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Stability limits for modes held in alternating trapping-expulsive potentials.

Zhihuan Luo1, Yan Liu1, Yongyao Li2

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Summary

We introduce trapping-expulsion management (TEM) to stabilize two-dimensional dynamical states against critical collapse. This method, using periodic potential modulation, is validated by numerical simulations and variational approximation, enhancing stability limits.

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Area of Science:

  • Nonlinear dynamics
  • Mathematical physics
  • Quantum optics
  • Bose-Einstein condensates

Background:

  • Critical collapse is a phenomenon in nonlinear systems driven by self-attraction.
  • Stabilizing two-dimensional dynamical states against collapse is crucial for applications.
  • Previous methods lacked precise control over collapse thresholds.

Purpose of the Study:

  • To develop and analyze a trapping-expulsion management (TEM) scheme for stabilizing two-dimensional dynamical states.
  • To investigate the effectiveness of TEM against critical collapse driven by cubic self-attraction.
  • To determine the stability boundaries of the system under TEM and compare with theoretical predictions.

Main Methods:

  • Numerical simulations were employed to model the system's behavior.
  • Variational approximation (VA) was used to derive analytical insights.
  • The dynamics were reduced to a nonlinear Ermakov equation, equivalent to a linear Mathieu equation.

Main Results:

  • TEM effectively stabilizes two-dimensional dynamical states against critical collapse.
  • Variational approximation accurately predicts stability limits, including parametric resonance onset.
  • Numerical simulations confirmed that TEM can increase the collapse threshold by approximately 1.5%.

Conclusions:

  • The trapping-expulsion management (TEM) scheme offers a viable method for enhancing the stability of nonlinear systems.
  • The interplay between periodic potential modulation and self-attraction is well-described by the derived mathematical framework.
  • TEM provides a controllable mechanism to modify universal constants like the collapse threshold, with implications for optics and BEC.