On the equation of the $p$-adic open string for the scalar tachyon field

TL;DR

This paper analyzes solutions of a non-linear pseudodifferential equation modeling the $p$-adic open string for scalar tachyon fields, focusing on zeros of solutions, discontinuities, and solution expansions using Hermite polynomials.

Contribution

It introduces a novel analysis of the $p$-adic string equation, including solution structure, zeros, discontinuities, and a Hermite polynomial expansion method, especially for the case p=2.

Findings

Discontinuous solutions can occur if p is even.

Hermite polynomial expansion links solution coefficients to conservation laws.

For p=2, an infinite nonlinear system in Hermite coefficients is constructed.

Abstract

We study the structure of solutions of the one-dimensional non-linear pseudodifferential equation describing the dynamics of the $p$-adic open string for the scalar tachyon field $p^{\frac12\partial^2_t}\Phi=\Phi^p$. We elicit the role of real zeros of the entire function $\Phi^p(z)$ and the behaviour of solutions $\Phi(t)$ in the neighbourhood of these zeros. We point out that discontinuous solutions can appear if $p$ is even. We use the method of expanding the solution $\Phi$ and the function $\Phi^p$ in the Hermite polynomials and modified Hermite polynomials and establish a connection between the coefficients of these expansions (integral conservation laws). For $p=2$ 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 the non-linear boundary-value problem for the heat equation.