# Localized Analytical Solutions and Parameters Analysis in the Nonlinear Dispersive Gross-Pitaevskii Mean-Field GP (m,n) Model with Space-Modulated Nonlinearity and Potential

## TL;DR

This paper explores new localized solutions in a nonlinear dispersive Gross-Pitaevskii model with space-dependent parameters, revealing diverse structures and solutions for different parameter regimes using self-similar transformations.

## Contribution

It introduces novel envelope compacton-like and spikon-like solutions in the GP(m,n) equation with space-modulated nonlinearity and potential, expanding the understanding of localized structures.

## Key findings

- Existence of novel localized solutions in the GP(m,n) model.
- Diverse structures obtained by choosing different self-similar functions.
- Solutions differ significantly from traditional compacton and spikon solutions.

## Abstract

The novel nonlinear dispersive Gross-Pitaevskii (GP) mean-field model with the space-modulated nonlinearity and potential (called GP(m, n) equation) is investigated in this paper. By using self-similar transformations and some powerful methods, we obtain some families of novel envelope compacton-like solutions spikon-like solutions to the GP(n, n) (n>1) equation. These solutions possess abundant localized structures because of infinite choices of the self-similar function X(x). In particular, we choose X(x) as the Jacobi amplitude function am(x,k) and the combination of linear and trigonometric functions of space x so that the novel localized structures of the GP(2,2) equation are illustrated, which are much different from the usual compacton and spikon solutions reported. Moreover, it is shown that GP(m,1) equation with linear dispersion also admits the compacton-like solutions for the case 0<m<1 and spikon-like solutions for the case m<0.