Exceptional discretizations of the sine-Gordon equation

Abstract

The method of one-dimensional maps was recently introduced as a means of generating exceptional discretizations of the phi(4) theory, i.e., discrete phi(4) models which support kinks centered at a continuous range of positions relative to the lattice. In this paper, we employ this method to obtain exceptional discretizations of the sine-Gordon equation (i.e., exceptional Frenkel-Kontorova chains). We also use one-dimensional maps to construct a discrete sine-Gordon equation supporting kinks which move with arbitrary velocities without emitting radiation.

‘‘Tomography’’ of the Cluster Structure of Light Nuclei via Relativistic Dissociation

“…The nuclear track emulsion technique at the JINR Synchrophasotron began to be used in the 50s with irradiations by 10 GeV protons [32]. Analysis of inelastic interactions of protons with nuclei of NTE composition pointed to the significant role of peripheral interactions.…”

Non-linear waves in lattices: past, present, future

“…Understanding such properties either on a model-specific basis, or, ideally, based on more general/fundamental principles is an important open direction for these nonlinear dynamical lattices. At this point, we should also mention in passing the very interesting recent work of Barashenkov & van Heerden (2008). There, the examination of exceptional discretizations [bearing an effective translational invariance through the map type approach of Pelinovsky (2006) and Barashenkov et al (2005)] led to some ingenious suggestions on how to discretize so as to preserve genuinely traveling localized excitations i.e., kinks in the discrete sine-Gordon and discrete φ 4 settings.…”

Discrete Variants of the $$\phi ^4$$ Model: Exceptional Discretizations, Conservation Laws and Related Topics

“…A moving discrete kink typically (i.e., in the case of generic discretizations) radiates small-amplitude wave packets losing its energy and being initially decelerated and eventually trapped by the lattice in a well of the PNp [16]. There exist numerous 1 attempts to analyze and reduce the radiation from a moving kink [15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]. The radiation is attributed to the resonance of moving kinks with small-amplitude phonons.…”

“…Nevertheless it has been shown that the radiation can vanish for a set of selected kink velocities [15,16,20,22,24,25,31]. There exists an example of kink in a non-integrable lattice, radiating no energy while moving with an arbitrary speed [18]. Kink solutions with oscillating background (also known as nanopterons) have been found as permanent profile traveling waves moving with constant velocity [24].…”