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As is well known, the linear one-dimensional wave equation

$\displaystyle -\phi_{tt}+\phi_{xx} = 0$, (1)

where $\phi: {\Bbb R} \times {\Bbb R} \to {\Bbb R}$ is the unknown field (which, for simplicity, we assume to be smooth), can be solved explicitly; indeed, the general solution to (1) takes the form

$\displaystyle \phi(t,x) = f( t+x ) + g(t-x)$ (2)

for some arbitrary (smooth) functions $f, g: {\Bbb R} \to {\Bbb R}$.  (One can of course determine f and g once one specifies enough initial data or other boundary conditions, but this is not the focus of my post today.)

When one moves from linear wave equations to nonlinear wave equations, then in general one does not expect to have a closed-form solution such as (2).  So I was pleasantly surprised recently while playing with the nonlinear wave equation

$\displaystyle -\phi_{tt}+\phi_{xx} = e^\phi$, (3)

to discover that this equation can also be explicitly solved in closed form.  (I hope to explain why I was interested in (3) in the first place in a later post.)

A posteriori, I now know the reason for this explicit solvability; (3) is the limiting case $a = 0, b \to -\infty$ of the more general equation

$\displaystyle -\phi_{tt}+\phi_{xx} = e^{\phi+a} - e^{-\phi+b}$

which (after applying the simple transformation $\phi = \frac{b-a}{2} + \psi( \sqrt{2} e^{\frac{a+b}{4}} t, \sqrt{2} e^{\frac{a+b}{4}} x)$) becomes the sinh-Gordon equation

$\displaystyle -\psi_{tt} + \psi_{xx} = \sinh(\psi)$

(a close cousin of the more famous sine-Gordon equation $-\phi_{tt} + \phi_{xx} = \sin(\phi)$), which is known to be completely integrable, and exactly solvable.  However, I only realised this after the fact, and stumbled upon the explicit solution to (3) by much more classical and elementary means.  I thought I might share the computations here, as I found them somewhat cute, and seem to serve as an example of how one might go about finding explicit solutions to PDE in general; accordingly, I will take a rather pedestrian approach to describing the hunt for the solution, rather than presenting the shortest or slickest route to the answer.

[The computations do seem to be very classical, though, and thus presumably already in the literature; if anyone knows of a place where the solvability of (3) is discussed, I would be very happy to learn of it.]  [Update, Jan 22: Patrick Dorey has pointed out that (3) is, indeed, extremely classical; it is known as Liouville’s equation and was solved by Liouville in J. Math. Pure et Appl. vol 18 (1853), 71-74, with essentially the same solution as presented here.]