14 - Nonlinear theories of p-adic generalized functions

Summary

Introduction

In standard mathematical physics (in the real setting) there are problems which require the definition of products of distributions (generalized functions) [66], [68], [86], [97], [194]. Such problems appear in quantum mechanics [53], [9], [27], quantum field theory, some problems of gas dynamics, elasticity theory, and also in the description of, e.g., shock waves, δ-shock waves, and typhoons. In the framework of the approaches connected to problems of multiplications of distributions, a theory of singular solutions of non-linear equations has been developed [8], [68]–[71], [149], [150], [195], [220]. Solving problems of this kind requires the development of special analytical methods, the construction of algebras containing the space of distributions, and the development of a technique for constructing singular asymptotics. As a result, the demand arises for a construction of a nonlinear theory of generalized functions. Besides, the development of nonlinear theories of distributions is of great interest in itself.

Since p-adic mathematical physics is a relatively young science, p-adic analogs of the above mentioned problems have not been studied so far (to the best of our knowledge). The problems of p-adic analysis related to the theory of p-adic distributions which have been solved up to now are of the linear type. To deal with nonlinear singular problems one needs some additional technique similar to that developed in the usual real mathematical physics mentioned above.