On the dependence of the nonlinear Schrodinger flow upon the power of the nonlinearity

TL;DR

This work analyzes how the flow of the defocusing energy-subcritical nonlinear Schrödinger equation depends on the nonlinearity power $\sigma$, establishing local-in-time continuity in $H^1$ for all admissible powers and global-in-time continuity when $\sigma$ is sufficiently large. A key novelty is the uniform control of rescaled density profiles in the Wasserstein distance $W_1$, which remains valid even in long-range regimes, highlighting robustness to dispersion. The authors further investigate the limit $\sigma\to 0$, showing convergence, in the rescaled variables, to the logarithmic Schrödinger equation $i\partial_t u + \frac{1}{2}\Delta u = u\ln|u|^2$, via a rigorous analysis that couples perturbed porous-medium dynamics with a harmonic Fokker–Planck operator and Duhamel-type perturbations. An Ehrenfest-time-type estimate precedes global convergence results, and the paper provides detailed rates and interpolation results bridging power-type and logarithmic nonlinearities. Overall, the work clarifies the continuous dependence of nonlinear Schrödinger dynamics on the nonlinearity power and builds a rigorous link between power and logarithmic regimes under a unified rescaled framework.

Abstract

We prove continuity properties for the flow map associated to the defocusing energy-subcritical power-like nonlinear Schr{ö}dinger equation, when the power varies. We show local in time continuity in the energy space for any power, and global in time continuity for sufficiently large powers. When the linear dispersive rate is counterbalanced by a time-dependent rescaling, we show a uniform in time continuity of the squared modulus of this rescaled function, in Kantorovich distance, for any power, including long range cases in terms of scattering. The most difficult result addresses the convergence of suitably renormalized solutions to the solution of the logarithmic Schr{ö}dinger equation, when the power goes to zero, uniformly in time, in Kantorovich distance. The proof relies on estimates for perturbed porous medium equations, involving the harmonic Fokker-Planck operator.

Theorems & Definitions (32)

• Remark 1.1
• Remark 1.2
• Theorem 1.3: Local in time continuity
• Theorem 1.4: Global continuity in the (very) short range case
• Corollary 1.5: Continuity of the scattering operator
• Theorem 1.6: Global continuity for the rescaled modulus
• Corollary 1.7
• Theorem 1.8
• Theorem 1.9
• Definition 2.1