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Nonlinear localization in discrete and continuum systems: applications for optical waveguide arrays
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics. Linköping University, The Institute of Technology.
2005 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

An introcludion to the formation of coherent structures in spatially discrete and continuous systems is given. Of special interest is the phenomenon of nonlinear localization, where the self-focusing of an excitation is balanced by a dispersive process. This leads to the existence of localized waves (solitons) in continuous systems, and under very general conditions to intrinsic localized modes (discrete breathers) in systems of coupled anharmonic oscillators. Focus is set on nonlinear Schrödinger equations. A discrete equation, describing the propagation of the electric field in an array of coupled optical waveguides embedded in a material with a nonlinear index of refraction of the Kerr-type, is derived. The equation also describes the evolution of weakly coupled Bose-Einstein condensates in a periodic potential. The model contains nonlinear coupling terms and an effort is made to understand the novel features introduced by these terms as well as the nonlinear dispersion arising from taking the continuum limit of the discrete equation.

Important contributions in the papers are the discovery of inversion of stability between stationary excitations localized, respectively, on and in between sites in the lattice model for waveguide arrays, leading to an enhanced mobility of highly localized modes. As these can be controlled by simple perturbations. they may have an important applicat ion for optical multiport switching. The nonlinear coupling terms also lead to existence of discrete breathers with compact support and to a new type of stationary, complex, phase-twisted modes not previously reported. Of interest is also the possibility of controlling the magnitude and direction of the norm (Poynting power) current flowing across the waveguide array by simple non-symmetry-breaking perturbations. For the continuum equation, the nonlinear dispersion leads to the formation of exotic solitons, i.e., localized waves with discontinuous derivatives. The emergence of short-wavelength instabilities due to the simultaneous presence of nonlocal and nonlinea.r dispersion is also explained.

Place, publisher, year, edition, pages
Linköping: Linköpings universitet , 2005. , 56 p.
Series
Linköping Studies in Science and Technology. Thesis, ISSN 0280-7971 ; 1153
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:liu:diva-28393Local ID: 13529ISBN: 91-85297-79-8 (print)OAI: oai:DiVA.org:liu-28393DiVA: diva2:249199
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2013-11-27
List of papers
1. Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability
Open this publication in new window or tab >>Enhanced mobility of strongly localized modes in waveguide arrays by inversion of stability
2003 (English)In: Physical Review E, ISSN 1063-651X, Vol. 67, no 5, 056606-1--056606-8 p.Article in journal (Refereed) Published
Abstract [en]

A model equation governing the amplitude of the electric field in an array of coupled optical waveguides embedded in a material with Kerr nonlinearities is derived and explored. The equation is an extended discrete nonlinear Schrödinger equation with intersite nonlinearities. Attention is turned towards localized solutions and investigations are made from the viewpoint of the theory of discrete breathers (DBs). Stability analysis reveals an inversion of stability between stationary one-site and symmetric or antisymmetric two-site solutions connected to bifurcations with a pair of asymmetric intermediate DBs. The stability inversion leads to the existence of high-intensity narrow mobile solutions, which can propagate essentially radiationless. The direction and transverse velocity of the mobile solutions can be controlled by appropriate perturbations. Such solutions may have an important application for multiport switching, allowing unambiguous selection of output channel. The derived equation also supports compact DBs, which in some sense yield the best possible solutions for switching purposes.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14237 (URN)10.1103/PhysRevE.67.056606 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2014-01-13
2. Nonlocal and nonlinear dispersion in a nonlinear Schrödinger-type equation: exotic solitons and short-wavelength instabilities
Open this publication in new window or tab >>Nonlocal and nonlinear dispersion in a nonlinear Schrödinger-type equation: exotic solitons and short-wavelength instabilities
2004 (English)In: Physica D: Nonlinear Phenomena, ISSN 0167-2789, Vol. 198, no 1-2, 29-50 p.Article in journal (Refereed) Published
Abstract [en]

We study the continuum limit of a nonlinear Schrödinger lattice model with both on-site and inter-site nonlinearities, describing weakly coupled optical waveguides or Bose–Einstein condensates. The resulting continuum nonlinear Schrödinger-type equation includes both nonlocal and nonlinear dispersion. Looking for stationary solutions, the equation is reduced to an ordinary differential equation with a rescaled spectral parameter and a single parameter interpolating between the nonlocality and the nonlinear dispersion. It is seen that these two effects give a similar behaviour for the solutions. We find smooth solitons and, beyond a critical value of the spectral parameter, also nonanalytic solitons in the form of peakons and capons. The existence of the exotic solitons is connected to the special properties of the phase space of the equation. Stability is investigated numerically by calculating eigenvalues and eigenfunctions of the linearized problem, and we particularly find that with both nonlocal and nonlinear dispersion simultaneously present, all solutions are unstable with respect to a break-up into short-wavelength oscillations.

Keyword
Nonlinear Schrödinger, Nonlinear dispersion, Nonlocal dispersion, Exotic solitons, Modulational instability
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14238 (URN)10.1016/j.physd.2004.08.007 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2014-01-13
3. Phase twisted modes and current reversals in a lattice model of waveguide arrays with nonlinear coupling
Open this publication in new window or tab >>Phase twisted modes and current reversals in a lattice model of waveguide arrays with nonlinear coupling
2005 (English)In: Physical Review E. Statistical, Nonlinear, and Soft Matter Physics: Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics, ISSN 1063-651X, E-ISSN 1095-3787, Vol. 71, no 2, 025601-1--025601-4 p.Article in journal (Refereed) Published
Abstract [en]

We consider a lattice model for waveguide arrays embedded in nonlinear Kerr media. Inclusion of nonlinear coupling results in many phenomena involving complex, phase-twisted, stationary modes. The norm (Poynting power) current of stable plane-wave solutions can be controlled in magnitude and direction, and may be reversed without symmetry-breaking perturbations. Also stable localized phase-twisted modes with zero current exist, which for particular parameter values may be compact and expressed analytically. The model also describes coupled Bose-Einstein condensates.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14239 (URN)10.1103/PhysRevE.71.025601 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2017-12-13

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