The World of Mathematical Equations

IPM Logo

Exact Solutions Methods Software For Authors Math Forums

EqArchive: Add Equation/Solution > View Equation

 English only

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
\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$.
The general solution in parametric form
($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
Statistic information
Submission date:Tue 03 Feb 2009 11:57
Edits by author:0

Edit (Only for author/contributor)

The EqWorld website presents extensive information on solutions to various classes of ordinary differential equations, partial differential equations, integral equations, functional equations, and other mathematical equations.

Copyright © 2006-2011 Andrei D. Polyanin, Alexei I. Zhurov and Alexander L. Levitin