# Parabolic solutions for the planar $N$-centre problem: multiplicity and scattering

## TL;DR

This paper proves the existence of special parabolic trajectories in the planar N-centre problem with prescribed asymptotic directions and topological properties, extending understanding of solutions in gravitational systems with multiple centers.

## Contribution

It introduces new methods to establish the existence of entire parabolic solutions with specific asymptotic and topological features in the N-centre problem.

## Key findings

- Existence of entire parabolic trajectories with prescribed asymptotic directions.
- Construction of solutions with specific topological characteristics.
- Extension of solution classes for the planar N-centre problem.

## Abstract

For the planar $N$-centre problem $$ \ddot x = - \sum_{i=1}^N \frac{m_i (x-c_i)}{| x - c_i|^{\alpha+2}}, \qquad x \in \mathbb{R}^2 \setminus \{ c_1,\ldots,c_N \}, $$ where $m_i > 0$ for $i=1,\ldots,N$ and $\alpha \in [1,2)$, we prove the existence of entire parabolic trajectories, having prescribed asymptotic directions for $t \to \pm\infty$ and prescribed topological characterization with respect to the set of the centres.