Modified Scattering and Asymptotics for Perturbed One-Dimensional Cubic NLS

Abstract

We study the long-time dynamics of small solutions to the one-dimensional nonlinear Schr\"odinger equation \[ i\partial_t v+\partial_x^2v-\beta\abs{v}^2v +\cW(x)\abs{v}^4v+\gamma i\partial_xv=0, \] where $\cW$ is spatially localized. The cubic nonlinearity is long range and produces the logarithmic phase correction, whereas the localized quintic term is short range at leading order. We construct a global forward modified wave operator for small complex asymptotic profiles and prove quantitative final-state estimates. For small data in the weighted energy space, we also establish global existence, sharp $t^{-1/2}$ decay, and forward modified scattering with a unique asymptotic profile. The principal new phenomenon occurs beyond this leading law. The exact Duhamel tail generated by the localized quintic term admits a quantitative inner scaling limit at the distinguished frequency $\zeta=-\gamma/2$ on the scale $\abs{\zeta+\gamma/2}\sim t^{-1/2}$. Its universal shape is explicit and depends on the value of the scattering profile on the distinguished ray and on the zeroth moment of $\cW$. When both quantities are nonzero, the limit is nontrivial, belongs optimally to $C^{2,1}_{\mathrm{loc}}$, and is not $C^3$ at the center. Away from the corresponding self-similar ray $\xi=-\gamma$, we construct rigorously defined higher-order outer expansions through every integer order allowed by the decay of $\cW$, and to each fixed finite order when $\cW$ is rapidly decreasing.