# Approximate analytic solutions to coupled nonlinear Dirac equations

## Abstract

Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g _{1} ^{2}/2($$\bar{ψ}$$ψ) ^{2} + g _{2} ^{2}/2($$\bar{Φ}$$Φ) ^{2} + g ^{2} _{3}($$\bar{ψ}$$ψ)($$\bar{Φ}$$Φ) as well as vector–vector interactions g _{1} ^{2}/2($$\bar{ψ}$$γμψ)($$\bar{ψ}$$γμψ) + g ^{2} _{2}/2($$\bar{Φ}$$γμΦ)($$\bar{Φ}$$γμΦ) + g ^{2} _{3}($$\bar{ψ}$$γμψ)($$\bar{Φ}$$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e ^{–iω1t}R _{1}cosθ,R _{1}sinθΦ=e ^{–iω2t}R _{2}cosη,R _{2}sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g _{3} ^{2}/g ^{2} _{2} and g _{3} ^{2}/g _{1} ^{2}. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.

- Authors:

- 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:
- 1342859

- Alternate Identifier(s):
- OSTI ID: 1412556

- Report Number(s):
- LA-UR-16-21471

Journal ID: ISSN 0375-9601; TRN: US1700910

- Grant/Contract Number:
- AC52-06NA25396

- Resource Type:
- Journal Article: Accepted Manuscript

- Journal Name:
- Physics Letters. A

- Additional Journal Information:
- Journal Volume: 381; Journal Issue: 12; Journal ID: ISSN 0375-9601

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

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

### Citation Formats

```
Khare, Avinash, Cooper, Fred, and Saxena, Avadh.
```*Approximate analytic solutions to coupled nonlinear Dirac equations*. United States: N. p., 2017.
Web. doi:10.1016/j.physleta.2017.01.018.

```
Khare, Avinash, Cooper, Fred, & Saxena, Avadh.
```*Approximate analytic solutions to coupled nonlinear Dirac equations*. United States. doi:10.1016/j.physleta.2017.01.018.

```
Khare, Avinash, Cooper, Fred, and Saxena, Avadh. Mon .
"Approximate analytic solutions to coupled nonlinear Dirac equations". United States.
doi:10.1016/j.physleta.2017.01.018. https://www.osti.gov/servlets/purl/1342859.
```

```
@article{osti_1342859,
```

title = {Approximate analytic solutions to coupled nonlinear Dirac equations},

author = {Khare, Avinash and Cooper, Fred and Saxena, Avadh},

abstractNote = {Here, we consider the coupled nonlinear Dirac equations (NLDEs) in 1+11+1 dimensions with scalar–scalar self-interactions g12/2($\bar{ψ}$ψ)2 + g22/2($\bar{Φ}$Φ)2 + g23($\bar{ψ}$ψ)($\bar{Φ}$Φ) as well as vector–vector interactions g12/2($\bar{ψ}$γμψ)($\bar{ψ}$γμψ) + g22/2($\bar{Φ}$γμΦ)($\bar{Φ}$γμΦ) + g23($\bar{ψ}$γμψ)($\bar{Φ}$γμΦ). Writing the two components of the assumed rest frame solution of the coupled NLDE equations in the form ψ=e–iω1tR1cosθ,R1sinθΦ=e–iω2tR2cosη,R2sinη, and assuming that θ(x),η(x) have the same functional form they had when g3 = 0, which is an approximation consistent with the conservation laws, we then find approximate analytic solutions for Ri(x) which are valid for small values of g32/g22 and g32/g12. In the nonrelativistic limit we show that both of these coupled models go over to the same coupled nonlinear Schrödinger equation for which we obtain two exact pulse solutions vanishing at x → ±∞.},

doi = {10.1016/j.physleta.2017.01.018},

journal = {Physics Letters. A},

number = 12,

volume = 381,

place = {United States},

year = {Mon Jan 30 00:00:00 EST 2017},

month = {Mon Jan 30 00:00:00 EST 2017}

}

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