No description available.

No description available.

Les milieux Kerr soumis à des faisceaux optiques contrapropageants provenant, soit d'une cavité Pérot-Fabry plan-plan ou d'une boucle de rétro-action, ont été très étudiés pour la génération de structures transverses. De nombreux développements théoriques dans la configuration de la cavité ont prédit l'existence de structures localisées (solitons). Cependant aucune mise en évidence expérimentale de ces structures n'a été réalisée à ce jour. Dans la configuration de la boucle de rétro-action optique, les régimes fortement non linéaires présentent des structures rares et intenses, très localisées aussi bien spatialement que temporellement. C'est dans ce cadre que nous nous intéressons expérimentalement à la structuration localisée de la lumière dans ces deux systèmes. La première partie de cette thèse est consacrée à la mise en évidence expérimentale de solitons spatiaux dans une cavité Pérot-Fabry plan-plan Kerr à diffraction positive. Ensuite, nous étudions cette cavité quand elle est soumise à de la diffraction négative. Des parois de domaines propagatives sont alors observées. Celles-ci sont bloquées par le forçage spatial lié à l'inhomogénéité du profil de pompage optique gaussien qui donne lieu à une localisation de la lumière. Nous étudions succinctement le cas limite d'une cavité sans diffraction dans laquelle les parois de domaines survivent. Pour finir nous effectuons une étude du régime très fortement non linéaire dans le dispositif de rétro-action optique grâce à une approche statistique. Nous observons l'apparition de structures localisées scélérates associées à l'émission d'un supercontinuum spectral spatial.

In this study, vertical profiles of velocity and hydrographic properties measured by the Lowered ADCP and CTD, respectively are used to calculate the vertical eddy diffusivity K based on small-scale turbulence theory. Two methods are used to estimate K, that is, the Thorpe scale analysis method (designated as Kz) and vertical wave number shear spectral method (designated as Ksh). Four different experiments with different flow conditions and bathymetry, i.e., internal tides, deep open-ocean, nonlinear internal waves and Kuroshio, are conducted and their K values are estimated and discussed. The internal tides at the mouth of Kao-Ping Submarine Canyon (KPSC) are observed during July and December (spring tide) of 2008. In each cruise the LADCP/CTD casts are repeated every two hours and last 27 and 40 hours, respectively. The results indicate the existence of strong, semi-diurnal internal tides with vertical displacement of 50~100 m and the nature of first baroclinic mode. Turbulent mixing during flood is significantly stronger than that during ebb. Note that in the winter experiments the Kz can reach 0.01 m2 s-1, which is even larger than the reported Kz values in other submarine canyons of the world, suggesting strong mixing processes are taking place in the KPSC. From the LADCP/CTD data of the joint hydrographic survey on May 2008 at SEATS station of the South China Sea, the estimated average values of Kz and Ksh in the upper 3000 m are about 3¡Ñ10-4 m2 s-1 and 1.8¡Ñ10-4 m2 s-1, respectively. The average value of Kz near the ocean bottom increases to 2.5¡Ñ10-3 m2 s-1. These estimated Kz are somewhat larger than the reported values in the open ocean. On the other hand, Kz values between 300 and 700 m deep are almost zero, indicating that turbulent mixing is inhibited in the stratified layer. Nonlinear internal waves are tracked in the South China Sea during May 2007. Our results show that after the internal solitons passed in the deep waters, the Kz profiles change significantly, surface mixing is weak, and Kz increases gradually from 400 m deep to the ocean bottom. In the shallow water region, shoaling effect of the nonlinear internal waves lead to enhanced energy dissipation and higher values of Kz, with the maximum value reaches 1 m2 s-1 near 180m depth. The flow structure of Kuroshio current between Taiwan and Lanyu is observed in October 2007. The results show that Kz in the surface layer is high (~10-2 m2 s-1), obviously due to strong Kuroshio flows there. At the 3000 m deep submarine trench near Lanyu, the Kz in the bottom layer is also very high (~ 1 m2 s-1 ), indicating that effective turbulent mixing in the bottom layer is mainly due to topography, which has similar level as the nonlinear internal waves.

Short Pulse equation derived by Schafer and Wayne is a nonlinear partial differential equation that describes ultra short laser propagation in a dispersive optical medium such as optical fibers. Some properties of this equation e.g. traveling wave solution and its soliton structure and some of its conserved quantities were investigated. Conserved quantities were obtained by mass conservation law, lax pair method and transformation between Sine-Gordon and short pulse equation. As a result, loop soliton characteristic and six conserved quantities were found.

No description available.

No description available.

In the paper, the beam propagation method is used to analyze the characteristics and the applications of nonlinear optical waveguide structures. The nonlinear optical waveguide is a medium whose refractive index changes with the electric field intensity. Based on the mode theory, the propagating envelop of optical light waves in the three-layers nonlinear waveguide with the nonlinear cladding, the nonlinear substrate and the linear guiding film can be solved. Not only the dispersion relation curve is described, but also the affection of input power to the electric field distribution is observed. In the application of nonlinear optical waveguide structure, the three-layers nonlinear waveguide structure and the local nonlinear Mach-Zehnder waveguide interferometer structure will be discussed: In the three-layers nonlinear waveguide structure, by launching the symmetric and antisymmetric modes, various characteristics of spatial optical solitons will be observed. Based on the interaction property between spatial optical solitons, a new all-optical 1¡ÑN switching device will be proposed; In the local nonlinear Mach-Zehnder waveguide interferometer structure, by fixing the input signal power and changing the control power, output signal beam will show the switching property. Besides, by changing the local nonlinear distributions, the nonlinear Mach-Zehnder interferometer will show various logic functions. The numerical results show that the proposed structures could function as all-optical switch devices and all-optical logic gates.

Many physical phenomena can be described by nonlinear models. The last few decades have seen an enormous growth of the applicability of nonlinear models and of the development of related nonlinear concepts. This has been driven by modern computer power as well as by the discovery of new mathematical techniques, which include two contrasting themes: (i) the theory of dynamical systems, most popularly associated with the study of chaos, and (ii) the theory of integrable systems associated, among other things, with the study of solitons. In this dissertation, we study two nonlinear models. One is the 1-dimensional vibrating string satisfying wtt − wxx = 0 with van der Pol boundary conditions. We formulate the problem into an equivalent first order hyperbolic system, and use the method of characteristics to derive a nonlinear reflection relation caused by the nonlinear boundary conditions. Thus, the problem is reduced to the discrete iteration problem of the type un+1 = F (un). Periodic solutions are investigated, an invariant interval for the Abel equation is studied, and numerical simulations and visualizations with different coefficients are illustrated. The other model is the Korteweg-de Vries-Burgers (KdVB) equation. In this dissertation, we proposed two new approaches: One is what we currently call First Integral Method, which is based on the ring theory of commutative algebra. Applying the Hilbert-Nullstellensatz, we reduce the KdVB equation to a first-order integrable ordinary differential equation. The other approach is called the Coordinate Transformation Method, which involves a series of variable transformations. Some new results on the traveling wave solution are established by using these two methods, which not only are more general than the existing ones in the previous literature, but also indicate that some corresponding solutions presented in the literature contain errors. We clarify the errors and instead give a refined result.

Η διακριτοποίηση των μερικών διαφορικών εξισώσεων (ΜΔΕ) αποτελεί κεντρικό βήμα στην αριθμητική τους επίλυση, και ως εκ τούτου είναι ένα από τα βασικά θέματα στα σύγχρονα μαθηματικά. Η μετάβαση από τη συνεχή ΜΔΕ στο αντίστοιχο διακριτό σύστημα μπορεί να γίνει με διάφορες αριθμητικές μεθόδους, ωστόσο δεν είναι όλες οι μέθοδοι εξίσου κατάλληλες και οφείλουμε πάντα να αναζητήσουμε την αρμόζουσα διακριτοποίηση για το εκάστοτε πρόβλημα. Στο 1ο κεφάλαιο γίνεται φανερό, μέσω του απλού παραδείγματος της λογιστικής εξίσωσης, πως μια αφελής διακριτοποίηση δύναται να αλλάξει δραματικά τη φύση του προβλήματος και των λύσεών του. Ιδιαίτερη προσοχή απαιτεί η διατήρηση (πριν και μετά τη διακριτοποίηση) των συμμετριών και των αναλλοίωτων μεγεθών του προβλήματος. Στην παρούσα διπλωματική εργασία μελετάμε την περίπτωση της εξίσωσης sine-Gordon, εστιάζοντας στις σολιτονικές της λύσεις. Στο 2ο κεφάλαιο παρουσιάζεται αναλυτικά η εξίσωση αυτή. Στο 3ο κεφάλαιο μέσω δύο διαφορετικών μεθόδων διακριτοποίησης, δείχνουμε τί ακριβώς πρέπει να προσέξει κανείς έτσι ώστε να δέχεται και το διακριτό σύστημα σολιτονικές λύσεις. Ως γνωστόν οι σολιτονικές λύσεις οφείλουν να πληρούν την ιδιότητα να παραμένουν αναλλοίωτες, διατηρώντας την ταχύτητα και το πλάτος τους πριν και μετά την αλληλεπίδρασή τους. Στο 4ο κεφάλαιο παρουσιάζονται συνοπτικά τα συμπεράσματα της παρούσας εργασίας ενώ συγκρίνουμε και τις δύο μεθόδους αριθμητικής επίλυσης που αναφέραμε. / The discretization of partial differential equations (PDEs) is a key step in their numerical solution, and therefore is one of the main issues in modern mathematics. The transition from continuous PDEs to their discrete counterparts can be done by various numerical methods, though not all methods are equally suitable; for this reason one should be careful to use an appropriate discretization method for each specific problem. In the first chapter it becomes clear, through the simple example of the logistic equation, that a naive discretization may dramatically change the nature of the problem and its solutions. Particular attention needs to be paid to the preservation (before and after the discretization) of the symmetries and invariant quantities of the problem. In the present work we study the case of the famous sine-Gordon equation, focusing on its soliton solutions. The second chapter presents a step-by-step derivation of the aforementioned equation. In the third chapter we show, by means of two different discretization schemes, which conditions must be met in order to guarantee that also the discrete system will admit soliton solutions. As is well known, soliton solutions are required to remain unchanged when they interact with each other, maintaining their speed and amplitude before and after the interaction. In the fourth chapter we summarize the conclusions of this work and draw a comparison between the two numerical schemes we have studied.