SUBCRITICAL LOCALIZATION IN THE DISCRETE NONLINEAR SCHRODINGER-EQUATION WITH ARBITRARY POWER NONLINEARITY

Discretizing the continuous nonlinear Schrodinger 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 coarse 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.