Wave collapse in physics: principles and applications to light and plasma waves

The collapse of self-focusing waves described by the nonlinear Schrödinger (NLS) equation and the Zakharov equations in nonlinear optics and plasma turbulence is reviewed. Special attention is paid to the blow-up properties of solutions to the NLS equation with a power nonlinearity. Conditions for blow-up and global existence of these solutions, together with criteria for the stability of stationary waves are recalled. The variational approach, the so-called exactly and quasi- self-similar analyses employed for modelling wave collapse are then compared. Besides, the influence of the group-velocity dispersion and deviations from the spatio-temporal envelope approximations are investigated as structural perturbations of the NLS standard models, which can strongly alter the blow-up dynamics. Finally, a detailed description of wave collapses in materials with quadratic and cubic responses and a map-making of the interaction regimes between two light cells in Kerr media complete this review.