I'm searching for throughout references that — in the long term — can help me gradually gain a solid background and firm foundations to understand the main methods and theorems to deal with nonlinear problems (in particular, wave equations, solitary wave solutions (solitons) , nonlinear elliptic and hyperbolic PDEs, periodic solutions of Lagrangian and Hamiltonian Systems, etc.) that arise in science (specifically, mathematical and theoretical physics).
The following textbooks caught my attention:
- Zdzislaw Denkowski, Stanislaw Migórski, and Nikolaos S. Papageorgiou, An
Introduction to Nonlinear Analysis: Theory; - Antonio Ambrosetti and Giovanni Prodi, A Primer of Nonlinear Analysis;
- Antonio Ambrosetti and David Arcoya, An Introduction to Nonlinear Functional Analysis and Elliptic Problems;
- Abdul-Majid Wazwaz, Partial differential equations and solitary waves theory;
- Herbert Koch, Daniel Tataru, and Monica Vişan, Dispersive Equations and Nonlinear Waves;
- Kung Ching Chang, Methods in Nonlinear Analysis.
I would like to receive some advice from the experienced researchers in nonlinear analysis and mathematical physics of Mathematics Stack Exchange:
Question: How should I go about learning nonlinear analysis? That is, assuming knowledge of real analysis, what resources and what kind of approach (and order) to read through them would you
recommend to build a solid knowledge of nonlinear analysis?
Best Answer
Nonlinear analysis is a very large field, and you'd be hard-pressed to find a resource that deals with its many methods in a comprehensive manner. Most resources I know of deal with a subset of methods or methods applied to particular situations (nonlinear elliptic equations, for example).
That said, if you're only looking for an introduction to the subject then there may be some nice books to get you started. Ambrosetti and Prodi's A Primer of Nonlinear Analysis is an introductory text in nonlinear functional analysis and bifurcation theory. I haven't read Primer myself, but I have read a sort-of-sequel by Ambrosetti and Malchiodi, Nonlinear Analysis and Semilinear Elliptic Equations, which builds on the material in Primer and discusses degree theory, fixed-point theory, critical point and Morse methods. My experience with the exposition in that book was very positive, so I think Primer should also live up to that standard.
Another book I might recommend is Chang's Methods in Nonlinear Analysis. It focuses on more topological methods and variational principles.
These books don't even come close to a comprehensive view of nonlinear analysis. If you're really interested in the subject then you'll probably come across problems and tools from all areas of mathematics.