New logarithmic power nonlinear Schr\"odinger equations with super-Gaussons

TL;DR

This paper introduces a new class of nonlinear Schrödinger equations with logarithmic-power nonlinearity, which admits exact super-Gaussian localized solutions called super-Gaussons, expanding the types of flat-top solitons possible in nonlinear media.

Contribution

The paper develops the logp-NLS model that generalizes the standard log-NLS, allowing for a broader family of flat-top solitons with tunable profiles and dynamics, supported by analytical and numerical analysis.

Findings

Super-Gaussons exhibit flat-top profiles with sharp edges.

The parameter p controls the flatness and decay sharpness of solitons.

Numerical simulations show how p influences soliton structure and collision behavior.

Abstract

We introduce a new class of nonlinear Schr\"odinger equations with a logarithmic-power nonlinearity that admits exact localized solutions of super-Gaussian form. The resulting stationary states possess flat-top profiles with sharp edges and are referred to as super-Gaussons, in analogy with the Gaussian Gaussons of the classical logarithmic NLS (log-NLS). The model, which we call the logarithmic-power NLS (logp-NLS), is parameterized by an exponent $p\geq1$ that controls the degree of flatness of the soliton core and the sharpness of its decay. Mathematically, $p$ interpolates between the standard log-NLS ($p=1$) and increasingly flat-top profiles as $p$ increases, while physically it governs the stiffness of an underlying logarithmic-power compressibility law. The proposed equation is constructed so as to admit super-Gaussian stationary states and can be interpreted a posteriori within a generalized pressure-law framework, thereby extending the log-NLS. We investigate the dynamics of super-Gaussons in one spatial dimension through numerical simulations for various values of $p$, demonstrating how this parameter regulates both the internal structure of the soliton and its collision dynamics. The logp-NLS thus generalizes the standard log-NLS by admitting a broader family of localized states with distinctive structural and dynamical properties, suggesting its relevance for flat-top solitons in nonlinear optics, Bose-Einstein condensates, and related nonlinear media.