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How close are integrable and nonintegrable models: A parametric case study based on the Salerno model.

Thudiyangal Mithun1, Aleksandra Maluckov2, Ana Mančić3

  • 1Department of Mathematics and Statistics, University of Massachusetts, Amherst, Massachusetts 01003-4515, USA.

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Summary

This study explores the Salerno model, bridging integrable and nonintegrable systems. It finds that Lyapunov exponents sensitively detect deviations from integrability in generic systems.

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Area of Science:

  • Nonlinear dynamics
  • Mathematical physics
  • Complex systems

Background:

  • The Salerno model serves as a bridge between the integrable Ablowitz-Ladik lattice and the nonintegrable discrete nonlinear Schrödinger model.
  • Investigating the transition from integrable to nonintegrable behavior is crucial for understanding complex physical systems.

Purpose of the Study:

  • To assess the proximity of integrable and nonintegrable models using generic initial data.
  • To determine how well conserved quantities remain constant in nonintegrable systems.
  • To identify a sensitive diagnostic for integrability breaking.

Main Methods:

  • Analysis of the Salerno model with generic initial conditions.
  • Examination of the constancy of conserved quantities under deviations from integrability.
  • Computation of the full spectrum of Lyapunov exponents.

Main Results:

  • Even minor deviations from integrability significantly affect conserved quantities in the Salerno model.
  • Measuring the constancy of conserved quantities requires complex physical and mathematical analysis.
  • The full spectrum of Lyapunov exponents proves to be a sensitive indicator of integrability breaking.

Conclusions:

  • The Salerno model effectively demonstrates the transition from integrable to nonintegrable dynamics.
  • Lyapunov exponents offer a more accessible and sensitive diagnostic for detecting integrability breaking compared to conserved quantities.
  • This research provides a valuable tool for analyzing complex systems exhibiting partial integrability.