This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

This dissertation deals with a class of nonlinear wave equations of the type discovered by R. Camassa and D. D. Holm which includes the Camassa-Holm, the Degasperis-Procesi, and the two component Camassa-Holm equations. All these equations admit certain non-smooth soliton-like solutions, called peakons as well as other non-smooth solutions like cuspons. We apply the techniques of the theory of distributions of L. Schwartz to study these solutions. In particular, every non-smooth traveling wave which is a distributional solution of the two component Camassa-Holm equation is a distributional solution of the Camassa-Holm equation if the set of points where the height of the wave equals its speed, is of measure zero. This includes peakon or cuspon traveling wave solutions.<p>We also develop a suitable modification of the classical Lax pair formalism to deal with singular solutions. We show that the Lax pair formalism can be extended to a distributional weak Lax pair which is appropriate for dealing with the peakon solutions of the Camassa-Holm equation.

Dans cette thèse on étudie la stabilité orbitale des ondes solitaires de deux types d’équations d’évolution non linéaires: l’équation de Degasperis-Procesi (DP), qui est une équation du type Camassa-Holm, et l’équation de Kawahara généralisée (gKW), qui correspond à une équation de Korteweg-de Vries généralisée (gKdV) supplémentée d’un terme d’ordre 5. Sur le modèle DP on apporte une amélioration significative de la preuve de la stabilité d’un peakon donnée par Lin et Liu. Puis, en utilisant la méthode de Martel-Merle-Tsai adaptée par El Dika-Molinet dans le cas de l’équation de Camassa-Holm, on montre que la somme de N peakons, de vitesses croissantes et suffisamment distants les uns des autres à l’instant initial, est orbitalement stable. Sur le modèle de Kawahara généralisé, on prouve l’existence de deux branches d’ondes solitaires : l’une construite en appliquant le théorème des fonctions implicites au voisinage d’une onde solitaire explicite de gKW découverte par Dey. al., l’autre construite en résolvant un problème de minimisation sur R, avec une contrainte qui force la famille à converger vers le soliton explicite de l’équation de Korteweg-de Vries généralisée (gKdV) lorsque le coefficient devant l’opérateur d’ordre 5 tend vers 0. Par remise à l’échelle, on obtient ainsi une branche constituée d’ondes solitaires voyageant à faibles vitesses. On prouve ensuite que les ondes solitaires constituant ces deux branches sont orbitalement stables en appliquant la méthode spectrale introduite par Benjamin et des arguments de continuité. / No summary available

Orientador: Prof. Dr. Igor Leite Freire / Tese (doutorado) - Universidade Federal do ABC, Programa de Pós-Graduação em Matemática, 2016. / Neste trabalho estudamos diversos aspectos de algumas classes de equações ou sistemas de equações. Simetrias de Lie, de Noether, leis de conservação derivadas do Teorema de Noether e soluções invariantes são obtidas para uma classe de equações diferenciais ordinárias. Também consideramos equações e sistemas do tipo Camassa-Holm, alguns dos quais foram obtidos como soluções de um problema inverso. Para todos são encontradas as simetrias de Lie e, para alguns, obtemos leis de conservação utilizando o Teorema de Ibragimov. Além disso, para casos particulares das equações deduzidas via problema inverso, investigamos a existência de soluções peakon e multipeakon. Finalmente, consideramos uma família de equações evolutivas, a qual admite soluções peakon e membros integráveis. / In this work we study several aspects of some families of differential equations and systems. Lie point symmetries, Noether symmetries, conservation laws obtained from Noether Theorem and invariant solutions are derived for a class of ordinary differential equations. We also consider Camassa-Holm type equations and systems, some of which deduced from an inverse problem. For all of them we obtain Lie point symmetry classifications and, for some, conservation laws using Ibragimov¿s Theorem. Furthermore, for particular cases of the equations obtained as an inverse problem, we investigate the existence of peakon and multipeakon solutions. Finally, we consider a family of evolution equations, which admits peakon solutions and integrable members.

In this thesis we employ two recent analytical approaches to investigate the possible classes of traveling wave solutions of some members of recently derived integrable family of generalized Camassa-Holm (GCH) equations. In the first part, a novel application of phase-plane analysis is employed to analyze the singular traveling wave equations of four GCH equations, i.e. the possible non-smooth peakon, cuspon and compacton solutions. Two of the GCH equations do no support singular traveling waves. We generalize an existing theorem to establish the existence of peakon solutions of the third GCH equation. This equation is found to also support four segmented, non-smooth M-wave solutions. While the fourth supports both solitary (peakon) and periodic (cuspon) cusp waves in different parameter regimes. In the second part of the thesis, smooth traveling waves of the four GCH equations are considered. Here, we use a recent technique to derive convergent multi-infinite series solutions for the homoclinic and heteroclinic orbits of their traveling-wave equations, corresponding to pulse and front (kink or shock) solutions respectively of the original PDEs. Unlike the majority of unaccelerated convergent series, high accuracy is attained with relatively few terms. Of course, the convergence rate is not comparable to typical asymptotic series. However, asymptotic solutions for global behavior along a full homoclinic/heteroclinic orbit are currently not available.