I would like to mention that I am a mathematician and not a physicist, so I apologize in advance if my question seems obvious.
Considering any linear PDE, it is common to understand the behavior of the solution by taking the Fourier transform of the equation. For instance, one can think about a linear Schrödinger equation $$ i\partial_t u=\partial_x^2u\,. $$ By taking the Fourier transform in the space variable $x$, $$ i\partial_t \hat{u}(\xi)=-\xi^2 \hat{u}(\xi)\,.$$ Hence, one obtains a differential equation that can be easily solved, and the solution $u$ can be deduced by taking the inverse Fourier transform. However, this approach may not be helpful when dealing with nonlinear PDEs. In these cases, alternative methods are necessary to analyze and understand the solution...
I recently observed that some mathematicians (for example, I came across this article) limit their study of nonlinear PDEs to solutions that have only positive frequencies, for instance, by assuming that the solution belongs to the Hardy space, i.e. $u$ is an analytic solution in the Hardy space, if and only if, $\hat{u}(\xi)=0$ for $\xi<0$.
My question: I'm curious about the physical significance of studying solutions in the Hardy space, especially in cases where signal theory is not involved. Specifically, I am wondering how the use of the Hardy space is physically relevant to the equation studied in the article mentioned above.
Badreddine seems to be dealing with a nonlocal nonlinear Schrodinger equation, which, if I have understood well, was initially obtained by Abanov and his collaborators, by considering the classical Calogero-Sutherland system in the thermodynamic regime.
Edit:
- For the abstract of Badreddine: click here
- For the abstract of Abanov-Bettelheim-Wiegmann: click here
