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

英文摘要：

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 $p^{\frac12\partial^2_t}Φ=Φ^p$. We elicit the role of real zeros of the entire function $Φ^p(z)$ and the behaviour of solutions $Φ(t)$ 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 $Φ$ and the function $Φ^p$ 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.