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A system of ODEs for a perturbation of a minimal mass soliton. (English) Zbl 1205.35295

Summary: We study soliton solutions to the nonlinear Schrödinger equation (NLS) with a saturated nonlinearity. A 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 A. Comech and D. Pelinovsky [Commun. Pure Appl. Math. 56, No. 11, 1565–1607 (2003; Zbl 1072.35165)] 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 D. Pelinovsky et al. [Phys. Rev. E 53, No. 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.

MSC:

35Q55 NLS equations (nonlinear Schrödinger equations)
35C08 Soliton solutions
35B35 Stability in context of PDEs
35A24 Methods of ordinary differential equations applied to PDEs
37K40 Soliton theory, asymptotic behavior of solutions of infinite-dimensional Hamiltonian systems

Citations:

Zbl 1072.35165

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