# Pseudo Spectral Transform for Schr\"odinger--Poisson Equations

## TL;DR

This paper introduces a novel pseudo spectral transform method to find exact stationary solutions to the Schr"odinger--Poisson equations, addressing technical challenges through a generalized spectral approach and integral convergence analysis.

## Contribution

It develops a new pseudo spectral transform technique for solving Schr"odinger--Poisson equations, extending the Korteweg--de Vries kernel to overcome previous methodological limitations.

## Key findings

- Derived exact stationary solutions using the pseudo spectral transform.
- Established convergence of the integral representation.
- Compared the new method with homotopy analysis approach.

## Abstract

Here we present exact, stationary, parametric solutions to the Schr\"odinger--Poisson system. We confront two images: on one hand, we draw on the homotopy analysis method which leads us to a nonlinear integral scheme. Indeed, this approach might be simplified by looking for sufficiently smooth solutions vanishing asymptotically. However, since our system possesses stiffness an additional analysis has to be considered. On the other hand, we seek for exact solutions over the inverse scattering method by introducing a pseudo spectral transform. In fact, this pseudo spectral method generalises Korteweg--de Vries family's kernel and let us to circumvent some technical difficulties originally arisen in our first approach although, again, we come to an integral representation, which we test for convergence.