Nonlinear dispersive equations: classical and new frameworks

• Published in: São Paulo Journal of Mathematical Sciences Vol. 16; no. 1; pp. 171 - 255
• Main Author: Angulo Pava, Jaime
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.05.2022; Springer Nature B.V
• Subjects: Graphs; Mathematical models; Mathematics; Mathematics and Statistics; Nonlinear evolution equations; Nonlinear phenomena; Operators; Orbital stability; Perturbation theory; Scientific papers; Solitary waves; Special issue commemorating the Golden Jubilee of the Institute of Mathematics and Statistics of the University of São Paulo; Spectral theory; Traveling waves; Variational methods; Wave dispersion; Variational methods; Primary 35Q51; 35J61; Dispersive equations; Traveling waves; Schrödinger models; 35Q53; Metric graphs; Sturm–Liouville theory; Linear instability; Extension theory of symmetric operators; Orbital stability; Secondary 47E05; Point interactions; Analytic perturbation; BBM models; Korteweg–de Vries models
• ISSN: 1982-6907; 2316-9028; 2306-9028
• DOI: 10.1007/s40863-020-00195-z

The purpose of this manuscript associated to the Golden Jubilee of the IME-USP is to present selected material from the author’s scientific contribution dealing with nonlinear phenomena of dispersive type. That study has been a source for modern research in the dynamic of traveling wave solutions of different type and it has been disseminated through publication of scientific articles and/or books. Here, we provide the reader with information about current research in stability theory for nonlinear dispersive equations and possible developments. Also, I hope it inspires future developments in this important and fascinating subject. In this manuscript we consider the following topics: stability theory of solitary waves and the applicability of the concentration–compactness principle, the existence and orbital (in)stability of periodic traveling wave solutions for nonlinear dispersive models, nonlinear Schrödinger and Korteweg–de Vries models on star-shaped metric graphs. The use of tools of the theory of spaces of Hilbert, the spectral theory for unbounded self-adjoint operators, Sturm–Liouville’s theory, variational methods, analytic perturbation theory of operators, and the extension theory of symmetric operators are pieces fundamental in our study. The methods presented in this manuscript have prospect for the study of the dynamic of solutions for nonlinear evolution equations around of different traveling waves profiles which may appear in non-standard environments such as star-shaped metric graphs.