The equation of the -adic open string for the scalar tachyon field

Abstract

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.