Kinetic theory of nonlinear diffusion in a weakly disordered nonlinear Schrödinger chain in the regime of homogeneous chaos

Abstract

We study the discrete nonlinear Schröinger equation with weak disorder, focusing on the regime when the nonlinearity is, on the one hand, weak enough for the normal modes of the linear problem to remain well resolved, but on the other, strong enough for the dynamics of the normal mode amplitudes to be chaotic for almost all modes. We show that in this regime and in the limit of high temperature, the macroscopic density $\rho$ satisfies the nonlinear diffusion equation with a density-dependent diffusion coefficient, $D(\rho )=D_0{\rho }^2$. An explicit expression for $D_0$ is obtained in terms of the eigenfunctions and eigenvalues of the linear problem, which is then evaluated numerically. The role of the second conserved quantity (energy) in the transport is also quantitatively discussed.

24 pages, 13 figures

References in corpus (16)

• Destruction of Anderson localization by a weak nonlinearity
• Universal spreading of wavepackets in disordered nonlinear systems
• Absence of Wavepacket Diffusion in Disordered Nonlinear Systems
• The crossover from strong to weak chaos for nonlinear waves in disordered systems
• Band-center anomaly of the conductance distribution in one-dimensional Anderson localization
• Nonequilibrium chaos of disordered nonlinear waves
• Nonlinear waves in disordered chains: probing the limits of chaos and spreading
• The nonequilibrium discrete nonlinear Schroedinger equation
• Statistical mechanics of general discrete nonlinear Schr{ö}dinger models: Localization transition and its relevance for Klein-Gordon lattices
• KAM tori in 1D random discrete nonlinear Schrödinger model?
• Scaling and the center of band anomaly in a one-dimensional Anderson model with diagonal disorder
• Spreading in Disordered Lattices with Different Nonlinearities
• Excitations of the One Dimensional Bose-Einstein Condensates in a Random Potential
• On the local nature and scaling of chaos in weakly nonlinear disordered chains
• Do nonlinear waves in random media follow nonlinear diffusion equations?
• Effective theory for the propagation of a wave-packet in a disordered and nonlinear medium

Cited by in corpus (20)

• Universal Nonlinear Disordered Wave Packet Subdiffusion: 12 Decades
• Nonlinear lattice waves in heterogeneous media
• Characteristics of chaos evolution in one-dimensional disordered nonlinear lattices
• Controlling Optical Beam Thermalization via Band-Gap Engineering
• Long-time expansion of a Bose-Einstein condensate: Can Anderson localization be observed?
• Destruction of Anderson localization by nonlinearity in kicked rotator at different effective dimensions
• Nonlinear Lattice Waves in Random Potentials
• Identifying localized and spreading chaos in nonlinear disordered lattices by the Generalized Alignment Index (GALI) method
• Classical and quantum systems: transport due to rare events
• Quantum subdiffusion with two- and three-body interactions
• Density Resolved Wave Packet Spreading in Disordered Gross-Pitaevskii Lattices
• Enhancement of chaotic subdiffusion in disordered ladders with synthetic gauge fields
• Spreading, Nonergodicity, and Selftrapping: a puzzle of interacting disordered lattice waves
• Random matrix model of Kolmogorov-Zakharov turbulence
• Prethermalization and wave condensation in a nonlinear disordered Floquet system
• Thermalization of isolated harmonic networks under conservative noise
• Chaotic behaviour of disordered nonlinear lattices
• Equilibrium and Non-equilibrium Gross-Pitaevskii Lattice Dynamics: Interactions, Disorder, and Thermalization
• Deconfinement of classical Yang-Mills color fields in a disorder potential
• Nonlinear dynamics and chaos in multidimensional disordered Hamiltonian systems