Nonlinear stability of the constant solution in the focusing case for large periods

Prove the nonlinear stability of the nonzero constant solution u(x,t) ≡ 1 to the focusing nonlocal derivative NLS equation i u_t = u_{xx} − u (i + H) (|u|^2)_x with respect to periodic perturbations in H^1_per((0,L), C) for every spatial period L ∈ [π, ∞).

Background

The authors prove nonlinear stability of the nonzero constant background in the defocusing case for all periods and, in the focusing case, for periods L ∈ (0, π), using a Lyapunov functional built from conserved quantities. They also identify a linear resonance and instability in the focusing case for the 2π-periodic setting, indicating challenges for larger periods.

Consequently, extending nonlinear stability to larger spatial periods in the focusing case is unresolved. The remark explicitly states that proving nonlinear stability for L ≥ π remains open, a regime that includes periods multiple of 2π where linear instability is present.