Nonautonomous Solitons in External Potentials

Abstract

Novel soliton solutions for the nonautonomous nonlinear Schrödinger equation models with linear and harmonic oscillator potentials substantially extend the concept of classical solitons and generalize it to the plethora of nonautonomous solitons that interact elastically and generally move with varying amplitudes, speeds, and spectra adapted both to the external potentials and to the dispersion and nonlinearity variations. The nonautonomous soliton concept can be applied to different physical systems, from hydrodynamics and plasma physics to nonlinear optics and matter waves, and offer many opportunities for further scientific studies.

Manipulating matter rogue waves and breathers in Bose-Einstein condensates

“…During the past several years considerable interest has been shown in exploring localized nonlinear waves in the variable coefficient nonlinear Schrödinger (NLS) equation and its generalizations [1][2][3]. The motivation comes from the fact that NLS equation and its variants appear in several branches/topics of physics, including nonlinear optics [4,5] and Bose-Einstein condensates (BECs) [6][7][8], etc.…”

“…Further, it has also been explained that the above structure (5) arises due to a modulation instability (MI) [16,20] of the plane wave solution of the constant coefficient NLS equation. Very recently higher-order RWs (HRWs) which correspond to the higher-order rational solutions of the NLS equation (2) have been deduced [21]. The explicit expressions for the second and third-order RWs of the NLS equation are given in appendix A.…”

“…Several theoretical studies on the dynamics of RWs in nonlinear fiber optics [25,26], plasma physics [27], laser-plasma interactions [28], and even econophysics [29], described by scalar NLS equation, have been made in recent times. Now, as mentioned in the beginning, the dynamics of a cigar shaped BEC at absolute zero temperature is usually described by the mean-field GP equation (1), which is a generalized form of the ubiquitous constant coefficient NLS equation (2), for the wave function of the condensate. Since the NLS equation (2) admits breather and RW solutions, it is natural to expect that RWs and breathers may also be found in BEC systems as well.…”

“…(1) to the standard NLS equation (2). Following the standard procedure [11,41] we find that the required similarity transformation should be of the form…”

“…and U (X, T ) is the solution of the standard NLS equation (2). In the above b, r 0 are arbitrary constants and the modulational functions R(t) and β(t) 2 should satisfy the following condition…”