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I’ve just uploaded to the arXiv my paper “The high exponent limit $p \to \infty$ for the one-dimensional nonlinear wave equation“, submitted to Analysis & PDE.  This paper concerns an under-explored limit for the Cauchy problem

$\displaystyle -\phi_{tt} + \phi_{xx} = |\phi|^{p-1} \phi; \quad \phi(0,x) = \phi_0(x); \quad \phi_t(0,x) = \phi_1(x)$ (1)

to the one-dimensional defocusing nonlinear wave equation, where $\phi: {\Bbb R} \times {\Bbb R} \to {\Bbb R}$ is the unknown scalar field, $p > 1$ is an exponent, and $\phi_0, \phi_1: {\Bbb R} \to {\Bbb R}$ are the initial position and velocity respectively, and the t and x subscripts denote differentiation in time and space.  To avoid some (extremely minor) technical difficulties let us assume that p is an odd integer, so that the nonlinearity is smooth; then standard energy methods, relying in particular on the conserved energy

$\displaystyle E(\phi)(t) = \int_{\Bbb R} \frac{1}{2} |\phi_t(t,x)|^2 + \frac{1}{2} |\phi_x(t,x)|^2 + \frac{1}{p+1} |\phi(t,x)|^{p+1}\ dx$, (2)

on finite speed of propagation, and on the one-dimensional Sobolev embedding $H^1({\Bbb R}) \subset L^\infty({\Bbb R})$, show that from any smooth initial data $\phi_0, \phi_1$, there is a unique global smooth solution $\phi$ to the Cauchy problem (1).

It is then natural to ask how the solution $\phi$ behaves under various asymptotic limits.  Popular limits for these sorts of PDE include the asymptotic time limit $t \to \pm \infty$, the non-relativistic limit $c \to \infty$ (where we insert suitable powers of c into various terms in (1)), the small dispersion limit (where we place a small factor in front of the dispersive term $+\phi_{xx}$), the high-frequency limit (where we send the frequency of the initial data $\phi_0, \phi_1$ to infinity), and so forth.

Tristan Roy recently posed to me a different type of limit, which to the best of my knowledge has not been explored much in the literature (although some of the literature on limits of the Ginzburg-Landau equation has a somewhat similar flavour): the high exponent limit $p \to \infty$ (holding the initial data $\phi_0, \phi_1$ fixed).  From (1) it is intuitively plausible that as p increases, the nonlinearity gets “stronger” when $|\phi| > 1$ and “weaker” when $|\phi| < 1$; the “limiting equation”

$\displaystyle -\phi_{tt} + \phi_{xx} = |\phi|^{\infty} \phi; \quad \phi(0,x) = \phi_0(x); \quad \phi_t(0,x) = \phi_1(x)$ (3)

would then be expected to be linear when $|\phi| < 1$ and infinitely repulsive when $|\phi| > 1$ (i.e. in the limit, the solution should be confined to range in the interval [-1,1], much as is the case with linear wave and Schrödinger equations with an infinite barrier potential; though with the key difference that the nonlinear barrier in (3) is confining the range of $\phi$ rather than the domain.).

Of course, the equation (3) does not make rigorous sense as written; we need to formalise what an “infinite nonlinear barrier” is, and how the wave $\phi$ will react to that barrier (e.g. will it reflect off of it, or be absorbed?).  So the questions are to find the correct description of the limiting equation, and to rigorously demonstrate that solutions to (1) converge in some sense to that equation.

It is natural to require that $\phi_0$ stays away from the barrier, in the sense that $|\phi_0(x)| < 1$ for all x; in particular this implies that the energy (2) stays (locally) bounded as $p \to \infty$; it also ensures that (1) converges in a satisfactory sense to the free wave equation for sufficiently short times.  For technical reasons we also have to make a mild assumption that either of the null energy densities $\phi_1 \pm \partial_x \phi_0$ vanish on a set with at most finitely many connected components.  The main result is then that as $p \to \infty$, the solution $\phi = \phi^{(p)}$ to (1) converges locally uniformly to a Lipschitz, piecewise smooth limit $\phi = \phi^{(\infty)}$, which is restricted to take values in [-1,1], with $-\phi_{tt}+\phi_{xx}$ (interpreted in a weak sense) being a negative measure supported on $\{ \phi=+1\}$ plus a positive measure supported on $\{\phi = -1\}$.  Furthermore, we have the reflection conditions

$\displaystyle (\partial_t \pm \partial_x) |\phi_t \mp \phi_x| = 0$.

It turns out that the above conditions uniquely determine $\phi$, and one can even solve for $\phi$ explicitly for any given data; such solutions start off smooth but pick up an increasing number of (Lipschitz continuous) singularities over time as they reflect back and forth across the nonlinear barriers $\{\phi=+1\}$ and $\{\phi=-1\}$.  (An explicit example of such a reflection is given in the paper.)

[The above conditions vaguely resemble entropy conditions, as appear for instance in kinetic formulations of conservation laws, though I do not know of a precise connection in this regard.]

In the remainder of this post I would like to describe the strategy of proof and one of the key a priori bounds needed.  I also want to point out the connection to Liouville’s equation, which was discussed in the previous post.

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.]