A System of ODEs for a Perturbation of a Minimal Mass Soliton

We study soliton solutions to the nonlinear Schrödinger equation (NLS) with a saturated nonlinearity. NLS with such a nonlinearity is known to possess a minimal mass soliton. We consider a small perturbation of a minimal mass soliton and identify a system of ODEs extending the work of Comech and Pelinovsky (Commun. Pure Appl. Math. 56:1565–1607, 2003), which models the behavior of the perturbation for short times. We then provide numerical evidence that under this system of ODEs there are two possible dynamical outcomes, in accord with the conclusions of Pelinovsky et al. (Phys. Rev. E 53(2):1940–1953, 1996). Generically, initial data which supports a soliton structure appears to oscillate, with oscillations centered on a stable soliton. For initial data which is expected to disperse, the finite dimensional dynamics initially follow the unstable portion of the soliton curve. Communicated by M.I. Weinstein. J.L. Marzuola Department of Applied Mathematics, Columbia University, 200 S. W. Mudd, 500 W. 120th St., New York City, NY 10027, USA e-mail: [email protected] S. Raynor ( ) Mathematics Department, Wake Forest University, P.O. Box 7388, 127 Manchester Hall, Winston-Salem, NC 27109, USA e-mail: [email protected] G. Simpson Mathematics Department, University of Toronto, 40 St. George St., Toronto, Ontario M5S 2E4, Canada e-mail: [email protected] 426 J Nonlinear Sci (2010) 20: 425–461