🌀 nonlinear sciences

New logarithmic power nonlinear Schrödinger equations with super-Gaussons

This paper introduces the logarithmic-power nonlinear Schrödinger equation (logp-NLS), a generalized model parameterized by an exponent $p$ that admits exact "super-Gausson" solitons with flat-top profiles and sharp edges, thereby extending the classical logarithmic NLS to describe a broader family of localized states relevant to nonlinear optics and Bose-Einstein condensates.

Original authors: Hadi Susanto

Published 2026-02-12

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CC BY 4.0

Original authors: Hadi Susanto

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are watching a ripple in a pond. Usually, these ripples spread out and fade away. But in the world of advanced physics, there are special, self-contained waves called solitons that can travel long distances without losing their shape. They are like perfect, solitary surfers who never fall off their board.

For decades, scientists have studied a specific type of wave equation (the Nonlinear Schrödinger equation) that describes these waves. One famous version of this equation produces waves that look like a smooth, rounded hill—mathematically known as a Gaussian shape (think of a perfect bell curve). These were called "Gaussons."

The Problem:
In the real world, not all waves look like smooth hills. Sometimes, in things like lasers or super-cold gases (Bose-Einstein condensates), waves have a flat top with very sharp, cliff-like edges. They look more like a table or a cylinder than a hill. The old math couldn't describe these "flat-top" waves perfectly.

The Solution: The "Super-Gausson"
Hadi Susanto, a mathematician at Khalifa University, has invented a new mathematical model to fix this. He calls it the logarithmic-power NLS (or logp-NLS for short).

Here is the simple breakdown of what he did:

1. The "Dial" (The Parameter $p$ )

Think of the old model as a camera with a fixed lens that only takes pictures of smooth, round hills. Susanto added a zoom dial (called the exponent $p$ ) to the camera.

When the dial is set to 1: You get the old, smooth, round hill (the classic Gaussian).
When you turn the dial higher (p > 1): The hill starts to flatten out. The top becomes a flat plateau, and the sides become steeper cliffs.
The Result: A new type of wave called a "Super-Gausson." It looks like a floating table or a flat-topped mountain.

2. The "Stiffness" of the Wave

Why does the wave flatten out?
Imagine the wave is made of a special kind of jelly.

In the old model, the jelly is soft and squishy everywhere. If you push it, it deforms easily, creating a smooth curve.
In Susanto's new model, the jelly has a magic property: inside the center of the wave, it becomes infinitely stiff. It acts like a solid block of concrete. You can't squish the middle down; it stays perfectly flat.
However, at the very edges, the jelly suddenly becomes soft again, allowing the wave to drop off sharply. This creates that "flat-top with sharp edges" look.

3. How They Crash Into Each Other

The paper also looked at what happens when two of these waves crash into each other.

The Old Way (Smooth Hills): When two smooth waves collide, they bounce off each other like billiard balls, but they lose a little energy and wiggle a bit. They aren't "perfect" bouncers.
The New Way (Flat Tables):
- Slow Collisions: If two "table" waves move slowly toward each other, they overlap for a long time. Because the middle is so stiff, they get very excited, wiggling and shaking (breathing) as they pass. They lose a lot of energy to this shaking.
- Fast Collisions: If they zoom past each other very quickly, the stiffness helps! They act like two rigid blocks of ice sliding past one another. They barely deform and bounce off with very little energy loss.

Why Does This Matter?

This isn't just a math game. These "flat-top" waves actually exist in nature:

Lasers: High-power laser beams often have flat tops rather than round centers.
Super-cold Gases: In Bose-Einstein condensates (matter cooled to near absolute zero), atoms can form these flat, table-like structures.
Optics: Designing better fiber optics and lenses requires understanding these flat shapes.

In Summary:
Susanto took a classic equation that only knew how to make round waves and gave it a "flatness knob." This new equation explains how nature creates those flat-topped, sharp-edged waves we see in advanced technology and physics. It's like upgrading from a model that only knows how to draw circles to one that can draw perfect squares and tables, too.

1. Problem Statement

The standard Nonlinear Schrödinger (NLS) equation with cubic nonlinearity is a fundamental model in nonlinear science but fails to describe physical regimes where self-trapped states exhibit flat-top profiles rather than Gaussian or sech-shaped profiles. While the classical logarithmic NLS (log-NLS) admits exact localized solutions called "Gaussons" (Gaussian profiles), it is strictly limited to these Gaussian shapes.

Many physical systems, including nonlinear optics and Bose-Einstein condensates (BECs), exhibit states with nearly constant amplitude over a finite core and sharp decay at the edges (super-Gaussian profiles). The paper addresses the lack of a mathematical model that:

Admits exact analytical solutions for these flat-top/super-Gaussian states.
Generalizes the log-NLS to allow for tunable "flatness" of the soliton core.
Maintains a Hamiltonian structure and physical interpretability via generalized compressibility laws.

2. Methodology

The author employs a combination of analytical derivation, variational calculus, and numerical simulation to develop and analyze a new class of equations.

Analytical Derivation: The paper constructs a new equation, the logarithmic-power NLS (logp-NLS), by reverse-engineering the nonlinearity required to admit a specific super-Gaussian ansatz as an exact stationary solution.
- Ansatz: $\psi(x, t) = e^{-r^{2p}} e^{i\theta}$ , where $r = |x|$ and $p \geq 1$ .
- Equation Formulation: The nonlinearity is derived to balance the radial Laplacian acting on the ansatz, resulting in a term involving $(\ln^2 |\psi|)^{(p-1)/2p}$ .
Variational and Hydrodynamic Analysis:
- The model is placed within a Hamiltonian framework, deriving the Lagrangian and energy functionals.
- Using Madelung variables ( $\psi = \sqrt{\rho}e^{iS}$ ), the equation is interpreted as a compressible quantum fluid. The author derives the associated enthalpy and pressure law, showing that the model corresponds to a fluid with a logarithmic-power compressibility law.
- The behavior of the pressure derivative near the plateau ( $\rho \approx 1$ ) is analyzed to explain the formation of flat-top profiles (infinite stiffness).
Linear Stability Analysis:
- A perturbation analysis is performed on the stationary solutions ( $\psi = \psi_0 + \epsilon(v+iw)$ ).
- This leads to a linear eigenvalue problem involving coupled Schrödinger-type operators with anharmonic and singular potentials.
Numerical Simulations:
- Regularization: Due to the singular derivative of the nonlinearity at $|\psi|=1$ , a regularization scheme is introduced: $\ln_\delta(|\psi|) = \frac{1}{2}\ln(|\psi|^2 + \delta)$ .
- Dynamics: Direct numerical simulations (using Fourier pseudospectral methods and RK4 time integration) are conducted in 1D to study collision dynamics and breathing modes for various $p$ values.

3. Key Contributions

Introduction of the logp-NLS: A new equation parameterized by an exponent $p \geq 1$ that interpolates between the standard log-NLS ( $p=1$ ) and increasingly flat-top profiles ( $p > 1$ ).
Discovery of "Super-Gaussons": The identification of exact localized stationary solutions in the form of super-Gaussian functions. These solutions feature a flat core and sharp edges, distinct from the Gaussian Gaussons of the classical model.
Physical Interpretation: The derivation of a generalized pressure law where the exponent $p$ governs the stiffness of the medium. The paper demonstrates that as $p$ increases, the medium becomes locally incompressible in the soliton core, enforcing the flat-top structure.
Stability and Dynamics Characterization:
- Proof of linear stability (spectrum lies on the imaginary axis).
- Demonstration that the model is non-integrable (collisions are inelastic).
- Discovery of a counter-intuitive dynamic behavior: While low-velocity collisions become more inelastic as $p$ increases (due to strong internal mode excitation), high-velocity collisions become more elastic (due to the effective rigidity of the flat-top solitons acting like incompressible droplets).

4. Results

Structural Properties:
- For $p=1$ , the solution is the standard Gaussian Gausson.
- As $p \to \infty$ , the solution approaches a cylinder-like profile with a sharp cutoff at $r=1$ .
- The pressure derivative $P'(\rho)$ diverges as $\rho \to 1$ , creating a "stiff" restoring force that prevents density variations in the core, while the quantum pressure term regularizes the sharp edges.
Linear Stability: Numerical spectral analysis confirms that the linearized operator around the super-Gausson has a purely discrete spectrum on the imaginary axis, indicating linear stability for the tested cases (e.g., $p=2$ ).
Collision Dynamics:
- Inelasticity: Collisions between super-Gaussons are inelastic, resulting in energy loss to radiation and persistent "breathing" (oscillations in width/amplitude).
- Velocity Dependence:
  - Low Velocity: High $p$ leads to stronger inelasticity and larger residual breathing due to prolonged core overlap and strong nonlinear pressure gradients.
  - High Velocity: High $p$ leads to more elastic scattering. The "stiffness" of the flat-top core prevents deformation during rapid impacts, causing solitons to pass through each other with minimal energy exchange.
Numerical Challenges: The paper highlights the numerical stiffness inherent in the logarithmic-power term near $|\psi|=1$ , necessitating regularization ( $\delta \approx 10^{-15}$ ) and initial amplitudes slightly below unity to ensure stable long-time integration.

5. Significance

Theoretical Advancement: The logp-NLS provides a systematic Hamiltonian extension of the log-NLS, bridging the gap between Gaussian and flat-top solitons. It offers a mathematically tractable model for a broader class of localized states.
Physical Relevance: The model suggests new physical mechanisms for flat-top solitons in nonlinear optics, Bose-Einstein condensates, and granular media. The "stiffness" parameter $p$ offers a theoretical link to material properties where compressibility laws deviate from standard power laws.
Dynamical Insight: The findings on collision dynamics reveal a rich interplay between nonlinearity exponent and collision speed, challenging the intuition that "stiffer" solitons always behave more elastically. The results suggest that the nature of inelasticity is highly dependent on the interaction timescale relative to the internal restoring forces.
Future Directions: The paper sets the stage for investigating the analytical spectrum of the linearized operator, multi-soliton interactions, and experimental realization of these super-Gaussian states in physical media.

1. The "Dial" (The Parameter ppp)