Nonlinear waves in networks: a simple approach using the sine—Gordon equation

Abstract

To study how nonlinear waves propagate across Y- and T-type junctions, we consider the 2D sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a 1D effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when $v > 1 - omega$, where $v$ and $\omega$ are respectively its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.

Keywords

Bibliographic record

BibTeX Citation

@article{Caputo2014nonlinearwaves,
  author = {Caputo, J.-G. and Dutykh, D.},
  title = {Nonlinear waves in networks: a simple approach using the sine—Gordon equation},
  journal = {Phys. Rev. E},
  year = {2014},
  volume = {90},
  pages = {022912},
  doi = {10.1103/PhysRevE.90.022912},
  abstract = {To study how nonlinear waves propagate across Y- and T-type junctions, we consider the 2D sine-Gordon equation as a model and examine the crossing of kinks and breathers. Comparing energies for different geometries reveals that, for small widths, the angle of the fork plays no role. Motivated by this, we introduce a 1D effective model whose solutions agree well with the 2D simulations for kink and breather solutions. These exhibit two different behaviors: a kink crosses if it has sufficient energy; conversely a breather crosses when $v > 1 - omega$, where $v$ and $\omega$ are respectively its velocity and frequency. This methodology can be generalized to more complex nonlinear wave models.},
  keywords = {kink, breather, sine-Gordon equation, Josephson junction}
}