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Stability and Mobility of Localized and Extended Excitations in Nonlinear Schrödinger Models
Linköping University, Department of Physics, Chemistry and Biology, Theoretical Physics . Linköping University, The Institute of Technology.
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: urn:nbn:se:liu:diva-8091ISBN: 978-91-85715-83-1 (print)OAI: oai:DiVA.org:liu-8091DiVA: diva2:22971
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
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
4. Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2
Open this publication in new window or tab >>Stable stationary and quasiperiodic discrete vortex breathers with topological charge S=2
2006 (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. 73, no 6, 066608-1--066608-6 p.Article in journal (Refereed) Published
Abstract [en]

We demonstrate the stability of a stationary vortex breather with vorticity S=2 in the two-dimensional discrete nonlinear Schrödinger model for a square lattice and also discuss the effects of exciting internal sites in a vortex ring. We also point out the fundamental difficulties of observing these solutions with current experimental techniques. Instead, we argue that relevant initial conditions will lead to the formation of quasiperiodic vortex breathers.

National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-14240 (URN)10.1103/PhysRevE.73.066608 (DOI)
Available from: 2007-01-26 Created: 2007-01-26 Last updated: 2017-12-13
5. Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling
Open this publication in new window or tab >>Stability, mobility and power currents in a two-dimensional model for waveguide arrays with nonlinear coupling
2009 (English)In: Physica D: Non-linear phenomena, ISSN 0167-2789, E-ISSN 1872-8022, Vol. 238, no 1, 88-99 p.Article in journal (Refereed) Published
Abstract [en]

A two-dimensional nonlinear Schrodinger lattice with nonlinear coupling, modelling a square array of weakly coupled linear optical waveguides embedded in a nonlinear Kerr material, is studied. We find that despite a vanishing energy difference (Peierls-Nabarro barrier) of fundamental stationary modes the mobility of localized excitations is very poor. This is attributed to a large separation in parameter space of the bifurcation points of the involved stationary modes. At these points the stability of the fundamental modes is changed and an asymmetric intermediate solution appears that connects the points. The control of the power flow across the array when excited with plane waves is also addressed and shown to exhibit great flexibility that may lead to applications for power-coupling devices. In certain parameter regimes, the direction of a stable propagating plane-wave current is shown to be continuously tunable by amplitude variation (with fixed phase gradient). More exotic effects of the nonlinear coupling terms like compact discrete breathers and vortices, and stationary complex modes with nontrivial phase relations are also briefly discussed. Regimes of dynamical linear stability are found for all these types of solutions.

Keyword
Nonlinear coupling, Peierls-Nabarro potential, Mobility, Inversion of stability, Modulational instability, Power currents
National Category
Natural Sciences
Identifiers
urn:nbn:se:liu:diva-16345 (URN)10.1016/j.physd.2008.08.006 (DOI)
Available from: 2009-01-16 Created: 2009-01-16 Last updated: 2017-12-14

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