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Stability of weakly nonlinear localized states in attractive potentials
1Institute of Condensed Matter Theory and Solid State Optics, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, 07743 Jena, Germany. stefan@pinet.uni-jena.de
Summary
We derived a stability criterion for bound states in nonlinear Schrödinger equations with attractive potentials. This method simplifies stability analysis by focusing on linear modes, confirmed by optical fiber simulations.
Area of Science:
- Nonlinear Optics
- Mathematical Physics
Background:
- The nonlinear Schrödinger equation (NLSE) models various wave phenomena, including light propagation in optical fibers.
- Analyzing the stability of bound states is crucial for understanding the behavior of nonlinear systems.
Purpose of the Study:
- To derive a sufficient stability criterion for bound states governed by the NLSE with an attractive linear potential and cubic nonlinearity.
- To simplify stability analysis by relying solely on the potential's linear modes.
Main Methods:
- Analytical derivation of a stability criterion for NLSE bound states.
- Numerical verification using a step-index optical fiber model.
- Estimation of growth rates in the weak power limit.
Main Results:
- A sufficient condition for the stability of bound states was established.
- The derived criterion effectively uses only the linear modes of the potential.
- Numerical simulations confirmed the analytical findings for optical fiber systems.
- An estimate for the growth rate as a function of nonlinearity was determined for weak power regimes.
Conclusions:
- The study provides a simplified yet sufficient criterion for assessing the stability of bound states in NLSE.
- The findings are broadly applicable to systems described by the NLSE with attractive potentials.
- Numerical validation underscores the practical relevance and accuracy of the derived criterion.