skip to main content
OSTI.GOV title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: 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 four-parameter 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:
 [1];  [2];  [3];  [4]; ORCiD logo [5];  [5]
  1. Univ. of Bayreuth (Germany). Physikalisches Institut
  2. Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
  3. Pontifical Catholic Univ. of Chile, Region Metropolitana (Chile). Dept. de Fisica
  4. Savitribai Phule Pune Univ., Pune (India)
  5. 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):
LA-UR-16-23477
Journal ID: ISSN 2470-0045
Grant/Contract Number:
AC52-06NA25396
Resource Type:
Journal Article: Accepted Manuscript
Journal Name:
Physical Review E
Additional Journal Information:
Journal Volume: 94; Journal Issue: 3; Journal ID: ISSN 2470-0045
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 four-parameter 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
}

Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record

Citation Metrics:
Cited by: 1work
Citation information provided by
Web of Science

Save / Share:
  • Cited by 1
  • The quantum mechanics for N particles interacting via a delta-function potential in one space dimension and one time dimension is known. The second- quantized description of this system has for its Euler-Lagrange equations of motion the cubic Schrodinger equation. This nonlinear differential equation supports solitary wave solutions. A quantization of these solitons reproduces the weak-coupling limit to the known quantum mechanics. The phase shift for two- body scattering and the energy of the N-body bound state is derived in this approximation. The nonlinear Schrodinger equation is contrasted with the sine- Gordon theory in respect to the ideas which the classicalmore » solutions play in the description of the quantum states.« less
  • The dynamics of waves in weakly nonlinear dispersive media can be described by the nonlinear Schrödinger equation (NLSE). An important feature of the equation is that it can be derived in a number of different physical contexts; therefore, analogies between different fields, such as for example fiber optics, water waves, plasma waves and Bose–Einstein condensates, can be established. Here, we investigate the similarities between wave propagation in optical Kerr media and water waves. In particular, we discuss the modulation instability (MI) in both media. In analogy to the water wave problem, we derive for Kerr-media the Benjamin–Feir index, i.e. amore » nondimensional parameter related to the probability of formation of rogue waves in incoherent wave trains.« less
  • The one-to-one correspondence between a (3+1)-dimensional variable-coefficient nonlinear Schrödinger equation with linear and parabolic potentials and a standard nonlinear Schrödinger equation is given, and an exact superposed Akhmediev breather solution in certain parameter conditions is obtained. These precise expressions for the peak, width, center and phase indicate that diffraction and chirp factors influence the evolutional characteristics such as phase, center and width, while the gain/loss parameter only affects the evolution of the peak. Moreover, by modulating the relation between the terminal accumulated time T{sub e} or the maximum accumulated time T{sub m} and the accumulated time T{sub 0} based onmore » the maximum amplitude of Akhmediev breather, the controllability for the type of excitation such as postpone, maintenance and restraint of the superposed Akhmediev breather is discussed. -- Highlights: • Exact superposed AB solution in certain parameter conditions is obtained. • The controllable factors for AB are discussed. • The controllable behaviors for superposed AB are studied in PDAS and DDM.« less
  • We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system which arises in Bose-Einstein condensate: −ε{sup 2}Δu + a(x)u = μ{sub 1}u{sup 3} + βuv{sup 2} and −ε{sup 2}Δv + b(x)v = μ{sub 2}v{sup 3} + βu{sup 2}v in R{sup 3} with u, v > 0 and u(x), v(x) → 0 as |x| → ∞. Here, a, b are non-negative continuous potentials, and μ{sub 1}, μ{sub 2} > 0. We consider the case where the coupling constant β > 0 is relatively large. Then for sufficiently small ε > 0, we obtain positive solutions of this systemmore » which concentrate around local minima of the potentials as ε → 0. The novelty is that the potentials a and b may vanish at someplace and decay to 0 at infinity.« less