Which methods are best to analyse wave-packets?

Question

I would like to find several methods to analyse wave-packets. My primary aim is to find a way to decompose wave-packets into "sub-wave packets" that, when subjected to some operation give the original wavepacket.

In other words, let a wave-packet be a bounded sequence $\{x_n\}$, and find a subsequence $\{y_n\}\in \{x_n\}$, which $A\{y_n\}=\{x_n\}$, where $A$ can be any linear or nonlinear functional. Then repeat the procedure to find n-th degree subsequence $\{y_n^{(n)}\} \in\{x_n\} $, which gives $A_n\{y_n^{(n)}\}=\{x_n\}$ where $A_n$ is any linear functional which $A:X\longrightarrow X$, where $X$ is a normed linear space, with elements $x,y,..=f(t),g(t)...$ where $t\in\mathbb{R}$.

For instance, the Fourier transform operator is such a functional, $A$, and gives a supersequence $\{y\}$ with $x=f(t)$, $$\{y\}_\omega=Ax=\int_{-\infty}^\infty f(t)e^{-i\omega t}dt$$ of which $x=f(t) \in X$ . Hence, $f(t)$ would be of interest.

My question is , if we have a nonlinear wavepacket, which methods are most suitable to construct such functionals? I have already tried Fourier transform and I am looking for alternative methods.

Thanks

PS: Update

From the given paper, I give an example of the measurement of "subharmonic" and "superharmonic" wave-packets in ocean waves due to sudden depth transition.

"We observe the generation of free second-order sub- and superharmonic wavepackets due to the sudden depth transition, in addition to changes to the main (first-order) wavepacket and its second-order bound waves. "

https://www.cambridge.org/core/journals/journal-of-fluid-mechanics/article/surface-wavepackets-subject-to-an-abrupt-depth-change-part-2-experimental-analysis/1EEAF8FC7C7482B7E6551EEB529A67AD

This is an example of what I would like to extract from a wavepacket reading. But with which methods?

Thanks

Answer 1

The situation for non-linear wavepackets is far more complex in general. Because the Fourier transform is a linear operator that preserves linearity, it is best suited for linear wavepackets. However, there are generalizations of Fourier transforms which are called short-time Fourier transforms.

https://en.wikipedia.org/wiki/Short-time_Fourier_transform

These won't transform the entire time-dependent signal to its frequency spectrum like classical Fourier transform does; it will instead still have additionally information about time dependence of the signal after transformation. This is achieved by choosing an appropriate window function.

Nonlinear waves look in a short time interval like ordinary linear waves, so you might choose a window function around this small time interval and analyze frequency spectrum there.

The Gabor transform is also an example.