Marangell, R., Jones, C. K. R. T. (Christopher K. R. T.) and Susanto, H.. (2010) Localized standing waves in inhomogeneous Schrodinger equations. Nonlinearity, Vol.23 (No.9). pp. 2059-2080. ISSN 0951-7715

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Abstract

A nonlinear Schrodinger equation arising from light propagation down an inhomogeneous medium is considered. The inhomogeneity is reflected through a non-uniform coefficient of the nonlinear term in the equation. In particular, a combination of self-focusing and self-defocusing nonlinearity, with the self-defocusing region localized in a finite interval, is investigated. Using numerical computations, the extension of linear eigenmodes of the corresponding linearized system into nonlinear states is established, particularly nonlinear continuations of the fundamental state and the first excited state. The (in) stability of the states is also numerically calculated, from which it is obtained that symmetric nonlinear solutions become unstable beyond a critical threshold norm. Instability of the symmetric states is then investigated analytically through the application of a topological argument. Determination of instability of positive symmetric states is reduced to simple geometric properties of the composite phase plane orbit of the standing wave. Further the topological argument is applied to higher excited states and instability is again reduced to straightforward geometric calculations. For a relatively high norm, it is observed that asymmetric states bifurcate from the symmetric ones. The stability and instability of asymmetric states is also considered.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Divisions:	Faculty of Science > Mathematics
Journal or Publication Title:	Nonlinearity
Publisher:	Institute of Physics Publishing Ltd.
ISSN:	0951-7715
Official Date:	September 2010
Volume:	Vol.23
Number:	No.9
Number of Pages:	22
Page Range:	pp. 2059-2080
Identification Number:	10.1088/0951-7715/23/9/002
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	ONR, NSF
Grant number:	N000140910235, DMS-041026
URI:	http://wrap.warwick.ac.uk/id/eprint/5360

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