Failure of Fixed-Profile Modified Scattering at the Pure $L^2$ Endpoint for the 1D Defocusing Cubic NLS

Abstract

We prove that the standard fixed-profile modified-scattering ansatz fails at the unweighted $L^2$ endpoint for the one-dimensional defocusing cubic nonlinear Schr\"odinger equation. More precisely, there exists a real-valued datum $$q_* \in L^1(\mathbb{R}) \cap \bigcap_{k \geq 0} H^k(\mathbb{R}), \quad x q_* \notin L^2(\mathbb{R})$$ for which the corrected Fourier profile has no strong $L^2$ limit. The $L^2$ norm of the datum may be prescribed arbitrarily. The construction is an inductively chosen sum of disjoint smooth bumps. Exact composition of the Zakharov--Shabat transfer matrices inserts a high-frequency oscillation into the logarithm of the transmission coefficient, while the one-sided logarithmic operator in the Deift--Zhou phase amplifies an insertion of size $\epsilon_n$ by a factor of order $\log X_n$. Choosing $\epsilon_n\log X_n=\kappa$ produces a uniform separation between consecutive smooth asymptotic profiles even though the partial data converge in $L^1\cap L^2$. The obstruction is specific to a single time-independent profile and leaves open adaptive or scale-dependent renormalizations.