Exact Methods for Nonlinear PDEs

The book describes and illustrates how to apply effective analytical methods to obtain exact solutions to nonlinear differential equations of mathematical physics and other partial differential equations.

Nonlinear partial differential equations of the second and higher orders, including nonlinear equations of mathematical physics, are commonly encountered in various fields of mathematics, physics, mechanics, chemistry, and biology and in numerous applications. Finding a general solution to these equations is only possible in exceptional cases.

Therefore, one usually has to limit themselves to identifying and analyzing particular solutions, which are referred to as exact solutions. A solution is exact if, when substituted into the mathematical equation, it turns the equation into an identity. No approximations or simplifications of the equation are allowed, and no a priori assumptions are used.

Exact solutions to equations of mathematical physics greatly enhance our understanding of the qualitative features of many phenomena and processes in various fields of natural science. Simple solutions of linear and nonlinear differential equations are widely used to illustrate theoretical material and various applications in university and college courses on applied mathematics, theoretical physics, fluid dynamics, nonlinear optics, and other disciplines.

Exact solutions are also necessary to update and improve relevant sections of computer software for analytical calculations (such as computer algebra systems like Mathematica, Maple, MATLAB, and others). The book includes examples and exercises to help the reader better understand the material and develop practical skills in solving nonlinear differential equations. This book will be helpful for a wide range of researchers, university teachers, and engineers, as well as graduate postgraduate and PhD students.