Intrinsic localized modes for DNLS equation with competing nonlinearities: Bifurcations

Area of Science:

• Nonlinear Dynamics
• Condensed Matter Physics
• Mathematical Physics

Background:

• Discrete Nonlinear Schrödinger (DNLS) equations model complex phenomena in various physical systems.
• Competing nonlinearities introduce rich dynamical behaviors, including intrinsic localized modes (ILMs).
• Understanding ILMs is crucial for applications in Bose-Einstein condensates and other wave systems.

Purpose of the Study:

• To analyze intrinsic localized modes (ILMs) in DNLS equations with cubic-quintic, quadratic-cubic, and cubic-quartic nonlinearities.
• To investigate the influence of coupling (α) and nonlinearity balance (γ) parameters on ILM behavior.
• To identify universal features and bifurcations of ILMs across different competing nonlinearity models.

Main Methods:

• Numerical continuation techniques were employed, starting from the anti-continuum limit (α=0).
• Analysis focused on α-dependent ILM branches and their bifurcations as the parameter γ is varied.
• Comprehensive bifurcation analysis was performed for three distinct DNLS models.

Main Results:

• Identified common bifurcation patterns for ILMs across all three studied DNLS models up to specific bifurcation values.
• Discovered exactly two universal ∞-branches (connecting discrete and continuous limits) for ILMs within a specific γ range (0;γ∗).
• Demonstrated that these ∞-branches exhibit a universal sequence of bifurcations as γ varies.

Conclusions:

• The study reveals universal bifurcation properties of intrinsic localized modes in DNLS equations with competing nonlinearities.
• The existence and behavior of two fundamental ∞-branches are confirmed, offering insights into the transition from discrete to continuous regimes.
• Findings provide a unified understanding of nonlinear localized modes across different competing nonlinearity models.