### Abstract

We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrödinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrödinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley.

Original language | English (US) |
---|---|

Pages (from-to) | 1077-1115 |

Number of pages | 39 |

Journal | Journal of Statistical Physics |

Volume | 88 |

Issue number | 5-6 |

State | Published - Sep 1997 |

### Fingerprint

### Keywords

- Nonlinear schrödinger equation
- Random media

### ASJC Scopus subject areas

- Mathematical Physics
- Physics and Astronomy(all)
- Statistical and Nonlinear Physics

### Cite this

*Journal of Statistical Physics*,

*88*(5-6), 1077-1115.

**On the stability of time-harmonic localized states in a disordered nonlinear medium.** / Bronski, Jared C.; McLaughlin, David W.; Shelley, Michael J.

Research output: Contribution to journal › Article

*Journal of Statistical Physics*, vol. 88, no. 5-6, pp. 1077-1115.

}

TY - JOUR

T1 - On the stability of time-harmonic localized states in a disordered nonlinear medium

AU - Bronski, Jared C.

AU - McLaughlin, David W.

AU - Shelley, Michael J.

PY - 1997/9

Y1 - 1997/9

N2 - We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrödinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrödinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley.

AB - We study the problem of localization in a disordered one-dimensional nonlinear medium modeled by the nonlinear Schrödinger equation. Devillard and Souillard have shown that almost every time-harmonic solution of this random PDE exhibits localization. We consider the temporal stability of such time-harmonic solutions and derive bounds on the location of any unstable eigenvalues. By direct numerical determination of the eigenvalues we show that these time-harmonic solutions are typically unstable, and find the distribution of eigenvalues in the complex plane. The distributions are distinctly different for focusing and defocusing nonlinearities. We argue further that these instabilities are connected with resonances in a Schrödinger problem, and interpret the earlier numerical simulations of Caputo, Newell, and Shelley, and of Shelley in terms of these instabilities. Finally, in the defocusing case we are able to construct a family of asymptotic solutions which includes the stable limiting time-harmonic state observed in the simulations of Shelley.

KW - Nonlinear schrödinger equation

KW - Random media

UR - http://www.scopus.com/inward/record.url?scp=0040513014&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040513014&partnerID=8YFLogxK

M3 - Article

VL - 88

SP - 1077

EP - 1115

JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

SN - 0022-4715

IS - 5-6

ER -