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I am currently taking a course on Numerical PDE. The course covers the following topics listed below.

Chapter 1: Solutions to Partial Dierential Equations:

Chapter 2: Introduction to Finite Elements:

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Numerical PDEs by J.W. Thomas might be a good book. If you check its table of contents, they have the majority of the topics you listed.

Here is one example of professor's lecture notes based on the same book.

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A book I like, but I am probably biased since he was my professor is Dr. Jianke Yang's book Nonlinear Waves in Integrable and Non-integrable systems

It covers:

  1. Derivation of non-linear waves
  2. Integrable theory for the non-linear Schrodinger equation
  3. Theories for integrable equations with order scattering operators
  4. Soliton perturbation theory and applications
  5. Theories for non-integrable systems
  6. Nonlinear waves phenomena in periodic media
  7. Numerical methods for non-linear wave equations

Also, Matlab code is provided in the book.

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