Multi-focussing and sustained dissipation in the cubic Schrödinger equation and a dissipative regularization

Brenton leMesurier

College of Charleston, Mathematics, Room 203, Maybank Hall,
165 Calhoun St Charleston, SC 29424-0001, USA

Abstract: The possibility of physically relevant singular solutions of the Cubic Schrödinger Equation having sustained dissipation into a point singularity is considered, through numerical study of a nonlinear dissipative regularization and its small dissipation limit. A new form of such dissipative solutions involving a multi-focussing mechanism is conjectured for certain parameter ranges where this behaviour was previously not expected, including the two dimensional case of laser self-focusing. The space and time structure of such solutions for very small values of the nonlinear dissipation parameter is studied numerically and compared to a conjectured mechanism related to a new family of stationary singular solutions of the CSE.