Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials
Abstract
Here in this paper, we discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a fourparameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.
 Authors:
 Univ. of Bayreuth (Germany). Physikalisches Institut
 Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Pontifical Catholic Univ. of Chile, Region Metropolitana (Chile). Dept. de Fisica
 Savitribai Phule Pune Univ., Pune (India)
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Publication Date:
 Research Org.:
 Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
 Sponsoring Org.:
 USDOE Laboratory Directed Research and Development (LDRD) Program
 OSTI Identifier:
 1414120
 Report Number(s):
 LAUR1623477
Journal ID: ISSN 24700045
 Grant/Contract Number:
 AC5206NA25396
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Physical Review E
 Additional Journal Information:
 Journal Volume: 94; Journal Issue: 3; Journal ID: ISSN 24700045
 Publisher:
 American Physical Society (APS)
 Country of Publication:
 United States
 Language:
 English
 Subject:
 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; Mathematics
Citation Formats
Mertens, Franz G., Cooper, Fred, Arevalo, Edward, Khare, Avinash, Saxena, Avadh, and Bishop, Alan R. Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials. United States: N. p., 2016.
Web. doi:10.1103/PhysRevE.94.032213.
Mertens, Franz G., Cooper, Fred, Arevalo, Edward, Khare, Avinash, Saxena, Avadh, & Bishop, Alan R. Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials. United States. doi:10.1103/PhysRevE.94.032213.
Mertens, Franz G., Cooper, Fred, Arevalo, Edward, Khare, Avinash, Saxena, Avadh, and Bishop, Alan R. 2016.
"Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials". United States.
doi:10.1103/PhysRevE.94.032213. https://www.osti.gov/servlets/purl/1414120.
@article{osti_1414120,
title = {Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials},
author = {Mertens, Franz G. and Cooper, Fred and Arevalo, Edward and Khare, Avinash and Saxena, Avadh and Bishop, Alan R},
abstractNote = {Here in this paper, we discuss the behavior of solitary wave solutions of the nonlinear Schrödinger equation (NLSE) as they interact with complex potentials, using a fourparameter variational approximation based on a dissipation functional formulation of the dynamics. We concentrate on spatially periodic potentials with the periods of the real and imaginary part being either the same or different. Our results for the time evolution of the collective coordinates of our variational ansatz are in good agreement with direct numerical simulation of the NLSE. We compare our method with a collective coordinate approach of Kominis and give examples where the two methods give qualitatively different answers. In our variational approach, we are able to give analytic results for the small oscillation frequency of the solitary wave oscillating parameters which agree with the numerical solution of the collective coordinate equations. We also verify that instabilities set in when the slope dp(t)/dv(t) becomes negative when plotted parametrically as a function of time, where p(t) is the momentum of the solitary wave and v(t) the velocity.},
doi = {10.1103/PhysRevE.94.032213},
journal = {Physical Review E},
number = 3,
volume = 94,
place = {United States},
year = 2016,
month = 9
}
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