Endless spreading of wave packets in disordered nonlinear lattices
In plain words
In a disordered lattice without interactions, waves stay trapped in place (Anderson localization). Weak nonlinearity lets a packet slowly spread, and it is unknown whether the spreading continues forever or eventually stops.
Precise statement
For the disordered discrete nonlinear Schrodinger equation i dpsi_n/dt = eps_n psi_n + beta |psi_n|^2 psi_n - (psi_(n+1) + psi_(n-1)) with i.i.d. random eps_n, and the analogous Klein-Gordon chain, determine whether the second moment $m_2(t)$ of an initially localized packet diverges as $t \to \infty$ (simulations show $m_2 \sim t^{1/3}$ over many decades) or saturates once the packet density falls below a chaos threshold. Answer: yes or no, with the asymptotic law.
What would settle it
A proof of unbounded or bounded spreading, or simulations long enough to show a clear departure from or persistence of subdiffusion with a theory for the observed law.