New logarithmic power nonlinear Schrödinger equations with super-Gaussons

We introduce a new class of nonlinear Schrödinger (NLS) 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≥1 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 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 affects 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.