Collapse dynamics for the discrete nonlinear Schr\"odinger equation with gain and loss

TL;DR

This paper analyzes finite-time collapse in a discrete nonlinear Schrödinger equation with gain and loss, providing analytical bounds for collapse time and exploring how initial conditions and parameters influence collapse dynamics.

Contribution

It offers the first analytical bounds for collapse time in a DNLS with gain and loss, and characterizes the transition between different collapse regimes.

Findings

Analytical upper and lower bounds for collapse time are established.

A critical linear loss parameter separates finite-time collapse from energy decay.

Numerical simulations confirm bounds and reveal two collapse types: extended and localized.

Abstract

We discuss the finite-time collapse, also referred as blow-up, of the solutions of a discrete nonlinear Schr\"{o}dinger (DNLS) equation incorporating linear and nonlinear gain and loss. This DNLS system appears in many inherently discrete physical contexts as a more realistic generalization of the Hamiltonian DNLS lattice. By using energy arguments in finite and infinite dimensional phase spaces (as guided by the boundary conditions imposed), we prove analytical upper and lower bounds for the collapse time, valid for both the defocusing and focusing cases of the model. In addition, the existence of a critical value in the linear loss parameter is underlined, separating finite time-collapse from energy decay. The numerical simulations, performed for a wide class of initial data, not only verified the validity of our bounds, but also revealed that the analytical bounds can be useful in identifying two distinct types of collapse dynamics, namely, extended or localized. Pending on the discreteness /amplitude regime, the system exhibits either type of collapse and the actual blow-up times approach, and in many cases are in excellent agreement, with the upper or the lower bound respectively. When these times lie between the analytical bounds, they are associated with a nontrivial mixing of the above major types of collapse dynamics, due to the corroboration of defocusing/focusing effects and energy gain/loss, in the presence of discreteness and nonlinearity.