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On the adiabatic invariance of the trapped wave's action

Ekaterina V. Shishkina, Serge N. Gavrilov

Abstract

Recently, it has been shown (Gavrilov et al., Nonlinear Dyn, 112, 2024) that in a linear continuum system with time-varying parameters, the amplitude of a strongly localized mode can be calculated as a function of current parameter values and does not depend on the history of parameters change. This fact allows one to introduce the adiabatic invariant for such a system according to the general definition as a quantity that remains approximately constant in a system with several slowly time-varying parameters. In this paper, we show that, introduced in such a way, the adiabatic invariant can be calculated as the ratio of localized mode energy and its frequency. This yields a dramatically simplified way to solve a class of problems concerning localized oscillation of continuum systems with discrete inclusions, although it is not always clear enough what "the mode energy" is. The observed fact allows one to consider newly introduced adiabatic invariants as a straightforward generalization of a notion known to Hamiltonian systems.

On the adiabatic invariance of the trapped wave's action

Abstract

Recently, it has been shown (Gavrilov et al., Nonlinear Dyn, 112, 2024) that in a linear continuum system with time-varying parameters, the amplitude of a strongly localized mode can be calculated as a function of current parameter values and does not depend on the history of parameters change. This fact allows one to introduce the adiabatic invariant for such a system according to the general definition as a quantity that remains approximately constant in a system with several slowly time-varying parameters. In this paper, we show that, introduced in such a way, the adiabatic invariant can be calculated as the ratio of localized mode energy and its frequency. This yields a dramatically simplified way to solve a class of problems concerning localized oscillation of continuum systems with discrete inclusions, although it is not always clear enough what "the mode energy" is. The observed fact allows one to consider newly introduced adiabatic invariants as a straightforward generalization of a notion known to Hamiltonian systems.
Paper Structure (15 sections, 72 equations)

This paper contains 15 sections, 72 equations.

Table of Contents

  1. Introduction
  2. A discrete mass-spring inclusion at a fixed position
  3. Mathematical formulation
  4. Stationary unperturbed problem
  5. Asymptotic solution of the non-stationary unperturbed problem by the method of stationary phase
  6. Asymptotic solution of the non-stationary perturbed problem by the space-time ray method
  7. The adiabatic invariance of the trapped wave's action
  8. A moving discrete mass-spring inclusion
  9. Mathematical formulation
  10. Stationary unperturbed problem
  11. Asymptotic solution of the non-stationary unperturbed problem by the method of stationary phase
  12. Asymptotic solution of the non-stationary perturbed problem by the space-time ray method
  13. The alternative solution using the adiabatic invariance of the trapped wave's action
  14. Discussion
  15. Calculation of the false invariant $J^{(1)}$

Theorems & Definitions (5)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark A.1