Asymmetric bound states of differential equations in nonlinear optics
type: Item
instance of: scholarly article
title: Asymmetric bound states of differential equations in nonlinear optics
zbMATH Open document ID: 0922.34020
author: A. Ambrosetti
author: David Arcoya
author: José Luis Gámez
published in: Rendiconti del Seminario Matematico della Università di Padova
publication date: 1999-10-14T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/108459
review text: Consider the boundary value problem \[ -u''(x)+\beta^2 u(x)=n^2 \bigl(x,u^2(x)\bigr) u(x),x\in \mathbb{R},\lim_{| x|\to+\infty} u(x)= \lim_{ | x|\to +\infty}u'(x)=0. \tag{1} \] It is known that at certain value \(\beta_0\) of \(\beta\) a family of asymmetric solutions bifurcates from the branch of symmetric ones. The authors investigate the same phenomenon for a class of equations (1) which cannot be integrated directly. The nonlinearity \(n^2\) is taken of the form \[ n^2(x,s)=q^2+ c^2h(x/ \varepsilon)+s-\alpha(x/ \varepsilon,s) \] where \(\varepsilon\) is a small positive parameter; \(q,c \in \mathbb{R}\), \(h\) and \(\alpha\) are given functions. The authors use a variational method related to the Poincaré-Melnikov theory of homoclinics.
reviewed by: Abdelkader Boucherif
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 37G99
Mathematics Subject Classification ID: 34L40
Mathematics Subject Classification ID: 37C75
Mathematics Subject Classification ID: 35Q60
zbMATH DE Number: 1279137
zbMATH Keywords: Schrödinger equation
zbMATH Keywords: bifurcation
zbMATH Keywords: nonlinear optics
zbMATH Keywords: asymmetric solutions
zbMATH Keywords: Poincaré-Melnikov theory
zbMATH Keywords: homoclinics
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1279137