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Semilinear dispersive and wave equations, of which the defocusing nonlinear wave equation

$\displaystyle -\partial_{tt} u + \Delta u = |u|^{p-1} u \ \ \ \ \ (1)$

is a typical example (where ${p>1}$ is a fixed exponent, and ${u: {\bf R}^{1+n} \rightarrow {\bf R}}$ is a scalar field), can be viewed as a “tug of war” between a linear dispersive equation, in this case the linear wave equation

$\displaystyle -\partial_{tt} u + \Delta u = 0 \ \ \ \ \ (2)$

and a nonlinear ODE, in this case the equation

$\displaystyle -\partial_{tt} u = |u|^{p-1} u. \ \ \ \ \ (3)$

If the nonlinear term was not present, leaving only the dispersive equation (2), then as the term “dispersive” suggests, in the asymptotic limit ${t \rightarrow \infty}$, the solution ${u(t,x)}$ would spread out in space and decay in amplitude. For instance, in the model case when ${d=3}$ and the initial position ${u(0,x)}$ vanishes (leaving only the initial velocity ${u_t(0,x)}$ as non-trivial initial data), the solution ${u(t,x)}$ for ${t>0}$ is given by the formula

$\displaystyle u(t,x) = \frac{1}{4\pi t} \int_{|y-x|=t} u_t(0,y)\ d\sigma$

where ${d\sigma}$ is surface measure on the sphere ${\{ y \in {\bf R}^3: |y-x| = t \}}$. (To avoid technical issues, let us restrict attention to classical (smooth) solutions.) Thus, if the initial velocity was bounded and compactly supported, then the solution ${u(t,x)}$ would be bounded by ${O(1/t)}$ and would thus would decay uniformly to zero as ${t \rightarrow \infty}$. Similar phenomena occur for all dimensions greater than ${1}$.

Conversely, if the dispersive term was not present, leaving only the ODE (3), then one no longer expects decay; indeed, given the conserved energy ${\frac{1}{2} u_t^2 + \frac{1}{p+1} |u|^{p+1}}$ for the ODE (3), we do not expect any decay at all (and indeed, solutions are instead periodic in time for each fixed ${x}$, as can easily be seen by viewing the ODE (and the energy curves) in phase space).

Depending on the relative “size” of the dispersive term ${\Delta u}$ and the nonlinear term ${|u|^{p-1} u}$, one can heuristically describe the behaviour of a solution ${u}$ at various positions at times as either being dispersion dominated (in which ${|\Delta u| \gg |u|^p}$), nonlinearity dominated (in which ${|u|^p \gg |\Delta u|}$), or contested (in which ${|\Delta u|}$, ${|u|^p}$ are comparable in size). Very roughly speaking, when one is in the dispersion dominated regime, then perturbation theory becomes effective, and one can often show that the solution to the nonlinear equation indeed behaves like the solution to the linear counterpart, in particular exhibiting decay as ${t \rightarrow \infty}$. In principle, perturbation theory is also available in the nonlinearity dominated regime (in which the dispersion is now viewed as the perturbation, and the nonlinearity as the main term), but in practice this is often difficult to apply (due to the nonlinearity of the approximating equation and the large number of derivatives present in the perturbative term), and so one has to fall back on non-perturbative tools, such as conservation laws and monotonicity formulae. The contested regime is the most interesting, and gives rise to intermediate types of behaviour that are not present in the purely dispersive or purely nonlinear equations, such as solitary wave solutions (solitons) or solutions that blow up in finite time.

In order to analyse how solutions behave in each of these regimes rigorously, one usually works with a variety of function spaces (such as Lebesgue spaces ${L^p}$ and Sobolev spaces ${H^s}$). As such, one generally needs to first establish a number of function space estimates (e.g. Sobolev inequalities, Hölder-type inequalities, Strichartz estimates, etc.) in order to study these equations at the formal level.

Unfortunately, this emphasis on function spaces and their estimates can obscure the underlying physical intuition behind the dynamics of these equations, and the field of analysis of PDE sometimes acquires a reputation for being unduly technical as a consequence. However, as noted in a previous blog post, one can view function space norms as a way to formalise the intuitive notions of the “height” (amplitude) and “width” (wavelength) of a function (wave).

It turns out that one can similarly analyse the behaviour of nonlinear dispersive equations on a similar heuristic level, as that of understanding the dynamics as the amplitude ${A(t)}$ and wavelength ${1/N(t)}$ (or frequency ${N(t)}$) of a wave. Below the fold I give some examples of this heuristic; for sake of concreteness I restrict attention to the nonlinear wave equation (1), though one can of course extend this heuristic to many other models also. Rigorous analogues of the arguments here can be found in several places, such as the book of Shatah and Struwe, or my own book on the subject.