Vasilii S Vladimirov

The equation of the

We study the structure of solutions of the one-dimensional non-linear pseudodifferential equation describing the dynamics of the -adic open string for the scalar tachyon field . We explain the role of real zeros of the entire function and the behaviour of solutions in the neighbourhood of these zeros. We point out that discontinuous solutions can appear if is even. We use the method of expanding the solution and the function in Hermite polynomials and modified Hermite polynomials and establish a connection between the coefficients of these expansions (integral conservation laws). For we construct an infinite system of non-linear equations in the unknown Hermite coefficients and study its structure. We consider the 3-approximation. We indicate a connection between the problems stated and a non-linear boundary-value problem for the heat equation.

p

The future tube in ℂ n +1 is the unbounded domain τ+ = z = (z o,...,z n ) ∈ ℂ n +1 : (Im z o)2 > (Imz 1)2 +...+ (Im z n )2 Im z o > 0. In other words, τ+ is a tube domain over the future cone V + = y ∈ ℝ n +1 : y 2 0 > y 2 1 +...+ y 2 n , y 0 > 0. The domain τ+ is biholomorphically equivalent to a classical Cartan domain of the IVth type, hence to a bounded symmetric domain in ℂ n +1. The future tube τ+ in ℂ4 (n = 3) is important in mathematical physics, especially in axiomatic quantum field theory, being the natural domain of definition of holomorphic relativistic fields. These specific features of the future tube motivated its investigation by mathematicians and physicists. Beginning with Elie Cartan’s classification of bounded symmetric domains, these domains were examined in many papers where the complex structure of their boundaries, integral representations, boundary values of holomorphic functions and so on were considered. The proof of the “edge-of-the-wedge” theorem by N.N. Bogolubov generated the rapid development of applications of the theory of several complex variables to axiomatic quantum field theory. During this period the “C-convex hull” and “finite covariance” theorems were proved, the Jost-Lehmann-Dyson representation was found et cetera. Recently R. Penrose has proposed a transformation connecting holomorphic solutions of the basic equations of field theory with analytic sheaf cohomologies of domains in ℝℙ3. These two directions developed, to a large extent, independently from each other, and some important results obtained in axiomatic quantum field theory still remain unknown to specialists in several complex variables and differential geometry. One of the goals of this paper is to give a unified presentation of advances in complex analysis in the future tube and related domains achieved in both of these directions.

-adic open string for the scalar tachyon field

The paper is concerned with the construction of exact solutions for nonlinear pseudodifferential equations which describe tachyon dynamics of open-closed p-adic strings. Existence of continuous solutions and their properties are discussed.

Complex Analysis in the Future Tube

We present a brief review of the scientific work and achievements of Igor V. Volovich on the occasion of his 65th birthday.

On some exact solutions in p-adic open-closed string theory

This work is devoted to the mathematical description of the dynamics of tachyons of open, closed and open-closed p-adic strings. The questions of existence and nonexistence of continuous solutions and their properties, as well zero structure of solutions is discussed. New multidimensional nonlinear equations of ultraparabolic type are obtained. Some unsolved problems are listed.