Soliton stability criterion for generalized nonlinear Schrödinger equations.

Area of Science:

• Nonlinear dynamics
• Mathematical physics
• Soliton theory

Background:

• A conjectured stability criterion for solitons in the driven nonlinear Schrödinger equation (NLSE) relates instability to p'(v)<0 and stability to p'(v)>0.
• Previously, the p(v) curve was approximated by collective coordinate theory and validated via simulations.

Purpose of the Study:

• To calculate the p(v) curve exactly for generalized nonlinear Schrödinger equations (NLSE) under various potentials and driving conditions.
• To analytically or numerically confirm the conjectured stability criterion for these NLSE models.

Main Methods:

• Exact analytical and numerical calculation of the p(v) curve for diverse NLSE scenarios.
• Investigation of solitons in real potentials, time-dependent ramp potentials, and time-dependent quadratic potentials.
• Analysis of logarithmic and cubic NLSE with time-independent potentials, including bisoliton solutions.

Main Results:

• The p(v) curve was precisely determined for several classes and cases of the generalized NLSE.
• The stability criterion was confirmed across all investigated scenarios, including those with time-dependent coefficients and potentials.
• A bisoliton solution exhibiting oscillatory behavior was identified for the cubic NLSE in a quadratic potential well.

Conclusions:

• The exact calculation of the p(v) curve provides rigorous validation for the soliton stability criterion in the driven NLSE.
• This work extends the understanding of soliton dynamics and stability in complex, time-dependent nonlinear systems.
• The findings have implications for controlling and predicting the behavior of solitons in various physical contexts.