Long-wavelength amplification of a nonlinear Schrödinger equation conserved-charge violation in a one-dimensional Gross–Pitaevskii–Poisson field

Abstract

Can weak self-gravity be treated as a small, generic perturbation of the integrable one-dimensional cubic nonlinear Schrödinger equation (NLS)? We test this question by following the first nontrivial local NLS charge, Q3. In the Gross–Pitaevskii–Poisson model it obeys the exact identity Q˙3=κF3, where F3 is the instantaneous gravitational force on the charge. We quantify the initial violation through the normalized mean-square displacement of Q3. Its curvature coefficient, A3=Var⁡(F3)/Var⁡(Q3), measures the root-mean-square initial speed of the charge in units of the charge’s own ensemble width. For a UV-regulated, linearly stable Gaussian initial-data preparation, Wick contraction gives Var⁡(Q3)=O(L) and Var⁡(F3)=O(L2), hence A3=O(L). Direct Monte Carlo evaluation of the full nonlinear observables confirms these powers for 16⩽L⩽512. The linear growth of A3 means that admitting longer wavelengths makes nominally weak gravity increasingly effective at displacing the NLS charge. Along the Jeans-stable sequence κL∝L−2, the associated short-time curvature scale is L3/2 instead of the L2 benchmark of an infrared-uniform perturbation. This is an equal-time result, not a relaxation-rate or Kardar–Parisi–Zhang (KPZ) measurement. The local sound sector contains the Burgers self-coupling required for KPZ, but unscreened gravity also generates a relevant inverse-gradient coupling and, at fixed strength, a Jeans band. A dynamical connection to KPZ therefore requires a projected charge-relaxation calculation and a branch-resolved structure-factor test.