On the nonexistence of NLS breathers
Miguel Á. Alejo, Adán J. Corcho
TL;DR
This work establishes rigorous nonexistence results for breather solutions of nonlinear Schrödinger equations by leveraging virial-type functionals that link periodic, localized dynamics to the energy–mass structure of the equation. The approach yields comprehensive criteria across zero and nonzero backgrounds, covering focusing/defocusing regimes and subcritical, critical, and supercritical nonlinearities, and extends to cubic–quintic, biharmonic, derivative, and logarithmic NLS variants. Key contributions include precise regime-based prohibitions tied to conserved quantities (mass, energy, momentum) and ground-state thresholds, clarifying when breathers cannot occur and how this constrains long-time dynamics and potential soliton resolution. The results provide a roadmap for identifying or ruling out breather-type profiles in diverse NLS models, with implications for optical and matter-wave systems and for understanding the global dynamics of dispersive equations.
Abstract
In this work, a rigorous proof of the nonexistence of breather solutions for NLS equations is presented. By using suitable virial functionals, we are able to characterize the nonexistence of breather solutions, different from standing waves, by only using their inner energy and the power of the corresponding nonlinearity of the equation. We extend this result for several NLS models with different power nonlinearities and even the derivative and logarithmic NLS equations.