Ce travail est consacré à l'étude de l'équation de Novikov-Veselov, un analogue ( 2 + 1 )-dimensionnel de l'équation renommée de Korteweg-de Vries, intégrable via la transformée de la diffusion inverse pour l'équation de Schrödinger stationnaire en dimension 2 à énergie fixe. Nous commençons par étudier une classe spéciale de solutions rationnelles non singulières de l'équation de Novikov-Veselov à énergie positive, construites par Grinevich et Zakharov, et nous démontrons que ces solutions sont multisolitons. Les solutions de Grinevich-Zakharov sont localisées comme $ O( | x |^{ -2 } ) $, $ | x | \to \infty $, et dans le travail présent, nous prouvons que cette localisation est presque la plus forte possible pour les solitons de l'équation de Novikov-Veselov: nous montrons que l'équation de Novikov-Veselov à énergie non nulle ne possède pas de solitons localisés plus fort que $ O ( | x |^{ - 3 } ) $, $ | x | \to \infty $. Pour le cas d'énergie zéro, nous montrons que si les solitons de l'équation de Novikov-Veselov appartiennent à l'image des solutions de l'équation de Novikov-Veselov modifiée sous la transformation de Miura, dans ce cas, la localisation plus forte que $ O( | x |^{ -2 } ) $ n'est pas possible. Dans le travail présent, nous étudions également la question du comportement asymptotique des solutions du problème de Cauchy pour l'équation de Novikov-Veselov à énergie non nulle (pour le cas d'énergie positive, les solutions transparentes ou " reflectionless " sont considérées). Sous l'hypothèse de non singularité des données de diffusion des solutions nous obtenons que ces solutions décroissent avec le temps de façon uniforme comme $ O( t^{ -1 } ) $, $ t \to +\infty $, dans le cas d'énergie positive et comme $ O( t^{ -3/4 } ) $, $ t \to +\infty $, dans le cas d'énergie négative; dans ce dernier cas, nous démontrons également que l'estimation obtenue est optimale.

Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenüber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und Stabilitätsanalyse werden Instabilitäten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der Grenzübergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklärt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der Kalziumsignalübertragung in Zellen identifiziert.

In this thesis we investigate the formation of patterns in a simple activator-inhibitor model supplemented with an inhibitory nonlocal coupling term. This model exhibits a wave instability for slow inhibitor diffusion, while, for fast inhibitor diffusion, a Turing instability is found. For moderate values of the inhibitor diffusion these two instabilities occur simultaneously at a codimension-2 wave-Turing instability. We perform a weakly nonlinear analysis of the model in the neighbourhood of this codimension-2 instability. The resulting amplitude equations consist in a set of coupled Ginzburg-Landau equations. These equations predict that the model exhibits bistability between travelling waves and Turing patterns. We present a study of interfaces separating wave and Turing patterns arising from the codimension-2 instability. We study theoretically and numerically the dynamics of such interfaces in the framework of the amplitude equations and compare these results with numerical simulations of the model near and far away from the codimension-2 instability. Near the instability, the dynamics of interfaces separating small amplitude Turing patterns and travelling waves is well described by the amplitude equations, while, far from the codimension-2 instability, we observe a locking of the interface velocities. This locking mechanism is imposed by the absence of defects near the interfaces and is responsible for the formation of drifting pattern domains, i.e. moving localised patches of travelling waves embedded in a Turing pattern background and vice versa.

The central theme of development, how cells are organized into functional structures and assembled into whole organisms, is addressed by developmental biology. One important feature of embryonic development is pattern formation, which is the generation of a particular arrangement of cells in three-dimensional space at a given point of time. Central to this work is the model system of the zebrafish, Danio rerio. The aim of the first part of this study was to try to understand how a distinct part of the embryonic brain called midbrain-hindbrain boundary (MHB), a region that acts as an organizer for the adjacent brain regions, is established in vertebrates. spiel-ohne-grenzen (spg) is one mutant which interferes with MHB development. Here, I addressed the role of pou2 in brain development by molecular, phenotypical and functional analysis. By genetic complementation and mapping I could elucidate the molecular nature of this mutant and found that the pou2 gene encoding the POU domain transcription factor is affected in spg mutant embryos. By chromosomal syntenic conservation, phylogenetic sequence comparison, and expression and functional data I imply that pou2 is the orthologue of the mammalian Oct4 (Pou5F1) gene. I find by detailed expression and transplantation analysis that pou2 is cell autonomously required within the neuroectoderm to activate genes of the MHB and hindbrain primordium, like pax2.1, wnt1, gbx2 or krox20. By gain-of-function experiments I demonstrate that pou2 synergizes with Fgf8 signaling in order to activate particularly the hindbrain primordium. Since pou2 is already provided to the embryo by the mother, I generated embryos which lack maternal and zygotic pou2 function (MZspg) to reveal a possible earlier than neuroectodermal role of pou2. In the second part of this work I demonstrate that pou2 is a key factor controlling endoderm differentiation. By expression and gain-of-function analysis I suggest a cell autonomous function for Pou2 in the first step of endodermal differentiation. By gain-of-function experiments involving the gene encoding the HMG transcription factor Casanova (Cas) I show that both Cas and Pou2 are necessary to activate expression of the endodermal differentiation marker sox17 in a mutually dependent way, and that the ability of Cas to ectopically induce sox17 strictly requires Pou2. I conclude that both maternal and zygotic pou2 function is necessary for commitment of endodermal progenitor cells to differentiate into endodermal precursor cells.

The correct patterning of embryos requires the exchange of information between cells. This is in part achieved by the proper distribution of signaling molecules, many of which exert their function by establishing gradients of concentration. Because of this property they were named "morphogens", or "form giving" substances. Among these, proteins belonging to the Hedgehog (Hh) family have received much attention, owing to their unusual double lipid modification and their involvement in human disease, causing congenital birth defects and cancer. Great efforts have been made in order to elucidate the mechanisms by which Hh molecules are propagated in the embryo. However, no conclusive evidence exists to date to which structures these molecules localize and how they, despite their membrane association, establish a gradient of concentration. Therefore, I decided to study the distribution of the vertebrate Hh homolog, Sonic Hedgehog (Shh) in developing zebrafish embryos. By fluorescently tagging Shh proteins, I found that these localize to discrete punctate structures at the membranes of expressing cells. These were often regions from which filopodial protrusions emanated from the cells. Puctate deposits of Shh were also located outside of expressing cells. In dividing cells, Shh accumulated at the cleavage plane. Furthermore, by making use of confocal microscopy and time lapse analysis, I visualized Shh proteins moving in filopodial extensions present between cells. This suggests a novel mechanism of Shh distribution, which relies on the direct contact of cells by filopodia for the exchange of signaling proteins. In a second part of my thesis, I characterized genes implicated in regulating Shh protein distribution and signaling function. I cloned three zebrafish genes belonging to the Ext1 (exostosin) family of glycosyltransferases required for the synthesis of Heparan Sulfate Proteoglycans and established a tentative link of these genes to somitic Hh signaling. In addition, I characterized the developmental expression and function of zebrafish Rab23, a small GTPase, which acts as a negative regulator of the Shh signaling pathway. Performing knock-down experiments of zebrafish Rab23, I found that Rab23 functions in left-right axis specification, a process previously shown to depend on proper Shh signaling.

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