# Local smooth solutions of the nonlinear Klein-gordon equation

## TL;DR

This paper proves the local existence of smooth, non-vanishing solutions to the nonlinear Klein-Gordon equation and extends the method to certain nonlinear Dirac equations, using a pseudo-differential operator approach.

## Contribution

It introduces a novel method for establishing local smooth solutions of nonlinear Klein-Gordon and Dirac equations that do not vanish, expanding the understanding of these equations' solution spaces.

## Key findings

- Existence of arbitrarily smooth, non-vanishing solutions for Klein-Gordon equation.
- Application of pseudo-differential operator techniques to nonlinear PDEs.
- Extension of the method to nonlinear Dirac equations.

## Abstract

Given any $\mu_1, \mu_2\in {\mathbb C}$ and $\alpha >0$, we prove the local existence of arbitrarily smooth solutions of the nonlinear Klein-Gordon equation $\partial_{ tt } u - \Delta u + \mu_1 u = \mu_2 |u|^\alpha u$ on ${\mathbb R}^N$, $N\ge 1$, that do not vanish, i.e. $ |u (t,x) | >0 $ for all $x \in {\mathbb R}^N$ and all sufficiently small $t$. We write the equation in the form of a first-order system associated with a pseudo-differential operator, then use a method adapted from~[Commun. Contemp. Math. {\bf 19} (2017), no. 2, 1650038]. We also apply a similar (but simpler than in the case of the Klein-Gordon equation) argument to prove an analogous result for a class of nonlinear Dirac equations.