Exact asymptotic expansion below the threshold p = 1/d

Determine the exact large-time asymptotic expansion for solutions of the nonlinear Schrödinger equation i∂_t u + (1/2)Δu = |u|^p u in the regime 0 < p < 1/d, beyond the finite-order asymptotic profiles constructed in the paper.

Background

The paper studies modified wave operators for the mass-subcritical long-range nonlinear Schrödinger equation with power 0 < p < 2/d. For 1/d < p < 2/d, the authors construct an ODE-type modifier, while for 0 < p ≤ 1/d they construct finite-order Fuchsian amplitude–eikonal profiles and establish quantitative remainder estimates.

Although these constructions provide arbitrarily high finite-order profiles by increasing the truncation order, the paper explicitly distinguishes them from an exact asymptotic expansion. The unresolved issue is therefore to identify and justify the complete large-time expansion in the range 0 < p < 1/d, rather than merely any prescribed finite number of terms.