Self-action, gradient catastrophe, and wave collapse

Self-action effects of the wave packets and beams in nonequilibrium media have attracted wide attention of researchers back in the 1960s with the advent of lasers and rapid development of nonlinear optics.
A number of most important results belong to the IAP RAS scientists (A. G. Litvak, V. A. Mironov, A. M. Sergeev, V. I. Talanov, and G. M. Fraiman). A general theory of the wave self-action phenomena in media with different mechanisms of nonlinearity is developed, and on its basis a wide range of self-focusing and self-modulation effects of electromagnetic waves in dielectrics, liquids, gases, and plasmas has been explored. Namely,
• a theory of stationary and nonstationary self-focusing of wave beams is developed;
• the phenomena of filamentary and modulation instabilities of waves are discovered, and a theory of the nonlinear stage of these processes is constructed;
• the suppression of filamentary instability with self-action in the media with inertial nonlinearity is studied;
• nonlinear dynamics of ultra-short laser pulses in dispersive media is analyzed;
• large-amplitude breathers resulting from the modulation instability in the problem of rogue waves in nonlinear dispersive media are explored;
• spatio-temporal collapses of three-dimensional wave packets are examined.
These results are widely used in the development of high-power laser systems and plasma heating facilities, methods for creating ultra-short optical pulses, methods of precision optical measurements of the nonlinearity constants of different media and wave-front curvature of optical beams, in the construction of predictive models of the evolution of intense internal waves on the ocean shelf, and explanation of the rogue-wave phenomena.
The development of optical methods for generating attosecond and terahertz pulses raised the problem of theoretical study of the self-action dynamics of wave fields with a spectral width of the order of the carrier frequency. The IAP RAS researchers (A. G. Livak, V. A. Mironov, A. A. Balakin, and S. A. Skobelev) developed an approach to the study of such waves based on use of the Maxwell equations in reflectionless approximation. In particular, a sufficient condition of collapse in the system is found by the method of moments. The use of a self-similar transformation has shown that a key process in the system dynamics is the steepening of the longitudinal profile of the pulse. It has been proved that this process is slightly ahead of the self-focusing of the field. This results in the formation of a more complex feature of the field, in which a gradient catastrophe is accompanied by a wave collapse.

Dynamics of self-action of an ultra-short circularly polarized pulse in a medium with cubic nonlinearity. The formation of a feature (top) and the subsequent spreading of the pulse due to dissipation on the shock wave front (bottom)