  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.
