Non-linear singular problems in $p$-adic analysis: associative algebras of $p$-adic distributions

We propose an algebraic theory which can be used for solving both linear and non-linear singular problems of $p$-adic analysis related to $p$-adic distributions (generalized functions). We construct the $p$-adic Colombeau–Egorov algebra of generalized functions, in which Vladimirov's pseudo-differential operator plays the role of differentiation. This algebra is closed under Fourier transformation and associative convolution. Pointvalues of generalized functions are defined, and it turns out that any generalized function is uniquely determined by its pointvalues. We also construct an associative algebra of asymptotic distributions, which is generated by the linear span of the set of associated homogeneous $p$-adic distributions. This algebra is embedded in the Colombeau–Egorov algebra as a subalgebra. In addition, a new technique for constructing weak asymptotics is developed.