Algebraic methods in nonlinear perturbation theory / V. N. Bogaevski, A. Povzner

he perturbation theory of the book concerns nonlinear systems of ordinary differential equations of first order containing a small parameter. The authors turn over from such a nonlinear system to the corresponding partial differential equation, for which the given system is the characteristic one. Using for the linear equation the familiar transformations to matrix normal forms, they obtain a formalism for calculating the asymptotic expansion of the wanted solutions. The asymptotic problems are classified into regular ones, where the asymptotics are valid in the whole domain of the variables, and singular ones, where different asymptotics must be constructed in various subdomains and a matching procedure must be applied in the overlapping domain. The book begins with the matrix perturbation theory and ends with an extension of the formalism to some partial differential equations. Emphasis is laid on a series of examples, which both illustrate the theory in detail and show the usefulness of the method for the solution of concrete applications in different parts of physics. (zbMath)