Mathematical aesthetics is not usually discussed as a separate discipline, even though it is reasonable to suppose that the foundations of physics lie in mathematical aesthetics. This book presents a list of mathematical principles that can be classified as "aesthetic" and shows that these principles can be cast into a nonlinear set of equations. Then, with this minimal input, the book shows that one can obtain lattice solutions, soliton systems, closed strings, instantons and chaotic-looking systems as well as multi-wave-packet solutions as output. These solutions have the common feature of being nonintegrable, ie. the results of integration depend on the integration path. The topic of nonintegrable systems is discussed Ch. 1. Introduction -- Ch. 2. Mathematical description of fluids -- Ch. 3. Linear waves -- Ch. 4. Model equations for weakly nonlinear waves -- Ch. 5. Analytical methods for solving the classical model wave equations -- Ch. 6. Numerical methods for a scalar hyperbolic equations -- Ch. 7. Review of numerical methods for model wave equations -- Ch. 8. Numerical schemes for a system of one-dimensional hyperbolic equations -- Ch. 9. A hyperbolic system of two-dimensional equations -- Ch. 10. Numerical methods for the MHD equations -- Ch. 11. Numerical experiments -- Ch. 12. Summary of the book