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Nonlocal and nonlinear dispersion in a nonlinear Schrödinger-type equation: exotic solitons and short-wavelength instabilities
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
Informatics and Mathematical Modelling and Department of Physics, The Technical University of Denmark, Lyngby, Denmark.
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.ORCID iD: 0000-0001-6708-1560
Informatics and Mathematical Modelling and Department of Physics, The Technical University of Denmark, Lyngby, Denmark.
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.

Place, publisher, year, edition, pages
2004. Vol. 198, no 1-2, 29-50 p.
Keyword [en]
Nonlinear Schrödinger, Nonlinear dispersion, Nonlocal dispersion, Exotic solitons, Modulational instability
National Category
Natural Sciences
Identifiers
URN: urn:nbn:se:liu:diva-14238DOI: 10.1016/j.physd.2004.08.007OAI: oai:DiVA.org:liu-14238DiVA: diva2:22967
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2014-01-13
In thesis
1. Stability and Mobility of Localized and Extended Excitations in Nonlinear Schrödinger Models
Open this publication in new window or tab >>Stability and Mobility of Localized and Extended Excitations in Nonlinear Schrödinger Models
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is mainly concerned with the properties of some discrete nonlinear Schrödinger equations. These naturally arise in many different physical contexts as the limiting form of general dynamical lattice equations that incorporate nonlinearity and coupling. Interest is focused on theoretical models of coupled optical waveguides constructed from materials with a nonlinear index of refraction. In arrays of waveguides the overlap of the evanescent electric field of the modes in neighbouring waveguides provides a coupling and the nonlinearity of the material provides a mechanism to halt the discrete diffraction that otherwise would spread localized energy across the array. In particular, waveguide structures where also a nonlinear coupling is taken into account are studied. It is noted that the equation for the evolution of the complex amplitudes of the electric field along an array of waveguides also can be used to describe the dynamics of Bose-Einstein condensates trapped in a periodic optical potential. Possible excitations in arrays in both one and two dimensions are considered, with emphasis on the effects of the nonlinear coupling.

Localized excitations are considered from the viewpoint of the theory of discrete breathers, or intrinsic localized modes, i.e., solutions of the dynamical equations that are periodic in time and have a spatial localization. The general theory of such solutions, that appear under very general circumstances in nonlinear lattice equations, is reviewed. In an array of waveguides this means that light can propagate along the array confined essentially to one or a few waveguides. In general a distinction is made between excitations that are centred on a waveguide, or site in the lattice, and excitations that are centred inbetween waveguides. Usually only the former give stable propagation. When the localized beam can be displaced to neighbouring waveguides the array can operate as an optical switch. With the inclusion of nonlinear coupling between the sites, as in the model derived in this thesis, the stability of the site-centred and bond-centred solutions can be exchanged. It is shown how this leads to the existence of highly localized mobile solutions that can propagate transversely in the one-dimensional array of waveguides. The inversion of stability of stationary solutions occurs also in the two-dimensional array, but in this setting it fails to give good mobility of localized excitations. The reason for this is also explained.

In a two-dimensional lattice a discrete breather can have the form of a vortex. This means that the phase of the complex amplitude will vary on a contour around the excitation, such that the phase is increased by 2πS, where S is the topological charge, on the completion of one turn. Some ring-like vortex excitations are considered and in particular a stable vortex with S=2 is found. It is also noted that the effect of charge flipping, i.e., when the topological charge periodically changes between -S and S, is connected to the existence of quasiperiodic solutions.

The nonlinear coupling of the waveguide model will also give rise to some more exotic and novel properties of localized solutions, e.g., discrete breathers with a nontrivial phase. When the linear coupling and the nonlinear coupling have opposite signs, there can be a decoupling in the lattice that allows for compact solutions. These localized excitations will have no decaying tail. Of interest is also the flexibility in controlling the transport of power across the array when it is excited with a nonlinear plane wave. It is shown how a change of the amplitude of a plane wave can affect the magnitude and direction of power flow in the array.

Also the continuum limit of the one-dimensional discrete waveguide model is considered with an equation incorporating both nonlocal and nonlinear dispersion. In general continuum equations the balance between nonlinearity and dispersion can lead to the formation of localized travelling waves, or solitons. With nonlinear dispersion it is seen that these solitons can be nonanalytic and have discontinuous spatial derivatives. The emergence of short-wavelength instabilities due to the simultaneous presence of nonlocal and nonlinear dispersion is also explained.

Place, publisher, year, edition, pages
Institutionen för fysik, kemi och biologi, 2007
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1072
National Category
Other Physics Topics
Identifiers
urn:nbn:se:liu:diva-8091 (URN)978-91-85715-83-1 (ISBN)
Public defence
2007-02-16, Planck, Fysikhuset, 10:15 (English)
Opponent
Supervisors
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2014-01-13
2. Nonlinear localization in discrete and continuum systems: applications for optical waveguide arrays
Open this publication in new window or tab >>Nonlinear localization in discrete and continuum systems: applications for optical waveguide arrays
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:nbn:se:liu:diva-28393 (URN)13529 (Local ID)91-85297-79-8 (ISBN)13529 (Archive number)13529 (OAI)
Available from: 2009-10-09 Created: 2009-10-09 Last updated: 2013-11-27

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Öster, MichaelJohansson, Magnus

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