The asymptotic analysis to multi-breather solutions of the nonlocal space-shifted nonlinear Schrödinger equation on continuous and spatial periodic backgrounds

Abstract

This study delves into the asymptotic analysis and dynamics of multi-breather waveforms within the nonlocal space-shifted nonlinear Schrödinger equation on two distinct backgrounds: a continuous background represented by a plane wave, and a periodic background with periodicity solely along the spatial variable. These breathers are grouped into multiple pairs during the asymptotic analysis, wherein the speeds of two breathers are identical but opposite in directions. Our analysis reveals that the shifting parameter $x_0$ significantly influences the localization center of only one breather within each breather pair in space. The other breather in each pair remains unaffected by changes in $x_0$, except for the shifts in the position of the maximum amplitude point of this breather on the spatial periodic background. By scrutinizing the correlations between velocities or periodicities and the corresponding amplitudes, we uncover both similarities and differences between nonlocal breathers and their associated local counterparts. While both types of breathers exhibit identical relations between velocities or periodicities and their associated parameters, the relationship between amplitude and its parameters for local breathers represents a specific example within the broader spectrum observed in the case of nonlocal breathers. Hence, the correlations of velocities or periodicities with amplitudes for local breathers are considered a subset of those observed in nonlocal breathers. The findings shed light on the intricate dynamics of multi-breather waveforms, offering valuable insights into their behavior on different backgrounds.