# 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:

- Univ. of Bayreuth (Germany). Physikalisches Institut
- Santa Fe Inst. (SFI), Santa Fe, NM (United States); Los Alamos National Lab. (LANL), Los Alamos, NM (United States)
- Pontifical Catholic Univ. of Chile, Region Metropolitana (Chile). Dept. de Fisica
- Savitribai Phule Pune Univ., Pune (India)
- 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

- Alternate Identifier(s):
- OSTI ID: 1324864

- Report Number(s):
- LA-UR-16-23477

Journal ID: ISSN 2470-0045; TRN: US1800646

- 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. Thu .
"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 = {Thu Sep 15 00:00:00 EDT 2016},

month = {Thu Sep 15 00:00:00 EDT 2016}

}

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