More on Nonlinear X-waves

People coming from soliton theory (especially nonlinear optics) are used to associate self-localization to a nonlinear self-action. Actually, self-localized waves do even exist for linear systems, and, surprisingly, they have been mostly studied in recent years, AFTER solitons and bullets.

The typical argument against localized waves (to be intended as the linear waves) is that they carry infinite enegy. However, this holds true for many nonlinear waves, as for example dark solitons and vortices. Finite energy realizations of localized waves have been observed, as it has been for darks and vortices.

The situation is even more intricate when localized waves propagate in the presence of a nonlinear self-action. The result are the NONLINEAR X-WAVES. They aresolutions to 3D+1 nonlinear evolution equations carrying infinite energy and have been investigated both experimentally and theoretically. [see Conti C., Phys. Rev. E 70, 046613 (2004) and the book chapter  NonlinearXwaves (.pdf, 1.2MB) in the book Localized Waves (ISBN 0-470-10885-1) for an introduction]

In other words:

1D nonlinear waves with infinite energy = dark solitons

2D nonlinear waves with infinite energy = vortex solitons

3D nonlinear waves with infinite energy = nonlinear X waves (or X solitons)


The world of nonlinear X waves is still mainly unexplored; not only it is fascinating because it deals with 3D waves, which have a counterpart in Bose Einstein condensates, but also because they are expected to play a role in ultrafast laser propagation. Indeed, contrary to light bullets, X-waves are sustained by normal dispersion, which is always the case in the relevant experiments.