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Propagation of solitary waves and undular bores over variable topographyTiong, Wei K. January 2012 (has links)
Description of the interaction of a shallow-water wave with variable topography is a classical and fundamental problem of fluid mechanics. The behaviour of linear waves and isolated solitary waves propagating over an uneven bottom is well understood. Much less is known about the propagation of nonlinear wavetrains over obstacles. For shallow-water waves, the nonlinear wavetrains are often generated in the form of undular bores, connecting two different basic flow states and having the structure of a slowly modulated periodic wave with a solitary wave at the leading edge. In this thesis, we examine the propagation of shallow-water undular bores over a nonuniform environment, and also subject to the effect of weak dissipation (turbulent bottom friction or volume viscosity). The study is performed in the framework of the variable-coefficient Korteweg-de Vries (vKdV) and variable-coefficient perturbed Korteweg-de Vries (vpKdV) equations. The behaviour of undular bores is compared with that of isolated solitary waves subject to the same external effects. We show that the interaction of the undular bore with variable topography can result in a number of adiabatic and non-adiabatic effects observed in different combinations depending on the specific bottom profile. The effects include: (i) the generation of a sequence of isolated solitons -- an expanding large-amplitude modulated solitary wavetrain propagating ahead of the bore; (ii) the generation of an extended weakly nonlinear wavetrain behind the bore; (iii) the formation of a transient multi-phase region inside the bore; (iv) a nonlocal variation of the leading solitary wave amplitude; (v) the change of the characteristics wavelength in the bore; and (vi) occurrence of a ``modulation phase shift" due to the interaction. The non-adiabatic effects (i) -- (iii) are new and to the best of our knowledge, have not been reported in previous studies. We use a combination of nonlinear modulation theory and numerical simulations to analyse these effects. In our work, we consider four prototypical variable topography profiles in our study: a slowly decreasing depth, a slowly increasing depth , a smooth bump and a smooth hole, which leads to qualitatively different undular bore deformation depending on the geometry of the slope. Also, we consider (numerically) a rapidly varying depth topography, a counterpart of the ``soliton fission" configuration. We show that all the effects mentioned above can also be observed when the undular bore propagates over a rapidly changing bottom . We then consider the modification of the variable topography effects on the undular bore by considering weak dissipation due to turbulent bottom friction or volume viscosity. The dissipation is modelled by appropriate right-hand side terms in the vKdV equation. The developed methods and results of our work can be extended to other problems involving the propagation of undular bores (dispersive shock waves in general) in variable media.
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Contrôlabilité d'équations issues de la mécanique des fluidesChapouly, Marianne 23 June 2009 (has links) (PDF)
Dans cette thèse on étudie la contrôlabilité globale de quelques équations non linéaires issues de la mécanique des fluides, précisément des équations de type Burgers, une équation de Korteweg-de Vries, et un système de Navier-Stokes 2-D. La stratégie employée consiste, d'une part, à appliquer la méthode du retour de J.-M. Coron, et d'autre part, à jouer sur la non linéarité de l'équation considérée. <br />De cette manière, on montre dans la première partie la contrôlabilité globale exacte pour tout temps d'équations de type Burgers non visqueuses puis on utilise ensuite ce résultat pour obtenir un résultat de contrôlabilité globale approchée pour l'équation de Burgers visqueuse. Cette propriété, combinée avec un résultat de contrôlabilité locale, entraîne ainsi la contrôlabilité globale aux trajectoires de l'équation de Burgers visqueuse, pour tout temps. <br />Dans la deuxième partie, on procède d'une manière similaire pour obtenir la contrôlabilité globale exacte d'une équation de Korteweg-de Vries non linéaire, pour tout temps. <br />Enfin, dans la dernière partie on s'intéresse à un système de Navier-Stokes 2-D avec conditions aux bords de type Navier. On obtient, en utilisant cette fois des résultats sur l'équation d'Euler des fluides incompressibles, la contrôlabilité globale à zéro, pour tout temps.
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Dynamique et collision de solitons pour quelques équations dispersives nonlinéairesMuñoz, Claudio 23 June 2010 (has links) (PDF)
Dans cette thèse, nous étudions quelques propriétés dynamiques des solutions de type soliton de quelques équations dispersives nonlinéaires généralisées. La première partie de ce travail est consacrée à l'étude de l'existence, de l'unicité et du comportement global de solitons pour des équations de KdV généralisées, à variation lente. On donnera une description détaillée de la dynamique pour tout temps et on montrera la non-existence de solitons purs, ce qui est une très grande différence avec l'équation gKdV standard. Dans une deuxième partie, on étudiera le cas de l'équation de Schrödinger nonlinéaire. Pour cette équation, nous allons améliorer tous les résultats précédents en donnant une description précise pour tout temps de la dynamique du soliton dans le régime à variation lente. En plus, sous des hypothèses générales, on montrera ce résultat dans le cas 2-D. Finalement, on considère le problème de collision de deux solitons pour l'équation de KdV généralisée. Complétant les résultats récents de Martel et Merle, concernant le cas quartique, nous montrons que la seule possibilité d'avoir une collision de type élastique est donnée par les cas intégrables. La preuve de tous ces résultats sont des développements et des améliorations de la théorie de Martel et Merle pour la collision de deux solitons des équations gKdV sous différents régimes asymptotiques.
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Energy Preserving Methods For Korteweg De Vries Type EquationsSimsek, Gorkem 01 July 2011 (has links) (PDF)
Two well-known types of water waves are shallow water waves and the solitary waves. The former waves are those waves which have larger wavelength than the local water depth and the latter waves are used for the ones which retain their shape and speed after colliding with each other. The most well known of the latter waves are Korteweg de Vries (KdV) equations, which are widely used in many branches of physics and engineering. These equations are nonlinear long waves and mathematically represented by partial differential equations (PDEs). For solving the KdV and KdV-type equations, several numerical methods were developed in the recent years which preserve their geometric structure, i.e. the Hamiltonian form, symplecticity and the integrals. All these methods are classified as symplectic and multisymplectic integrators. They produce stable solutions in long term integration, but they do not preserve the Hamiltonian and the symplectic structure at the same time. This thesis concerns the application of energy preserving average vector field integrator(AVF) to nonlinear Hamiltonian partial differential equations (PDEs) in canonical and non-canonical forms. Among the PDEs, Korteweg de Vries (KdV) equation, modified KdV equation, the Ito&rsquo / s system and the KdV-KdV systems are discetrized in space by preserving the skew-symmetry of the Hamiltonian structure. The resulting ordinary differential equations (ODEs) are solved with the AVF method. Numerical examples confirm that the energy is preserved in long term integration and the other integrals are well preserved too. Soliton and traveling wave solutions for the KdV type equations are accurate as those solved by other methods. The preservation of the dispersive properties of the AVF method is also shown for each PDE.
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Analiticidade, na variável espacial, da solução do problema de Caucky para a equação de KdV.Miotto, Márcio Luís 13 March 2006 (has links)
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Previous issue date: 2006-03-13 / Financiadora de Estudos e Projetos / In this work we use bilinear estimates and point fix theorem to show that the solution to the initial value problem for the Korteweg-de Vries equation with analytic periodic initial data is analytic and periodic in the space variable. / Neste trabalho usamos estimativas bilineares e teorema do ponto fixo para mostrar que a solução do Problema de Cauchy para a equaçãoo de Korteweg-de Vries, com dado inicial analítico periódico, é analítica e periódica na variável espacial.
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On Traveling Wave Solutions of Linear and Nonlinear Wave Models (Seeking Solitary Waves)Moussa, Mounira 02 June 2023 (has links)
No description available.
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Mathematical modelling of nonlinear ring waves in a stratified fluidZhang, Xizheng January 2015 (has links)
Oceanic waves registered by satellite observations often have curvilinear fronts and propagate over various currents. In this thesis, we study long linear and weakly-nonlinear ring waves in a stratified fluid in the presence of a depth-dependent horizontal shear flow. It is shown that despite the clashing geometries of the waves and the shear flow, there exists a linear modal decomposition, which can be used to describe distortion of the wavefronts of surface and internal waves, and systematically derive a 2+1-dimensional cylindrical Korteweg-de Vries (cKdV)-type equation for the amplitudes of the waves. The general theory is applied to the case of the waves in a two-layer fluid with a piecewise-constant shear flow, with an emphasis on the effect of the shear flow on the geometry of the wavefronts. The distortion of the wavefronts is described by the singular solution (envelope of the general solution) of the nonlinear first order differential equation, constituting generalisation of the dispersion relation in this curvilinear geometry. There exists a striking difference in the shape of the wavefronts: the wavefront of the surface wave is elongated in the shear flow direction while the wavefront of the interfacial wave is squeezed in this direction. We solve the derived 2+1-dimensional cKdV-type equation numerically using a finite-difference scheme. The effects of nonlinearity and dispersion are studied by considering numerical results for surface and interfacial ring waves generated from a localised source with and without shear flow and the 2D dam break problem. In these examples, the linear and nonlinear surface waves are faster than interfacial waves, the wave height decreases faster at the surface, the shear flow leads to the wave height decreasing slower downstream and faster upstream, and the effect becomes more prominent as the shear flow strengthens.
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Contrôlabilité exacte d'équations dispersives issues de la mécanique.Crépeau, Emmanuelle 06 December 2002 (has links) (PDF)
Le sujet principal de cette thèse est l'étude de la contrôlabilité exacte de deux équations dispersives, l'équation de Korteweg-de Vries et la "bonne" équation de Boussinesq. En ce qui concerne l'équation de Korteweg-de Vrie, on étend un résultat de Rosier en montrant la contrôlabilité exacte en tout temps de l'équation non linéaire autour d'une solution stationnaire proche de zéro mais non nulle, ce pour des longueurs de domaine spatial critiques. Cette démonstration utilise en particulier la méthode d'unicité hilbertienne couplée avec la méthode des multiplicateurs et un théorème de point fixe. Ensuite, nous étudions le problème de la contrôlabilité exacte de l'équation de Boussinesq pour deux contrôles différents. On utilise également la méthode d'unicité hilbertienne pour ces problèmes en appliquant une inégalité de Ingham. On obtient ainsi un résultat de contrôlabilité exacte pour des temps arbitrairement petits. Nous implémentons ensuite cette méthode de facon numérique pour l'équation de Boussinesq avec un contrôle portant sur la dérivée seconde a droite, tant sur le problème linéaire que non linéaire.
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Inversion of Nonlinear Dispersive Wave and its Application in Determining Tsunami Wave SoureLi, Lieh-Yu 13 April 2011 (has links)
In this study, the method of deciding the water level of the initial tsunami is proposed by using spatial-temporal focusing (Coalescence) theory and waveform inversion reciprocal with Green function. Tsunami and earthquake are so closely bonded that the current tsunami numerical model is dependent on the parameters of the fault and the initial tsunami water level by calculating the theory of
half flexibility. But in fact, it is not easy to have the parameters of seabed fault so that the initial tsunami water level is very hard to get a accurate value. On the other hand, although the parameters of fault can be speculated by seismic waves, because ground is uneven medium, therefore, it is still a lot of improvement to get the parameters of fault by using seismic waves. For the tsunami simulation, if you have the value of the initial tsunami water level, the fault parameters can be estimated.Since the propagation of tsunami in the ocean is a linear behavior, the propagating process is affected by the topography of the ocean and the nonlinear effect
so minimal that it is to satisfy the linear shallow water equations and the requirement of reversibility;However, in fact, the values of the water level measured by the tide stations on the coast are influenced by the shoaling effect so that the reversibility of linear system can not be directly applied to Coastal areas.Therefore, the overall Inversion procedure on this study consists of two parts; the first one is that the usage of variable
coefficient Korteweg-de Vries (vKdV) equation and the Coalescence theory inverses the data gathered by the Coastal tide stations to the water level data where the depth is more than 50m on the linear region, and compares the above results with the stimulation and confirms the accuracy of the inversed waveform;The second one is that according to the reversibility of the linear system the use of least squares and least squares QR- decomposition (LSQR) method reproduce the initial tsunami wave source that compares with the initial tsunami wave source by stimulating and has a very good conformity. The seismic parameters can be easily decided by the above results.
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Représentation stochastique d'équations aux dérivées partielles d'ordre supérieur à 3 issues des neurosciences / Stochastic representation of high-order partial differential equations resulting from neurosciencesVigot, Alexis 29 November 2016 (has links)
Cette Thèse se divise en deux parties. Dans la partie mathématique, nous étudions différentes edp d'ordre supérieur à 3 issues des neurosciences avec un point de vue probabiliste. Nous démontrons une formule de FK pour une grande classe de solutions de KdV (pas seulement les n-solitons), à l'aide des déterminants de Fredholm et des transformées de Laplace d'intégrales de Skorohod itérées. Concernant les edp d'ordre supérieur à 3, les processus itérés qui consistent en la composition de deux processus indépendants, l'un correspondant à la position et l'autre au temps, sont liés à leurs solutions. En effet, nous montrons une formule de FK pour des solutions d'edp d'ordre supérieur à 3 basée sur des fonctionnelles de processus itérés, même dans le cas non Markovien, étendant ainsi les résultats existants. Nous proposons aussi un schéma numérique pour la simulation de trajectoires de diffusions itérées basé sur le schéma d'Euler, qui converge p.s., uniformément en temps, avec un taux de convergence d'ordre $1/4$. Une estimation de l'erreur est proposée. Dans la partie biologique, nous avons collecté plusieurs articles en neuroscience et d'autres domaines de biologie, où les edp précédentes sont utilisées. En particulier, on s'intéresse à la simulation et à la propagation du potentiel d'action lorsque la capacité de la membrane cellulaire n'est pas supposée constante. Ces articles ont en commun le fait qu'ils remettent en question le fameux modèle d'Hodgkin-Huxley datant des années cinquante. En effet, même si ce modèle a été très efficace pour la compréhension du signal neuronal, il ne prend pas en compte tous les phénomènes résultants de la propagation du potentiel d'action. / This Thesis consists of two parts. In the mathematical part we study Korteweg--de Vries (KdV) equation and high-order pdes with a probabilistic point of view in order to obtain Feynman-Kac (FK) type formulas. This study was motivated by recent biological models. We prove a FK representation for a larger class of solutions of KdV equation (not only n-solitons), using Fredholm determinants and Laplace transforms of iterated Skorohod integrals. Regarding higher order pdes, iterated processes that consist in the composition of two independent processes, one corresponding to position and the other one to time, are naturally related to their solutions. Indeed, we prove FK formulas for solutions of high order pdes based on functionals of iterated processes even in the non Markovian case, thus extending the existing results. We also propose a scheme for the simulation of iterated diffusions trajectories based on Euler scheme, that converges a.s., uniformly in time, with a rate of convergence of order $1/4$. An estimation of the error is proposed. In the biological part, we have collected several papers in neuroscience and other fields of biology where the previous types of pdes are involved. In particular, we are interested in the simulation of the propagation of the action potential when the capacitance of the cell membrane is not assumed to be constant. These papers have in common the fact that they question the famous Hodgkin Huxley model dating back to the fifties. Indeed this model even if it has been very efficient for the understanding of neuronal signaling does not take into account all the phenomena that occur during the propagation of the action potential.
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