Modulational instability for a logarithmic nonlinear Schrödinger equation
type: Item
instance of: scholarly article
title: Modulational instability for a logarithmic nonlinear Schrödinger equation
zbMATH Open document ID: 1358.35147
author: Takuya Yamano
published in: Applied Mathematics Letters
publication date: 2015-09-01T00:00:00Z
review text: The author considers the extended logarithmic nonlinear Schrödinger equation (NLSE) in one-dimensional case for the free particle Hamiltonian of mass \(m\): \[ i\hbar\left[\dfrac{\partial}{\partial t}+\kappa(\ln |\psi(x,t)|^2-\langle \ln |\psi(x,t)|^2\rangle)-\dfrac{\Gamma}{2}\left(\ln\dfrac{\psi(x,t)}{\overline{\psi}(x,t)}-\left\langle\ln\dfrac{\psi(x,t)}{\overline{\psi}(x,t)}\right\rangle\right) \right]\psi(x,t)=H\psi(x,t) \] Here \(\kappa\in \mathbb R\) and \(\Gamma\in \mathbb R\) are coupling strengths, the brackets \(\langle\rangle\) denote the mean with \(|\psi(x,t)|^2\) over the entire domain. Assuming the perturbations being carrier wave forms with a wave number \(\delta\) and an angular frequency \(\nu\), the author shows that the modulational instability arises when \(\nu=\dfrac{\hbar k}{m}\delta\pm \sqrt{\left(\dfrac{\hbar \delta}{2m}\right)^2-\Gamma^2}\) is a complex number. It means that the perturbation grows exponentially. If \(\delta^2\approx 0\), then the growth rate \(\Gamma\sqrt{1-\left(\dfrac{\hbar}{2m\Gamma}\right)^2\delta^2}\) approximately equals to the friction coefficient \(\Gamma\). The instability increases faster at lower frequencies of perturbation. It is indicated that the modulational instability feature of the present logarithmic NLSE differs from the cubic one, \(i\psi_t+p\psi_{xx}+q|\psi|^2\psi=0\). The presented situation is similar to the polarization instability of a light pulse propagating.
Mathematics Subject Classification ID: 35Q41
Mathematics Subject Classification ID: 81P15
Mathematics Subject Classification ID: 81Q05
Mathematics Subject Classification ID: 35Q40
zbMATH DE Number: 6477334
zbMATH Keywords: modulational instability
zbMATH Keywords: logarithmic nonlinear Schrödinger equation
zbMATH Keywords: plane wave
reviewed by: Artyom N. Andronov
MaRDI profile type: Publication
full work available at URL: j.aml.2015.03.020
OpenAlex ID: W2076860865
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DOI: 10.1016/J.AML.2015.03.020
description: scientific article; zbMATH DE number 6477334