Asymptotic localization of stationary states in the nonlinear Schrödinger equation

From power law to Anderson localization in nonlinear Schrödinger equation with nonlinear randomness

We study the propagation of coherent waves in a nonlinearly induced random potential and find regimes of self-organized criticality and other regimes where the nonlinear equivalent of Anderson localization prevails. The regime of self-organized criticality leads to power-law decay of transport [Y. Sharabi et al., Phys. Rev. Lett. 121, 233901 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.233901], whereas the second regime exhibits exponential decay.

A self-consistent theory of localization in nonlinear random media

The self-consistent theory of localization is generalized to account for a weak quadratic nonlinear potential in the wave equation. For spreading wave packets, the theory predicts the destruction of Anderson localization by the nonlinearity and its replacement by algebraic subdiffusion, while classical diffusion remains unaffected. In 3D, this leads to the emergence of a subdiffusion-diffusion transition in place of the Anderson transition. The accuracy and the limitations of the theory are discussed.

One-parameter scaling theory for stationary states of disordered nonlinear systems

We show, using detailed numerical analysis and theoretical arguments, that the normalized participation number of the stationary solutions of disordered nonlinear lattices obeys a one-parameter scaling law. Our approach opens a new way to investigate the interplay of Anderson localization and nonlinearity based on the powerful ideas of scaling theory.

Dynamics of wave packets for the nonlinear Schrödinger equation with a random potential

The dynamics of an initially localized Anderson mode is studied in the framework of the nonlinear Schrödinger equation in the presence of disorder. It is shown that the dynamics can be described in the framework of the Liouville operator. An analytical expression for a wave function of the initial time dynamics is found by a perturbation approach. As follows from a perturbative solution the initially localized wave function remains localized. At asymptotically large times the dynamics can be described qualitatively in the framework of a phenomenological probabilistic approach by means of a probability distribution function. It is shown that the probability distribution function may be governed by the fractional Fokker-Planck equation and corresponds to subdiffusion.

Observation of a localization transition in quasiperiodic photonic lattices

We report the observation of the signature of a localization phase transition for light in one-dimensional quasiperiodic photonic lattices, by directly measuring wave transport inside the lattice. Below the predicted transition point an initially narrow wave packet expands as it propagates, while above the transition expansion is fully suppressed. In addition, we measure the effect of focusing nonlinear interaction on the propagation and find it increases the width of the localized wave packets.

Lyapunov exponents in one-dimensional disordered systems with long-range memory

The Lyapunov exponents for Anderson localization are studied in a one-dimensional disordered system. A random Gaussian potential with the power-law decay approximately 1/|x|q of the correlation function is considered. The exponential growth of the moments of the eigenfunctions and their derivative is obtained. Positive Lyapunov exponents, which determine the asymptotic growth rate, are found.