# Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity

## Abstract

We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.

- Authors:

- National Research University of Electronic Technology MIET, Zelenograd, Moscow 124498 (Russian Federation)
- Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978 (Israel)
- (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 22596609

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Chaos (Woodbury, N. Y.); Journal Volume: 26; Journal Issue: 7; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; 97 MATHEMATICAL METHODS AND COMPUTING; BOSE-EINSTEIN CONDENSATION; DIPOLES; NONLINEAR OPTICS; NONLINEAR PROBLEMS; ONE-DIMENSIONAL CALCULATIONS; PERIODICITY; SCHROEDINGER EQUATION

### Citation Formats

```
Lebedev, M. E., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Alfimov, G. L., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Malomed, Boris A., E-mail: malomed@post.tau.ac.il, and Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101.
```*Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity*. United States: N. p., 2016.
Web. doi:10.1063/1.4958710.

```
Lebedev, M. E., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Alfimov, G. L., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Malomed, Boris A., E-mail: malomed@post.tau.ac.il, & Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101.
```*Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity*. United States. doi:10.1063/1.4958710.

```
Lebedev, M. E., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Alfimov, G. L., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com, Malomed, Boris A., E-mail: malomed@post.tau.ac.il, and Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101. Fri .
"Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity". United States.
doi:10.1063/1.4958710.
```

```
@article{osti_22596609,
```

title = {Stable dipole solitons and soliton complexes in the nonlinear Schrödinger equation with periodically modulated nonlinearity},

author = {Lebedev, M. E., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com and Alfimov, G. L., E-mail: gloriouslair@gmail.com, E-mail: galfimov@yahoo.com and Malomed, Boris A., E-mail: malomed@post.tau.ac.il and Laboratory of Nonlinear-Optical Informatics, ITMO University, St. Petersburg 197101},

abstractNote = {We develop a general classification of the infinite number of families of solitons and soliton complexes in the one-dimensional Gross-Pitaevskii/nonlinear Schrödinger equation with a nonlinear lattice pseudopotential, i.e., periodically modulated coefficient in front of the cubic term, which takes both positive and negative local values. This model finds direct implementations in atomic Bose-Einstein condensates and nonlinear optics. The most essential finding is the existence of two branches of dipole solitons (DSs), which feature an antisymmetric shape, being essentially squeezed into a single cell of the nonlinear lattice. This soliton species was not previously considered in nonlinear lattices. We demonstrate that one branch of the DS family (namely, which obeys the Vakhitov-Kolokolov criterion) is stable, while unstable DSs spontaneously transform into stable fundamental solitons (FSs). The results are obtained in numerical and approximate analytical forms, the latter based on the variational approximation. Some stable bound states of FSs are found too.},

doi = {10.1063/1.4958710},

journal = {Chaos (Woodbury, N. Y.)},

number = 7,

volume = 26,

place = {United States},

year = {Fri Jul 15 00:00:00 EDT 2016},

month = {Fri Jul 15 00:00:00 EDT 2016}

}