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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?

Andrews
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Dal
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    A somewhat ambiguous question: "non-linear analysis" doesn't mean the same thing in statistics as in pure mathematics, and I think there many also be usages in signal processing, physics, and some engineering fields. – Michael Hardy Nov 04 '14 at 20:21
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    A good source for nonlinear differential equations and Lyapunov stability theory can be found in Khalil's Nonlinear Systems book. It is very thorough. – Joel Nov 21 '14 at 23:39
  • In order to obtain answers on this topic, I'd suggest to do something with your Physics account as well. – Han de Bruijn Mar 19 '15 at 12:08
  • Let's say that it's possible that some physicists are more knowledgeable on non-linear analysis than some mathematicians. Maybe electronics is a good starting point. Though I've never been an expert in that area, I remember an excellent course on the subject. Wish I could find my old notes .. – Han de Bruijn Mar 19 '15 at 12:16
  • If you're looking to apply your knowledge to mathematical physics, then maybe you should take a look at Lie methods (reductions and symmetries) for PDEs and non-linear problems. These are often very useful for getting analytical and insightful solutions to hard problems. The other thing you might like to do is learn some numerics. – user27182 Mar 19 '15 at 12:34
  • Along the lines of what @user27182 says, OIver's books is a solid reference on the application of Lie Groups to DE's. http://www.amazon.com/Applications-Differential-Equations-Graduate-Mathematics/dp/0387950001/ref=sr_1_2?ie=UTF8&qid=1432903819&sr=8-2&keywords=olver+differential – Paul May 29 '15 at 12:51

2 Answers2

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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.

Gyu Eun Lee
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  • $+1$ for Ambrosetti and Prodi's A Primer of Nonlinear Analysis, which seems like a great book. – Dal Nov 23 '14 at 23:53
  • What background knowledge is assumed for reading Chang's book? – Dal Mar 09 '15 at 18:01
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    @Dal Depends on the chapter. All throughout, you should be familiar with Banach spaces and real analysis. If you know about Sobolev spaces then you are probably in a good position for much of the analysis prerequisites. Familiar with homology, say from an introductory algebraic topology course, could be useful, but I imagine not strictly necessary. At least some exposure to PDE at the standard graduate level (e.g. Evans) will be useful, as well as differential geometry/topology. If your institution has access to SpringerLink, you can legally access a pdf of the book for free. – Gyu Eun Lee Mar 13 '15 at 07:37
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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.