## 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 g ^{2}/κ + 1 (Ψ¯Ψ) ^{κ + 1} in the presence of external forces as well as damping of the form f(x) - iμγ ^{0}Ψ, 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:

- 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:

- 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 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. doi: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. doi: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}

}

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