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Discrete breathers: exact solutions in piecewise linear models

Lahiri1, Panda, Roy

  • 1Department of Physics, Vidyasagar Evening College, Calcutta 700 006, India.

Physical Review Letters
|October 6, 2000
PubMed
Summary

Exact breather solutions were found for piecewise linear (PWL) discrete nonlinear equations, offering insights into breather properties and stability. These findings stem from analyzing manifold intersections in 2D mappings.

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Area of Science:

  • Nonlinear Dynamics
  • Mathematical Physics

Background:

  • Discrete nonlinear equations, such as the discrete nonlinear Schrödinger (DNLS) and Klein-Gordon (DKG) equations, are fundamental in modeling various physical phenomena.
  • Breather solutions represent localized, oscillating energy packets within these systems.
  • Understanding the properties and stability of breathers is crucial for predicting system behavior.

Purpose of the Study:

  • To construct exact breather solutions for piecewise linear (PWL) versions of DNLS and DKG equations.
  • To investigate the underlying mathematical mechanisms governing these solutions.
  • To gain insights into breather properties, including their dynamical stability.

Main Methods:

  • Development of piecewise linear (PWL) models for discrete nonlinear equations.

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  • Analysis of associated 2D mappings to identify fixed points.
  • Investigation of the intersections between stable and unstable manifolds of these fixed points.
  • Exact construction of breather solutions based on these manifold intersections.

    • Main Results:

    • Successfully constructed exact breather solutions for PWL DNLS and DKG equations.
    • Demonstrated that these solutions arise from the intersection of stable and unstable manifolds.
    • The PWL nature of the models facilitates the exact construction of these solutions.
    • The derived solutions provide valuable insights into breather characteristics.

    Conclusions:

    • Exact breather solutions can be rigorously constructed in simplified, piecewise linear discrete nonlinear systems.
    • The geometric approach analyzing manifold intersections provides a powerful tool for understanding breather dynamics.
    • These findings lay the groundwork for future investigations into the dynamical stability of breathers.