On the Nonlinear Dynamical Equation in the p-adic String Theory

TL;DR

This paper investigates nonlinear pseudo-differential equations with infinite derivatives from p-adic string theory, establishing uniqueness, boundary conditions, and solutions including solitons and q-branes, with implications for string theory and cosmology.

Contribution

It provides a systematic mathematical analysis of these equations, including uniqueness theorems, boundary problem solutions, and multidimensional solutions relevant to string theory.

Findings

Uniqueness of solutions in a specific algebra of distributions.

Existence of space-homogeneous solutions for odd p.

Non-existence of continuous solutions for even p, suggesting possible discontinuous solutions.

Abstract

In this work nonlinear pseudo-differential equations with the infinite number of derivatives are studied. These equations form a new class of equations which initially appeared in p-adic string theory. These equations are of much interest in mathematical physics and its applications in particular in string theory and cosmology. In the present work a systematical mathematical investigation of the properties of these equations is performed. The main theorem of uniqueness in some algebra of tempored distributions is proved. Boundary problems for bounded solutions are studied, the existence of a space-homogenous solution for odd p is proved. For even p it is proved that there is no continuous solutions and it is pointed to the possibility of existence of discontinuous solutions. Multidimensional equation is also considered and its soliton and q-brane solutions are discussed.