Subcritical localization in the discrete nonlinear Schrodinger equation with arbitrary power nonlinearity

Abstract

Discretizing the continuous nonlinear SchrMinger equation with arbitrary power nonlinearity influences the time evolution of its ground state solitary solution. In the subcritical case, for grid resolutions above a certain transition value, depending on the degree of nonlinearity, the solution will oscillate smoothly with a frequency and amplitude that depend on both the resolution and the degree of nonlinearity. Thus in this region the discrete system will give a good reproduction of the dynamics in the continuum one. However, when the discretization gets too coaxe the solution will become localized in finite time, although the degree of nonlinearity is subcritical. Thus in this region the discrete system cannot reproduce the stable solitary solution found in the continuum system. Numerical studies of the phenomenon are performed and a variational approach is used to give a qualitative explanation of the dynamics in the discrete system.

Propagation and collisions of semidiscrete solitons in arrayed and stacked waveguides

“…This is explained by the fact that discrete solitons tend to be mobile when they are broad enough, which corresponds to the quasi-continuum limit [32,43], while, as mentioned above, the 2D continual counterpart of the SDS is the Townes soliton, that, in turn, is subject to the collapse, i.e., the catastrophic self-compression [23]. Thus, the onset of the collapse converts broad quasi-continual solitons back into the essentially discrete (narrow) ones [44]. The self-compression eventually arrests the collapse, making the SDSs strongly pinned to the underlying lattice structure, i.e., immobile.…”

Discretization effects in the nonlinear Schrödinger equation

“…Similarly, the consequences of, say, using secondorder discretization versus fourth-order one have not been analyzed. In fact, to the best of our knowledge the only study on discretization effects in the NLS is by Bang, Rasmussen and Christiansen [3] on stability of waveguides in the one-dimensional NLS. It is instructive to compare this lack of theory with the voluminous body of research on numerical methods for conservation laws.…”

Stability conditions for one-dimensional optical solitons in cubic-quintic-septimal media

“…2, where the third-and seventh-order nonlinearities are fixed to be 3 1 g = + and 7 1 g = + , respectively. In that figure, solid (dashed) lines correspond to regions of stable (unstable) soliton propagation, identified To extend these conclusions, numerical solutions of Eq.…”

“…Unstable soliton propagation is observed in (1+1)D when the system exhibits higher-order nonlinearities (HON) [7]. For example, in focusing quintic NL media the diffraction effect is not sufficient to balance the self-focusing, consequently the beam is subject to the critical collapse [8].…”