Collapse and generation of ultrashort optical pulses in a nonlinear optical cavity

Summary form only given. Wave collapse or singularity formation in a finite time in the solution of nonlinear partial differential equations describing systems of dispersive waves is a striking and general phenomenon of nonlinear physics. The nonlinear Schrodinger equation (NLS) is a universal model of weakly nonlinear wave evolution, and is known to lead to collapse when the dimensionality of the problem is at least two. Growth of amplitudes of collapsing waves is accompanied with dramatic contraction of the wave packet. In many cases the underlying physical system is dissipative so that it is natural to account for a source of energy. A generalized forced NLS (FNLS) that includes forcing and damping terms can be written in the form i(/spl delta//spl psi///spl delta/t)=b/spl psi/-/spl nabla//sup 2//spl psi/-|/spl psi/|/sup 2//spl psi/+E. Here b=b/sub r/+ib/sub i/ is a complex constant, E is real, and /spl nabla//sup 2/ operates in 1, 2 or 3 dimensions. When b=E=0 we regain the canonical NLS. The present work has been motivated by recent studies of three dimensional space-time focusing and structure formation in nonlinear optical cavities pumped by an external train of pulses.