In this work we will demonstrate that the Cauchy problem associated with the Korteweg-de Vries equation, denoted by KdV, and Korteweg-de Vries modified equation, denoted by mKdV, with initial data in the space of Sobolev Hs(|R), is locally well-posed on Hs(|R), with s>3/4 for KdV and s≥1/4 for mKdV, where the notion of well-posedness includes existence, uniqueness, persistence property of solution and continuous dependence of solution with respect to the initial data. This result is based on the works of Kenig, Ponce and Vega. The technique used to obtain these results is based on fixed point Banach theorem combined with the regularizantes effects of the group associated with the linear part. / Fundação de Amparo a Pesquisa do Estado de Alagoas / Neste trabalho demonstraremos que o problema de Cauchy associado as equações de Korteweg-de Vries, denotada por KdV, e de Korteweg-de Vries modificada, denotada por mKdV, com dado inicial no espaço de Sobolev Hs(|R), é bem posto localmente em Hs(|R), com s>3/4 para a KdV e s≥1/4 para a mKdV, onde a noção de boa postura inclui a existência, unicidade, a propriedade de persistência da solução e dependência contínua da solução com relação ao dado inicial. Este resultado é baseado nos trabalhos de Kenig, Ponce e Vega. A técnica utilizada para obter tais resultados se baseia no Teorema do Ponto Fixo de Banach combinada com os efeitos regularizantes do grupo associado com a parte linear.

Made available in DSpace on 2014-06-11T19:25:30Z (GMT). No. of bitstreams: 0 Previous issue date: 2005-02Bitstream added on 2014-06-13T18:53:35Z : No. of bitstreams: 1 ymai_lh_me_ift.pdf: 483788 bytes, checksum: 4f0df0fea1d850bbc366844622fd0b5f (MD5) / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES) / Apresentamos a formulação da hierarquia mKdV e sinh-Gordon supersimétrica com N = 1 baseada na superálgebra de Kac-Moody sl(2,1). Obtivemos soluções para até quatro vértices para ambos os modelos de dressing e também discutimos a transformação de Bäcklund para o modelo sinh-Gordon supersimétrico utilizando supercampos / We introduce the formulation of supersymmetric mKdV and sinh-Gordon hierarchy with N = 1 based on Kac-Moody sl(2,1) superalgebra. We ontained solutions to up to four vertex operator for both models using the dressing method and we also discuss the Bäcklund transformation for the supersymmetric sinh-Gordon model using superfields

The Fermi-Pasta-Ulam-Tsingou (FPUT) lattice became of great mathematical interest when it was observed that it exhibited a near-recurrence of its initial condition, despite it being a nonlinear system. This behavior was explained by showing that the Korteweg-de Vries (KdV) equation serves as a continuum limit for the FPUT and has soliton solutions. Much work has been done into analyzing the solitary wave solutions of the FPUT and the relationship between the lattice and its continuum limit. For certain potentials the modified KdV (mKdV) instead serves as the continuum limit for the FPUT. However, there has been little research done to examine how the defocusing mKdV can be used a modulation equation for the FPUT or how the kink solutions of the mKdV relate to solutions of the FPUT. This thesis first addresses the existence of kink-like solutions of the FPUT and shows that their profiles can be approximated by the profiles of the kink solutions of the mKdV. Next, it is shown that the defocusing mKdV can be used more widely as a modulation equation for small-amplitude, long-wavelength solutions of the FPUT lattice. Finally, the issue of stability of the kink-like solutions is discussed, and some results toward linear stability are given.

京都大学 / 0048 / 新制・課程博士 / 博士(理学) / 甲第21412号 / 理博第4432号 / 京都大学大学院理学研究科数学・数理解析専攻 / (主査)教授 堤 誉志雄, 教授 泉 正己, 教授 上 正明 / 学位規則第4条第1項該当 / Doctor of Science / Kyoto University / DFAM