TITLE: A Learning Roadmap to the "foundations" of Nonlinear Analysis (and certain specific topics)
QUESTION [8 upvotes]: 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?

REPLY [5 votes]: There are the four enormous tomes collectively titled Nonlinear Functional Analysis and Its Applications by Zeidler:

Part I - Fixed Point Theorems
Part II(a) - Linear Monotone Operators
Part II(b) - Non-linear Monotone Operators
Part III - Variational Methods and Optimization
Part IV - Applications to Mathematical Physics

The usual (linear) functional analysis prerequisites are contained in the appendix to the first part.
I have only skimmed (even that is an overstatement) over these books and they seem to be thoroughly peppered with real-world examples. They look amazing and I think they author has a nice style.