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

doi:10.1088/1751-8113/49/6/065402

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

Article Details
Other Related Research

In this paper, we consider the nonlinear Dirac (NLD) equation in (1 + 1) dimensions with scalar–scalar self interaction g ²/κ + 1 (Ψ¯Ψ) ^{κ + 1} in the presence of external forces as well as damping of the form f(x) - iμγ ⁰Ψ, 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 f _j = r _j exp(–iΚx) and again find agreement if we take into account a certainmore »

Authors:

Mertens, Franz G. ^[1] ; Cooper, Fred ^[2] ; Quintero, Niurka R. ^[3] ; Shao, Sihong ^[4] ; Khare, Avinash ^[5] ; Saxena, Avadh ^[6]

Univ. of Bayreuth (Germany). Inst. of Physics
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
Univ. of Seville (Spain). Dept. of Applied Physics I. Inst. of Mathematics of the Univ. of Seville (IMUS)
Peking Univ., Beijing (China). School of Mathematical Sciences. LMAM
Savitribai Phule Pune Univ., Pune (India). Physics Dept.
Los Alamos National Lab. (LANL), Los Alamos, NM (United States). Theoretical Division. Center for Nonlinear Studies

Publication Date:: 2016-01-05

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

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

Research Org:: Los Alamos National Lab. (LANL), Los Alamos, NM (United States)

Sponsoring Org:: USDOE; National Natural Science Foundation of China (NNSFC); 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 Orgs:: 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)

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Atomic and Nuclear Physics; Mathematics

OSTI Identifier:: 1325634


                    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.,  
        Web.  doi:10.1088/1751-8113/49/6/065402.

Copy to clipboard


                    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.  doi:10.1088/1751-8113/49/6/065402.

Copy to clipboard


                    Mertens, Franz G., Cooper, Fred, Quintero, Niurka R., Shao, Sihong, Khare, Avinash, and Saxena, Avadh. 2016.  
        "Solitary waves in the nonlinear Dirac equation in the presence of external driving forces".  United States.  
        doi:10.1088/1751-8113/49/6/065402.  https://www.osti.gov/servlets/purl/1325634.

Copy to clipboard


                    
@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}

}

Copy to clipboard

Free Publicly Accessible Full Text
Accepted Manuscript (DOE)
Publisher's Version of Record
10.1088/1751-8113/49/6/065402
https://doi.org/10.1088/1751-8113/49/6/065402

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

Save / Share this Record
Export Metadata
- Endnote
- RIS
- Excel
- CSV
- XML
Save to My Library
Send to Email

Send to Email

Email address:

Content:

Similar Records

Similar records in DOE PAGES and OSTI.GOV collections:

Erratum: Nonlinear Dirac equation solitary waves in external fields [Phys. Rev. E 86, 046602 (2012)]

Journal Article Mertens, Franz G. ; Quintero, Niurka R. ; Cooper, Fred ; ... May 2016 - Physical Review E

In Sec. IV of our original paper, we assumed a particular conservation law Eq. (4.6), which was true in the absence of external potentials, to derive some particular potentials for which we obtained solutions to the nonlinear Dirac equation (NLDE). Because the conservation law of Eq. (4.6) for the component T ¹¹ of the energy-momentum tensor is not true in the presence of these external potentials, the solutions we found do not satisfy the NLDEs in the presence of these potentials. Thus all the equations from Eq. (4.6) through Eq. (4.44) are not correct, since the exact solutions that followedmore »« less
Full Text Available
Solitary Waves of a $$\mathcal {P}$$ $$\mathcal {T}$$-Symmetric Nonlinear Dirac Equation

Journal Article Cuevas-Maraver, Jesus ; Kevrekidis, Panayotis G. ; Saxena, Avadh ; ... October 2015 - IEEE Journal of Selected Topics in Quantum Electronics

In our study we consider we consider a prototypical example of a mathcalP mathcalT-symmetric Dirac model. We discuss the underlying linear limit of the model and identify the threshold of the mathcalP mathcalT -phase transition in an analytical form. We then focus on the examination of the nonlinear model. We consider the continuation in the mathcalP mathcalT -symmetric model of the solutions of the corresponding Hamiltonian model and find that the solutions can be continued robustly as stable ones all the way up to the mathcalP mathcalT-transition threshold. In the latter, they degenerate into linear waves. We also examine themore »« less
Cited by 2Full Text Available
Erratum: Nonlinear Dirac equation solitary waves in external fields [Phys. Rev. E 86 , 046602 (2012)]

Journal Article Mertens, Franz G. ; Quintero, Niurka R. ; Cooper, Fred ; ... May 2016 - Physical Review E

Full Text Available
Variational approach to studying solitary waves in the nonlinear Schrödinger equation with complex potentials

Journal Article Mertens, Franz G. ; Cooper, Fred ; Arevalo, Edward ; ... September 2016 - Physical Review E

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 themore »« less
Cited by 2Full Text Available
Solitary waves in the nonlinear Dirac equation with arbitrary nonlinearity

Journal Article - in OSTI.gov collection Cooper, Fred ; Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 ; Khare, Avinash ; ... September 2010 - Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics (Print)

We consider the nonlinear Dirac equations (NLDE's) in 1+1 dimension with scalar-scalar self interaction (g{sup 2}/k+1)({Psi}{Psi}){sup k+1}, as well as a vector-vector self interaction (g{sup 2}/k+1)({Psi}{gamma}{sub {mu}}{Psi}{Psi}{gamma}{sup {mu}}{Psi})1/2(k+1). We find the exact analytic form for solitary waves for arbitrary k and find that they are a generalization of the exact solutions for the nonlinear Schroedinger equation (NLSE) and reduce to these solutions in a well defined nonrelativistic limit. We perform the nonrelativistic reduction and find the 1/2m correction to the NLSE, valid when |{omega}-m|<<2m, where {omega} is the frequency of the solitary wave in the rest frame. We discuss themore »« less
Full Text Available

Access journal articles and accepted manuscripts resulting from DOE research funding

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

Access journal articles and accepted manuscripts
resulting from DOE research funding