Wave equations

Non-linear wave equations

Nonlinear wave equations arise in physics from two major sources: relativity and elasticity.

All relativistic field equations in (classical) physics are variants of the free wave equation or Klein-Gordon equation on Minkowski space.

There are several ways to perturb this equation. There are linear perturbations, which include the addition of potential terms, connection terms, and drag terms, as well as the replacement of the flat Minkowski metric with a more general curved metric, or by placing obstacles or otherwise changing the topology of the domain manifold $R^{1+d}$.

Here we shall focus more on purely non-linear perturbations, which collapse to a constant-coefficient wave equation in the small amplitude limit. In the fullest generality, this would mean studying equations of the form

$F(f,Df,D_{}^{2}f)=0$

where $D$ denotes differentiation in space or time and the Taylor expansion of $F$ to first order is the free wave or Klein-Gordon equation. Such fully non-linear equations, though, are very difficult to study, and have only really been analyzed in the one-dimensional case (in which case it can be subsumed into the general theory of 1+1-dimensional hyperbolic systems). In higher dimensions the only known tool to analyze this case is to differentiate the equation, turning it into a quasi-linear system. As such we do not discuss fully non-linear wave equations here. Instead, we consider three less general types of equations, which in increasing order of complexity are the semi-linear, semi-linear with derivatives, and quasi-linear equations.

Non-linear wave equations are often the Euler-Lagrange equation for some variational problem. This usually generates the conserved stress-energy tensor, which is of fundamental importance in the analysis of such equations, especially for the global-in-time theory.

The principle of relativity asserts that the equations of physics are covariant with respect to the underlying geometry of spacetime. This can be exploited in a number of ways. One is via stress-energy tensor mentioned previously. Another is via conformal transformation of spacetime. A third is via finite speed of propagation. The covariance also generates some important null structures in the nonlinear components of the equation.

The perturbative theory for nonlinear wave equations rests on various linear, bilinear, and nonlinear estimates for the linear wave equation.

Dependence on dimension

The one-dimensional case $d=1$ is special for several reasons. Firstly, there is the very convenient null co-ordinate system $u=t+x,v=t-x$ which can be used to factorize $\Box$. Also, the stress-energy tensor often becomes trace-free, which leads to better conformal invariance properties. There are a vastly larger number of conformal transformations, indeed anything of the form $(u,v)\rightarrow (F(u),Y(v))$ is conformal. Also, the one-dimensional wave equation has no decay, local smoothing, or dispersion properties, and its solutions are essentially travelling waves. Finally, there are a much larger range of spaces beyond Sobolev spaces which are available for well-posedness theory, because the free wave evolution operator preserves all translation-invariant spaces. (In two and higher dimensions only $L^{2}$-based spaces such as Sobolev spaces H^s are preserved, because waves can focus at a point (or defocus from a point)).

The higher-dimensional case $d>1$ is usually quite different from the one-dimensional case, although in spherically symmetric situations one can obtain similar behaviour, especially when viewed in the null co-ordinates $u=t+r,v=t-r$. Indeed one can think of spherically symmetric wave equations as one-dimensional wave equations with a singular drag term $(n-1)f_{r}/r$.

Specific wave equations

• Semilinear wave equations (sine-Gordon, etc.)
• NLW with derivatives
  • linear-derivative nonlinear wave equations (Yang-Mills, Yang-Mills-Higgs, Maxwell-Klein-Gordon)
  • quadratic-derivative nonlinear wave equations (wave maps)
• Dirac-type equations
• Quasilinear wave equations (Einstein, minimal surface, etc.)