equation, Stud. Appl. Math., Vol. 70, No. 3, pp. 189-199, 1984.
2. Pavlov, M. V., The Calogero equation and Liouville-type equations, Theor. & Math. Phys., Vol. 128, No. 1, pp. 927-932, 2001.
3. Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton-London, 2004 (pp. 433-434).
Author/Contributor's Details
Last name:Kosovtsov
First name:Yurii
Country:Ukraine
City:Lvov
Statistic information
Submission date:Tue 03 Feb 2009 11:57
Edits by author:0

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