Stable/ unstable w.r.t. perturbations in $L^2(\mathbb{R})$

Question

Suppose we have some travelling wave solution $w(z)$ of a pde. What is then meant by "perturbations in $L^2(\mathbb{R})$?

Does this mean that $w(z)+v(z)$ with $v\in L^2(\mathbb{R}$ behaves like $w(z)$?

Answer 1

It means that you look at the behaviour of $w(x + c t) + v(x,t)$ when substituted in the PDE. Since $v(x,t)$ is considered to be 'small', you can then expand the nonlinear terms of the PDE in powers of $v$, and neglect terms of order $v^2$ or higher. In other words, you'll obtain a linear PDE for $v(x,t)$. Now, from standard PDE theory (or Fourier theory, whatever you prefer), it follows that you can write $v(x,t)$ as the product of a function of $t$ and a function of $x$, that is, $v(x,t) = f(t) g(x)$. The description 'perturbations in $L^2(\mathbb{R})$' now means that we take $g \in L^2(\mathbb{R})$, i.e. we assume that the `spatial part' of the perturbation $v$ is square integrable on $\mathbb{R}$.