Multiscale expansions avector solitons of a two‐dimensional nonlocal nonlinear Schrödinger system

Abstract

One-and two-dimensional solitons of a multicomponent nonlocal nonlinear Schrödinger (NLS) system are constructed. The model finds applications in nonlinear optics, where it may describe the interaction of optical beams of different frequencies. We asymptotically reduce the model, via multiscale analysis, to completely integrable ones in both Cartesian and cylindrical geometries; we thus derive a Kadomtsev-Petviashvili equation and its cylindrical counterpart, Johnson's equation. This way, we derive approximate soliton solutions of the nonlocal NLS system, which have the form of: (a) dark or antidark soliton stripes and (b) dark lumps in the Cartesian geometry, as well as (c) ring dark or antidark solitons in the cylindrical geometry. The type of the soliton, namely dark or antidark, is determined by the degree of nonlocality: dark (antidark) soliton states are formed for weaker (stronger) nonlocality. We perform numerical simulations and show that the derived soliton solutions do exist and propagate undistorted in the original nonlocal NLS system.

Extended shallow water wave equations

“…Furthermore, higher order KP equations, as well versions in cylindrical coordinatesfirst derived for water waves [46] and then used extensively in plasma physics [5]-have also appeared in many other contexts. Indeed, in higher dimensions, the defocusing nematic equations describing nonlinear optical beam propagation in nematic liquid crystals can be reduced to the KP equation [64,63,65] and the cKdV equation [66]. While the higher order corrections to these equations have not, as yet, been derived, these are anticipated to be of the same form as the extended KP (eKP) and extended cylindrical KdV (ecKdV) equations of the present review.…”

Transverse instability and dynamics of nonlocal bright solitons

“…[50]) is another interesting and relevant theme. This is due to the fact that that there exists a plethora of vector solitons in such settings [51][52][53], while studies on the transverse dynamics of solitons are mainly numerical ones [54]. It would, therefore, be particularly interesting to investigate the combined effect of nonlocality and soliton coupling on the soliton instability dynamics.…”

Dynamics of transformed nonlinear waves for the (2+1)-dimensional pKP-BKP equation: interactions and molecular waves

“…Nonlinear waves include breathers, rogue waves, and solitons. A lot of researchers in this subject have been interested in studying these nonlinear waves lately [27][28][29][30][31][32][33][34][35]. Breathers are periodic in space or time, representing a type of oscillating localized structure.…”