Schrödinger Equations in Nonlinear Systems

Abstract

of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Spatiotemporal engineering of matter-wave solitons in Bose–Einstein condensates

“…In most experiments, factor α(t) is typically fixed to a constant value, but adiabatic changes in the strength of the trap are experimentally feasible too. In the case of the cigar-shaped BEC, the aforementioned self-consistent reduction of the 3D GP equation to the 1D form with the external potential can be provided by means of a multiple-scale expansion [128] which exploits a small parameter δ 2 = (N a s /a 0 )α 1, where a s is the s-wave scattering length. Parameter δ evaluates the relative strength of the two-body interactions as compared to the kinetic energy of the atoms.…”

“…The object considered here is an effectively 1D BEC loaded in a cigar-shaped trap, which is elongated in x and tightly confined in the transverse plane (y, z) [128]. In the mean-field approximation, atoms in the hyperfine state with atomic spin F = 1 are described by a 1D three-component wave function,…”

“…The full GP system (220) is not integrable. Therefore, it is relevant to employ VA for the analysis of the soliton dynamics [128,162], making use of the Lagrangian…”

Comparison of the Effectiveness of Different Methods for Constructing Exact Solutions to Nonlinear PDEs. Generalizations and New Solutions

“…Nonlinear Schrödinger-type equations are often used to describe different processes in theoretical physics, including nonlinear optics, superconductivity, and plasma physics [85][86][87][88].…”

Compensation process and generation of chirped femtosecond solitons and double-kink solitons in Bose–Einstein condensates with time-dependent atomic scattering length in a time-varying complex potential

“…At ultralow temperatures, the properties of weakly interacting bosonic gases trapped in a potential V is usually described by the time-dependent, nonlinear, mean-field Gross-Pitaevskii equation for the order parameter Ψ(r, t) at position r and time t [5][6][7][8][9][10][11]…”