I'm looking for a textbook or review article about existing non-approximate methods for solving nonlinear partial differential equations.
Davis' book: http://www.amazon.com/Introduction-Nonlinear-Differential-Equations-Mathematics/dp/0486609715 might be a good starting point.
However the subject of nonlinear partial differential equations is vast, so the question needs to be more focused.
One example is: http://inside.mines.edu/~whereman/talks/WRI-00-Tanh-Sech-Methods-PDEs.pdf
another: http://silverdialogues.fas.nyu.edu/docs/IO/24458/fanghualin.pdf
I think that a very good handbook, devoted to the nonlinear partial differential equations, is: A. D. Polyanin and V. F. Zaitsev, Handbook of Nonlinear Partial Differential Equations, Chapman & Hall/CRC Press, Boca Raton–London, 2004 (first edition) and 2012 (second edition). Another very good handbook on nonlinear PDE's, is: Lokenath Debnath, Nonlinear Partial Differential Equations for Scientists and Engineers, 2012 (I have given here a lin to Springer website with this book online).
http://link.springer.com/book/10.1007%2F978-0-8176-8265-1
There is also a handbook by Zwillinger, and for nonlinear first-order PDEs there is also a classic book by Kamke in German, see the links below the text.
Also, to complement the answer of prof. Stepien, Zaitsev et al. have an online handbook giving solutions to a number of nonlinear PDEs at
http://eqworld.ipmnet.ru/ ; in particular, see
http://eqworld.ipmnet.ru/en/solutions/npde.htm.
Finally, an important class of PDEs possessing infinite families of exact solutions, are integrable systems (see e.g. https://en.wikipedia.org/wiki/Integrable_system for details). For these, see the online Encyclopedia of integrable systems at http://home.itp.ac.ru/~adler/E/e.pdf
Article Handbook of differential equations. With CD Rom. 3rd ed
Article Differentialgleichungen. Lösungsmethoden und Lösungen. II: P...
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