Auswahl der wissenschaftlichen Literatur zum Thema „1D-NLSE“

We consider an extended model of the model considered before with double-square potential, namely one-dimensional (1D) nonlinear Schrödinger equation (NLSE) with self-focusing nonlinearity and a 1D double-gauss potential. Spontaneous symmetry breaking has been presented in terms of the control parameter which is propagation constant in the case of optics and chemical potential in the of Bose-Einstein Condensate (BEC), correspondingly. The numerical simulations predict a bifurcation breaking the symmetry of 1D trapped in the double-gauss potential of the supercritical type as in the case of double-square potential. Furthermore we have constructed bifurcation diagrams considering the stability of solitons with three methods: the method using Vakhitov–Kolokolov (V-K) Stability Criterion, Pseudospectral Method and Method for Linear-Stability Eigenvalues. It will be shown that for our model the results obtained are the same for these three methods but the third one is the fastest.

In this paper, four compelling numerical approaches, namely, the split-step Fourier transform (SSFT), Fourier pseudospectral method (FPSM), Crank-Nicolson method (CNM), and Hopscotch method (HSM), are exhaustively presented for solving the 1D nonlinear Schrodinger equation (NLSE). The significance of this equation is referred to its notable contribution in modeling wave propagation in a plethora of crucial real-life applications such as the fiber optics field. Although exact solutions can be obtained to solve this equation, these solutions are extremely insufficient because of their limitations to only a unique structure under some limited initial conditions. Therefore, seeking high-performance numerical techniques to manipulate this well-known equation is our fundamental purpose in this study. In this regard, extensive comparisons of the proposed numerical approaches, against the exact solution, are conducted to investigate the benefits of each of them along with their drawbacks, targeting a broad range of temporal and spatial values. Based on the obtained numerical simulations via MATLAB, we extrapolated that the SSFT invariably exhibits the topmost robust potentiality for solving this equation. However, the other suggested schemes are substantiated to be consistently accurate, but they might generate higher errors or even consume more processing time under certain conditions.

AbstractWe solve the Nonlinear Schrödinger Equation (NLSE) in 1D in presence of one, two and several Dirac delta potentials. With the help of an equivalent central force problem we obtain the analytical solutions in terms of a biparametric family containing the Jacobi functions. Elliptic Jacobi functions are already reported in the literature but they have not been used in the context of a scattering problem under causal boundary conditions. In the simplest examples of one or two Dirac deltas we analyze how the nonlinear term of the equation affects the modulus and phase profiles of the wave function. We also study the transmission curves under the nonlinear modification of the tunneling behavior for the first time. For a Fabri-Perot configuration made of two deltas, we obtain the effect of nonlinear coupling in the positions of the local maxima (resonances). We lay the foundations for nonlinear Anderson localization of 1D BECs in a speckle field. Upon redefinition of parameters these novel results describe the dynamics of a stationary Higgs field in 1D. Finally, we discuss the conditions for soliton formation under the influence of a Dirac comb potential, giving rise to fully correlated locations and intensities of the defects.

L'étude de l'optique non-linéaire est d'importance pratique car cela désigne des systèmes tels que les fibres optiques et des cristaux liquides mais aussi d'importance théorique car la lumière non-linéaire possède des propriétés très similaires à l'hydrodynamique. Les systèmes optiques non-linéaires sont modélisés par une équation non-intégrable qui contient une physique riche. Dans cette thèse, nous explorons deux aspects de cette équation. Nous analysons d'abord la propagation de structures localisées dans ce système et nous concluons que ce dernier tend vers un état final qui agit comme un attracteur. Nous l'identifions à un état lié, une structure localisée qui oscille en amplitude et en largeur et qui se propage au milieu d'ondes faiblement non-linéaires. Cet état-lié est caractérisé avec différents outils dont le spectre non-linéaire. Ce spectre non-linéaire a été élaboré pour le système intégrable correspondant à notre système non-intégrable, et apparaît comme un outil pertinent pour caractériser notre système.Nous étudions aussi les cascades turbulentes de ce système non-intégrable avec l'aide d'un modèle réduit. Ce modèle permet une étude théorique en simplifiant les interactions entre modes qui sont responsables des cascades turbulentes. Avec ce modèle, nous pouvons déterminer les spectres de Kolmogorov-Zakharov des invariants qui cascadent. Pour l'un de ces invariants, l'action d'onde, la prédiction de Kolmogorov-Zakarhov s'avère être non réalisable à cause de flux divergents. Un modèle non-local permet d'obtenir un nouveau spectre. Ces prédictions théoriques sont confrontées à des simulations numériques. Les spectres obtenus numériquement confirment globalement ces prédictions théoriques. Lors de ces simulations, nous avons observé des solitons incohérents, qui sont des structures pouvant se propager dans des systèmes optiques non-intégrables avec la particularité d'avoir une enveloppe globale constante malgré des changements à échelle plus petite. Ces structures ont été observées en coexistence avec des cascades turbulentes
Studying non-linear optics systems is of practical importance because it applies to systems such as optical fibers and liquid crystals but also of theoretical importance because non-linear light exhibit properties very similar to hydrodynamics. Non-linear optics are modeled by an non-integrable equation which contains a rich physics. In this thesis, we explore two aspects of this equation. We first analyse the propagation of localized structures in this system and we conclude that the system tends to a final state which acts as a statistical attractor. We identify this attractor as a bound-state, a localized structure which oscillates in amplitude and in width and which propagates among weakly non-linear waves.We also study the turbulent cascades of this system with the help of an reduced model of the wave kinetics. This reduced model allows us to derive the Kolmogorov-Zakharov spectra of cascading quantities. The Kolmogorov-Zakharov spectrum for the wave-action is found to be non-local and replaced by a no-local prediction. These theoretical predictions are then compared to numerical simulations and show an overall good accordance with numerics, particularly for the non-local spectrum of wave-action. Such numerical simulations show the existence of Incoherent Solitons, which are localized structures propagating with an envelope approximately constant but with propagation of smaller structures inside it. Incoherent Solitons have been found in coexistence with cascade, but for different directions in the Fourier space

Since Zacharov and Shabat integrated one-dimensional Nonlinear Schrödinger Equation (1D NLSE) and found solitons [1,2], much have been done in one-dimensional, but very little in two-dimensional soliton physics. The integrability of 2D NLSE as well as existence of solitons in it is under the question to the time.

Silberberg1 has recently shown that, in a homogeneous nonlinear Kerr material exhibiting anomalous group-velocity dispersion (AGVD), the propagation of the slowly-varying envelope of the electric-field can be described by a 3+1D nonlinear Schrodinger equation (NLSE): which is written in a group-velocity coordinate frame. The AGVD has been used to make temporal dispersion isomorphic to spatial diffraction which in turn gives rise to the possibility of simultaneous two-dimensional, radially symmetric self-focusing and temporal pulse compression resulting in a 3D soliton or “light-bullet”. This light-bullet is fully confined by nonlinear effects alone but exhibits the behavior of both temporal solitons in fibers and spatial solitons in slab waveguides.