# The role of the massless phantom term in the stability of a non-topological soliton solution

## TL;DR

This paper introduces a modified Lagrangian with a massless phantom term that stabilizes a non-topological soliton solution, ensuring its stability and reducing to the nonlinear Klein-Gordon equations.

## Contribution

It proposes a novel Lagrangian density incorporating a phantom term to achieve stable non-topological solitons with standard Klein-Gordon dynamics.

## Key findings

- The phantom term stabilizes the soliton against perturbations.
- The resulting equations reduce to the nonlinear Klein-Gordon form.
- The approach guarantees classical stability of the soliton solution.

## Abstract

We intend to introduce classically a special Lagrangian density in such a way that, firstly, it leads to a special non-topological solitary wave solution, secondly, the stability of that is guaranteed properly, and thirdly, its dominant dynamical equations reduce to the standard nonlinear Klein-Gordon equations. For these purposes, we have to consider a new term in the Lagrangian density, whose role is like a massless phantom that surrounds the special solitary wave solution and resists any change in its internal structure.