Transverse
(symmetry-breaking) instabilities and self-focusing
of nonlinear waves in two spatial dimensions commonly occur
in lasers, plasmas, fluids, and water waves.
Nonlinear evolution equations describing
transverse self-focusing in Hamiltonian wave systems
are classified into three large groups:
Transverse instability problems can be solved analytically by using the asymptotic expansion methods. Two main asymptotic methods were developed for long-scaled and short-scaled transverse perturbations. Both the asymptotic methods are based on reduction of the underlying transverse instability problem to a chain of elliptic evolution equations. Decomposition of elliptic equations uses Laplace coordinates and brings about explicit asymptotic solution of the transverse instability problem.
