New logarithmic power nonlinear Schrödinger equations with super-Gaussons

TL;DR

The paper introduces the logp-NLS, a nonlinear Schrödinger equation with a logarithmic–power nonlinearity that supports exact stationary super-Gaussian solitons (super-Gaussons) with profiles $\psi(\mathbf{x},t)=e^{-r^{2p}}\,e^{i\theta}$ for $p\ge 1$. By balancing the radial Laplacian with a generalized, logarithmic–power enthalpy, the model provides a Hamiltonian framework and a hydrodynamic interpretation via a density-dependent pressure law, yielding flat-top cores whose sharp edges arise from a stiff restoring force away from the plateau. The authors analyze the 1D dynamics, establish linear spectral stability (discrete, purely imaginary spectrum) for various $p$, and investigate inelastic soliton collisions, revealing how increasing $p$ strengthens core flatness and modulates collision outcomes. They also address numerical stiffness near $|\psi|=1$ with a regularization strategy, outlining future directions in spectral analysis, long-time dynamics, and potential applications in nonlinear optics and Bose–Einstein condensates.

Abstract

We introduce a new class of nonlinear Schrödinger 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.

Figures (2)

• Figure 1: (a) Stationary super-Gausson profiles $\psi$ for several values of the exponent $p$, illustrating the transition from the Gaussian profile ($p=1$) to increasingly flat-top shapes as $p$ increases. (b) Spectrum of the linearized operator around the super-Gausson for $p=2$, shown in the complex plane. The spectrum consists solely of discrete eigenvalues, with no continuous spectrum present. All eigenvalues lie on the imaginary axis, indicating linear stability of the super-Gausson.
• Figure 2: Inelastic collisions of two super-Gaussons, initially separated, with $\omega=0.01$. Shown is the top view of $|\psi(x,t)|^{1/2}$.