scholarly journals Non-Reciprocal Supratransmission in Mechanical Lattices with Non-Local Feedback Control Interactions

Crystals ◽  
2021 ◽  
Vol 11 (2) ◽  
pp. 94
Author(s):  
Jack E. Pechac ◽  
Michael J. Frazier
Keyword(s):  
Wave Propagation ◽  
Signal Amplification ◽  
Energy Transport ◽  
Gordon Equation ◽  
Nonlinear Wave Propagation ◽  
Sine Gordon Equation ◽  
Passive System ◽  
Local Feedback ◽  
Non Local ◽  
Mechanical Logic

We numerically investigate the supratransmission phenomenon in an active nonlinear system modeled by the 1D/2D discrete sine-Gordon equation with non-local feedback. While, at a given frequency, the typical passive system exhibits a single amplitude threshold marking the onset of the phenomenon, we show that the inclusion of non-local feedback manifests additional thresholds that depend upon the specific boundary from which supratransmission is stimulated, realizing asymmetric (i.e., non-reciprocal) dynamics. The results illustrate a new means of controlling nonlinear wave propagation and energy transport for, e.g., signal amplification and mechanical logic.

Geometric theory of local and non-local conservation laws for the sine-Gordon equation

1981 ◽  
Vol 376 (1766) ◽  
pp. 401-433 ◽  
Keyword(s):  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Gordon Equation ◽  
Vries Equation ◽  
Geometric Theory ◽  
Nonlinear Evolution Equations ◽  
Sine Gordon Equation ◽  
Local Conservation ◽  
Complete Set ◽  
Non Local

In previous work one of the authors gave a geometric theory of those nonlinear evolution equations (n. e. es) that can be solved by the Zakharov & Shabat (1972) inverse scattering scheme as generalized by Ablowitz, Kaup, Newell & Segur (1973 b , 1974). In this paper we extend the geo­metric theory to include the Hamiltonian structure of those n. e. es solvable by the method, and we indicate the connection between the geometric theory and the theory of prolongation structures and pseudopotentials due to Wahlquist & Estabrook (1975, 1976). We exploit a ‘gauge’ in­variance of the geometric theory to derive both the well known polynomial conserved densities of the sine-Gordon equation and a non-local set of conserved densities. These act as Hamiltonian densities for a hierarchy of sine-Gordon equations which is analogous to that found by Lax (1968) for the Korteweg-de Vries equation and appears to be new. In an Appendix we derive an expression for the equation of motion for an arbitrary member of the sine-Gordon hierarchy by methods which can be applied in larger context. The results for the sine-Gordon equation lead to the conclusion that a complete set of conserved densities for an arbitrary n. e. e. solvable by the A. K. N. S. –Z. S. scheme must include non-local conserved densities.

scholarly journals Duality of positive and negative integrable hierarchies via relativistically invariant fields

2021 ◽  
Vol 2021 (7) ◽  
Author(s):  
S. Y. Lou ◽  
X. B. Hu ◽  
Q. P. Liu
Keyword(s):  
Integrable Hierarchies ◽  
Gordon Equation ◽  
Formal Series ◽  
Two Dimensional ◽  
Relativistic Invariance ◽  
Sine Gordon Equation ◽  
Recursion Operators ◽  
Symmetry Approach ◽  
Duality Relations ◽  
Duality Problem

Abstract It is shown that the relativistic invariance plays a key role in the study of integrable systems. Using the relativistically invariant sine-Gordon equation, the Tzitzeica equation, the Toda fields and the second heavenly equation as dual relations, some continuous and discrete integrable positive hierarchies such as the potential modified Korteweg-de Vries hierarchy, the potential Fordy-Gibbons hierarchies, the potential dispersionless Kadomtsev-Petviashvili-like (dKPL) hierarchy, the differential-difference dKPL hierarchy and the second heavenly hierarchies are converted to the integrable negative hierarchies including the sG hierarchy and the Tzitzeica hierarchy, the two-dimensional dispersionless Toda hierarchy, the two-dimensional Toda hierarchies and negative heavenly hierarchy. In (1+1)-dimensional cases the positive/negative hierarchy dualities are guaranteed by the dualities between the recursion operators and their inverses. In (2+1)-dimensional cases, the positive/negative hierarchy dualities are explicitly shown by using the formal series symmetry approach, the mastersymmetry method and the relativistic invariance of the duality relations. For the 4-dimensional heavenly system, the duality problem is studied firstly by formal series symmetry approach. Two elegant commuting recursion operators of the heavenly equation appear naturally from the formal series symmetry approach so that the duality problem can also be studied by means of the recursion operators.

Wave propagation improvement in two-dimensional bond-based peridynamics model

2021 ◽  
pp. 095440622098555
Author(s):  
Reza Alebrahim ◽  
Pawel Packo ◽  
Mirco Zaccariotto ◽  
Ugo Galvanetto
Keyword(s):  
Wave Propagation ◽  
Dynamic Performance ◽  
Numerical Dispersion ◽  
Motion Modeling ◽  
Minimization Method ◽  
Irregular Wave ◽  
Point Collocation ◽  
Potential Applications ◽  
Non Local ◽  
Set Up

In this study, methods to mitigate anomalous wave propagation in 2-D Bond-Based Peridynamics (PD) are presented. Similarly to what happens in classical non-local models, an irregular wave transmission phenomenon occurs at high frequencies. This feature of the dynamic performance of PD, limits its potential applications. A minimization method based on the weighted residual point collocation is introduced to substantially extend the frequency range of wave motion modeling. The optimization problem, developed through inverse analysis, is set up by comparing exact and numerical dispersion curves and minimizing the error in the frequency-wavenumber domain. A significant improvement in the wave propagation simulation using Bond-Based PD is observed.

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