In this work we present some of the general theory of shock waves and their stability properties. We examine the concepts of nonlinear stability and spectral stability, noting that for certain classes of equations the study of nonlinear stability is reduced to the analysis of the spectra of the linearized eigenvalue problem. A useful tool in the study of spectral stability is the Evans function, an analytic function whose zeros correspond to the eigenvalues of the linearized eigenvalue problem. We discuss techniques for numerical Evans function computation that ensure analyticity, allowing standard winding number arguments and rootfinding methods to be used to locate eigenvalues. The Evans function is then used to study the spectra of the high Lewis number combustion system, tracking eigenvalues in the right-half plane.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22863号 / 情博第742号 / 新制||情||127(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 矢ヶ崎 一幸, 教授 中村 佳正, 准教授 柴山 允瑠, 教授 國府 寛司 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

This thesis is concerned with the existence and spectral stability of periodic waves in the fractional Korteweg-de Vries (KdV) equation and the fractional modified Korteweg-de Vries (mKdV) equation. We study the existence of periodic travelling waves using various tools such as Green's function for fractional Laplacian operator, Petviashvili fixed point method, and a new variational characterization in which the periodic waves in fractional KdV and fractional mKdV are realized as the constrained minimizers of the quadratic part of the energy functional subject to fixed L3 and L4 norm respectively. This new variational framework allows us to identify the existence region of periodic travelling waves and to derive the criterion for spectral stability of the periodic waves with respect to perturbations of the same period. / Thesis / Doctor of Philosophy (PhD)