Pseudo Spectral Transform for Schrödinger--Poisson Equations

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Here we present exact, stationary, parametric solutions to the Schrödinger--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.