DOE PAGES title logo U.S. Department of Energy
Office of Scientific and Technical Information

Title: Solitary waves in the nonlinear Dirac equation in the presence of external driving forces

Abstract

In this paper, we consider the nonlinear Dirac (NLD) equation in (1 + 1) dimensions with scalar–scalar self interaction g2/κ + 1 (Ψ¯Ψ)κ + 1 in the presence of external forces as well as damping of the form f(x) - iμγ0Ψ, where both f and Ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also affected, because the soliton position oscillates around a mean trajectory. For κ = 1 we solve explicitly the CC equations of the variational approximation for slow moving solitary waves in a constant external force without damping and find reasonable agreement with solving numerically the CC equations. Finally, we then compare the results of the variational approximation with no damping with numerical simulations of the NLD equation for κ = 1, when the components of the external force are of the form fj = rj exp(–iΚx) and again find agreement if we take into account a certain linear excitation with specific wavenumbermore » that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved.« less

Authors:
 [1];  [2];  [3];  [4];  [5];  [6]
  1. Univ. of Bayreuth (Germany). Inst. of Physics
  2. Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division. Center for Nonlinear Studies
  3. Univ. of Seville (Spain). Dept. of Applied Physics I. Inst. of Mathematics of the Univ. of Seville (IMUS)
  4. Peking Univ., Beijing (China). School of Mathematical Sciences. LMAM
  5. Savitribai Phule Pune Univ., Pune (India). Physics Dept.
  6. Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division. Center for Nonlinear Studies
Publication Date:
Research Org.:
Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
Sponsoring Org.:
USDOE; National Natural Science Foundation of China (NSFC); Alexander von Humboldt Foundation (Germany); Ministry of Science and Innovation (Spain); Junta de Andalucia (Spain); Univ. of Seville (Spain); Dept. of Atomic Energy (India)
Contributing Org.:
Univ. of Bayreuth (Germany); Santa Fe Inst. (SFI), Santa Fe, NM (United States); Univ. of Seville (Spain); Peking Univ., Beijing (China); Savitribai Phule Pune Univ., Pune (India)
OSTI Identifier:
1325634
Report Number(s):
LA-UR-15-21438
Journal ID: ISSN 1751-8113
Grant/Contract Number:  
11471025; 91330110; 11421101; SPA 1146358 STP; FIS2014-54497-P; FQM207; P06-FQM-01735; P09-FQM-4643
Resource Type:
Accepted Manuscript
Journal Name:
Journal of Physics. A, Mathematical and Theoretical
Additional Journal Information:
Journal Volume: 49; Journal Issue: 6; Journal ID: ISSN 1751-8113
Publisher:
IOP Publishing
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Atomic and Nuclear Physics; Mathematics

Citation Formats

Mertens, Franz G., Cooper, Fred, Quintero, Niurka R., Shao, Sihong, Khare, Avinash, and Saxena, Avadh. Solitary waves in the nonlinear Dirac equation in the presence of external driving forces. United States: N. p., 2016. Web. doi:10.1088/1751-8113/49/6/065402.
Mertens, Franz G., Cooper, Fred, Quintero, Niurka R., Shao, Sihong, Khare, Avinash, & Saxena, Avadh. Solitary waves in the nonlinear Dirac equation in the presence of external driving forces. United States. https://doi.org/10.1088/1751-8113/49/6/065402
Mertens, Franz G., Cooper, Fred, Quintero, Niurka R., Shao, Sihong, Khare, Avinash, and Saxena, Avadh. Tue . "Solitary waves in the nonlinear Dirac equation in the presence of external driving forces". United States. https://doi.org/10.1088/1751-8113/49/6/065402. https://www.osti.gov/servlets/purl/1325634.
@article{osti_1325634,
title = {Solitary waves in the nonlinear Dirac equation in the presence of external driving forces},
author = {Mertens, Franz G. and Cooper, Fred and Quintero, Niurka R. and Shao, Sihong and Khare, Avinash and Saxena, Avadh},
abstractNote = {In this paper, we consider the nonlinear Dirac (NLD) equation in (1 + 1) dimensions with scalar–scalar self interaction g2/κ + 1 (Ψ¯Ψ)κ + 1 in the presence of external forces as well as damping of the form f(x) - iμγ0Ψ, where both f and Ψ are two-component spinors. We develop an approximate variational approach using collective coordinates (CC) for studying the time dependent response of the solitary waves to these external forces. This approach predicts intrinsic oscillations of the solitary waves, i.e. the amplitude, width and phase all oscillate with the same frequency. The translational motion is also affected, because the soliton position oscillates around a mean trajectory. For κ = 1 we solve explicitly the CC equations of the variational approximation for slow moving solitary waves in a constant external force without damping and find reasonable agreement with solving numerically the CC equations. Finally, we then compare the results of the variational approximation with no damping with numerical simulations of the NLD equation for κ = 1, when the components of the external force are of the form fj = rj exp(–iΚx) and again find agreement if we take into account a certain linear excitation with specific wavenumber that is excited together with the intrinsic oscillations such that the momentum in a transformed NLD equation is conserved.},
doi = {10.1088/1751-8113/49/6/065402},
journal = {Journal of Physics. A, Mathematical and Theoretical},
number = 6,
volume = 49,
place = {United States},
year = {2016},
month = {1}
}

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

Citation Metrics:
Cited by: 4 works
Citation information provided by
Web of Science

Save / Share:

Works referenced in this record:

Nonlinear Spinor Fields
journal, July 1951


Nonlinear Spinor Field
journal, September 1956


Quantum Theory of Fields and Elementary Particles
journal, July 1957


Vibrations and Oscillatory Instabilities of Gap Solitons
journal, June 1998


Photonic analog of Zitterbewegung in binary waveguide arrays
journal, January 2010


Classical Simulation of Relativistic Zitterbewegung in Photonic Lattices
journal, September 2010


Optical analogue of relativistic Dirac solitons in binary waveguide arrays
journal, January 2014


The nonlinear Dirac equation in Bose–Einstein condensates: Foundation and symmetries
journal, July 2009


Formal Analogy between the Dirac Equation in Its Majorana Form and the Discrete-Velocity Version of the Boltzmann Kinetic Equation
journal, October 2013


Nonlinear Spinor Fields and Its Role in Cosmology
journal, January 2012


On a conform-invariant spinor wave equation
journal, May 1956


Classical, Stable, Nonlinear Spinor Field with Positive Rest Energy
journal, May 1970


Dynamical symmetry breaking in asymptotically free field theories
journal, November 1974


Baglike solutions of a Dirac equation with fractional nonlinearity
journal, June 1984


A soluble relativistic field theory
journal, January 1958


Instability of stationary states for nonlinear spinor models with quartic self-interaction
journal, June 1983


Nonlinear model of c -number confined Dirac quarks
journal, March 1984


Exact localized solutions of a family of two‐dimensional nonlinear spinor fields
journal, October 1986

  • Stubbe, Joachim
  • Journal of Mathematical Physics, Vol. 27, Issue 10
  • DOI: 10.1063/1.527325

Transparent potential for the one-dimensional Dirac equation
journal, April 1992


A Remark on the Coupling of Gravitation and Electron
journal, March 1950


Exact localized solutions of two-dimensional field theories of massive fermions with Fermi interactions
journal, October 1975


Chiral confinement: An exact solution of the massive Thirring model
journal, June 1975


Compact solitons, bags, and radial excitations
journal, December 1985


Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity
journal, September 2010


Multi-hump solitary waves of a nonlinear Dirac equation
journal, January 2015

  • Xu, Jian; Shao, Sihong; Tang, Huazhong
  • Communications in Mathematical Sciences, Vol. 13, Issue 5
  • DOI: 10.4310/CMS.2015.v13.n5.a7

Soliton solutions for Dirac equations with homogeneous non-linearity in (1+1) dimensions
journal, November 1985


Interaction dynamics for the solitary waves of a nonlinear Dirac model
journal, November 1981


Interaction for the solitary waves of a nonlinear Dirac model
journal, September 2005



journal, January 2006

  • Shao, Sihong; Tang, Huazhong
  • Discrete & Continuous Dynamical Systems - B, Vol. 6, Issue 3
  • DOI: 10.3934/dcdsb.2006.6.623

Numerical methods for nonlinear Dirac equation
journal, July 2013


Dirac soliton stability and interaction in binary waveguide arrays
journal, January 2014

  • Tran, Truong X.; Nguyen, Xuan N.; Duong, Dũng C.
  • Journal of the Optical Society of America B, Vol. 31, Issue 5
  • DOI: 10.1364/JOSAB.31.001132

Solitary waves in a discrete nonlinear Dirac equation
journal, January 2015

  • Cuevas-Maraver, Jesús; Kevrekidis, Panayotis G.; Saxena, Avadh
  • Journal of Physics A: Mathematical and Theoretical, Vol. 48, Issue 5
  • DOI: 10.1088/1751-8113/48/5/055204

Stability under dilations of nonlinear spinor fields
journal, July 1986


Stability of the minimum solitary wave of a nonlinear spinorial model
journal, July 1986


Stability of nonlinear spinor fields with application to the Gross-Neveu model
journal, October 1987


On spinor soliton stability
journal, September 1979


Energetic Stability Criterion for a Nonlinear Spinorial Model
journal, April 1983


Spinorial solitary wave dynamics of a (1+3)-dimensional model
journal, May 1985


Comments on Nonlinear Wave Equations as Models for Elementary Particles
journal, September 1964

  • Derrick, G. H.
  • Journal of Mathematical Physics, Vol. 5, Issue 9
  • DOI: 10.1063/1.1704233

Stability of solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity
journal, September 2014


On linear instability of solitary waves for the nonlinear Dirac equation
journal, May 2014

  • Comech, Andrew; Guan, Meijiao; Gustafson, Stephen
  • Annales de l'Institut Henri Poincare (C) Non Linear Analysis, Vol. 31, Issue 3
  • DOI: 10.1016/j.anihpc.2013.06.001

Nonlinear Dirac soliton in an external field
journal, March 1995

  • Nogami, Y.; Toyama, F. M.; Zhao, Z.
  • Journal of Physics A: Mathematical and General, Vol. 28, Issue 5
  • DOI: 10.1088/0305-4470/28/5/025

Behaviour of the nonlinear Dirac soliton in an external field of the Lorentz scalar type
journal, March 1998

  • Toyama, F. M.
  • Canadian Journal of Physics, Vol. 76, Issue 3
  • DOI: 10.1139/p97-054

Article
journal, June 1998

  • Toyama, F. M.; Nogami, Y.
  • Canadian Journal of Physics, Vol. 76, Issue 7
  • DOI: 10.1139/p98-035

Nonlinear Dirac equation solitary waves in external fields
journal, October 2012


Refined empirical stability criterion for nonlinear Schrödinger solitons under spatiotemporal forcing
journal, August 2011


Soliton stability criterion for generalized nonlinear Schrödinger equations
journal, January 2015


Nonlinear Schrödinger equation with spatiotemporal perturbations
journal, January 2010


Nonlinear Schrödinger solitons oscillate under a constant external force
journal, March 2013


Works referencing / citing this record:

Exact stationary solutions of the parametrically driven and damped nonlinear Dirac equation
journal, September 2019

  • Quintero, Niurka R.; Sánchez-Rey, Bernardo
  • Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 29, Issue 9
  • DOI: 10.1063/1.5115505

Dynamics of Dirac solitons in networks
journal, September 2018

  • Sabirov, K. K.; Babajanov, D. B.; Matrasulov, D. U.
  • Journal of Physics A: Mathematical and Theoretical, Vol. 51, Issue 43
  • DOI: 10.1088/1751-8121/aadfb0

Externally driven nonlinear Dirac equation revisited: theory and simulations
journal, March 2019

  • Quintero, Niurka R.; Shao, Sihong; Alvarez-Nodarse, Renato
  • Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 15
  • DOI: 10.1088/1751-8121/ab0dd9

Length-scale competition in the parametrically driven nonlinear Dirac equation with a spatially periodic force
journal, June 2019

  • Quintero, Niurka R.; Sánchez-Rey, Bernardo; Cooper, Fred
  • Journal of Physics A: Mathematical and Theoretical, Vol. 52, Issue 28
  • DOI: 10.1088/1751-8121/ab2469

Nonlinear spinor field in isotropic space-time and dark energy models
journal, July 2016