Nonlinear waves and singularities in nonlinear optics, plasmas, hydrodynamics and biology
Many nonlinear systems of partial differential equations have a striking phenomenon of spontaneous formation of singularities in a finite time (blow up). Blow up is often accompanied by a dramatic contraction of the spatial extent of solution, which is called by collapse. Near singularity point there is usually a qualitative change in underlying nonlinear phenomena, reduced models loose their applicability with diverse singularity regularization mechanisms become important such as optical breakdown and formation of plasma in nonlinear optical media, excluded volume constraints in bacterial aggregation or dissipation of breaking water waves. Collapses occur in numerous physical and biological systems including a nonlinear Schrodinger equation, Keller-Segel equation and many others. Wavebreaking is another example of spontaneous formation of singularities corresponding to the breaking of initially smooth smooth fluid's free surface. The recent progress in collapse theory will be reviewed with multiple applications discussed ranging from laser fusion to optical beam combining. The 2D dynamics of fluid's free surface will be also addressed through the motion of singularities outside of fluid with wavebreaking resulting from the approach of these singularities to the fluid's surface. New exact solutions are found which have arbitrary number of poles coupled with branch cuts. The residues of these poles are integrals of motion. They commute with respect of the non-canonical Poisson bracket which provides strong arguments towards complete integrability of free surface hydrodynamics.