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A generalized Liouville equation and magnetic stability

Alireza Ataei, Douglas Lundholm, Dinh-Thi Nguyen

TL;DR

This work provides a rigorous, unified treatment of two nonlinear 2D problems anchored in mathematical physics: an explicit generalized Liouville equation solvable via Wronskians of coprime polynomials and a magnetic interpolation inequality with a self-generated field. The Liouville analysis yields a complete classification of weak solutions through ψ_{P,Q} = log(8) − 2 log(|P|^2+|Q|^2) with f = P'Q − PQ' and a clear U(2) symmetry on the solution space. In the magnetic setting, the authors establish sharp LGN-type bounds with a self-consistent Pauli-type energy, prove a supersymmetric factorization, and show β-quantization and minimizer structure: γ_*(β) = 2πβ for β ≥ 2 with minimizers expressed as u_{P,Q}, existing only for β ∈ 2ℕ. These results have direct implications for the stability of almost-bosonic anyon gases (Keller–Lieb–Thirring bounds) and provide a rigorous link between nonlinear Liouville densities and vortex solitons in Chern–Simons–Higgs-type theories. Overall, the paper delivers a robust analytical framework connecting conformal Liouville geometry, nonlocal magnetic interactions, and quantum stability in 2D systems, with explicit formulas and symmetry classifications for the minimizers.

Abstract

This work considers two related families of nonlinear and nonlocal problems in the plane $\mathbb{R}^2$. The first main result derives the general integrable solution to a generalized Liouville equation using the Wronskian of two coprime complex polynomials. The second main result concerns an application to a generalized Ladyzhenskaya-Gagliardo-Nirenberg interpolation inequality, with a single real parameter $β$ interpreted as the strength of a magnetic self-interaction. The optimal constant of the inequality and the corresponding minimizers of the quotient are studied and it is proved that for $β\ge 2$, for which the constant equals $2πβ$, such minimizers only exist at quantized $β\in 2\mathbb{N}$ corresponding to nonlinear generalizations of Landau levels with densities solving the generalized Liouville equation. This latter problem originates from the study of self-dual vortex solitons in the abelian Chern-Simons-Higgs theory and from the average-field-Pauli effective theory of anyons, i.e. quantum particles with statistics intermediate to bosons and fermions. An immediate application is given to Keller-Lieb-Thirring stability bounds for a gas of such anyons which self-interact magnetically (vector nonlocal repulsion) as well as electrostatically (scalar local/point attraction), thus generalizing the stability theory of the 2D cubic nonlinear Schrödinger equation.

A generalized Liouville equation and magnetic stability

TL;DR

This work provides a rigorous, unified treatment of two nonlinear 2D problems anchored in mathematical physics: an explicit generalized Liouville equation solvable via Wronskians of coprime polynomials and a magnetic interpolation inequality with a self-generated field. The Liouville analysis yields a complete classification of weak solutions through ψ_{P,Q} = log(8) − 2 log(|P|^2+|Q|^2) with f = P'Q − PQ' and a clear U(2) symmetry on the solution space. In the magnetic setting, the authors establish sharp LGN-type bounds with a self-consistent Pauli-type energy, prove a supersymmetric factorization, and show β-quantization and minimizer structure: γ_*(β) = 2πβ for β ≥ 2 with minimizers expressed as u_{P,Q}, existing only for β ∈ 2ℕ. These results have direct implications for the stability of almost-bosonic anyon gases (Keller–Lieb–Thirring bounds) and provide a rigorous link between nonlinear Liouville densities and vortex solitons in Chern–Simons–Higgs-type theories. Overall, the paper delivers a robust analytical framework connecting conformal Liouville geometry, nonlocal magnetic interactions, and quantum stability in 2D systems, with explicit formulas and symmetry classifications for the minimizers.

Abstract

This work considers two related families of nonlinear and nonlocal problems in the plane . The first main result derives the general integrable solution to a generalized Liouville equation using the Wronskian of two coprime complex polynomials. The second main result concerns an application to a generalized Ladyzhenskaya-Gagliardo-Nirenberg interpolation inequality, with a single real parameter interpreted as the strength of a magnetic self-interaction. The optimal constant of the inequality and the corresponding minimizers of the quotient are studied and it is proved that for , for which the constant equals , such minimizers only exist at quantized corresponding to nonlinear generalizations of Landau levels with densities solving the generalized Liouville equation. This latter problem originates from the study of self-dual vortex solitons in the abelian Chern-Simons-Higgs theory and from the average-field-Pauli effective theory of anyons, i.e. quantum particles with statistics intermediate to bosons and fermions. An immediate application is given to Keller-Lieb-Thirring stability bounds for a gas of such anyons which self-interact magnetically (vector nonlocal repulsion) as well as electrostatically (scalar local/point attraction), thus generalizing the stability theory of the 2D cubic nonlinear Schrödinger equation.
Paper Structure (21 sections, 43 theorems, 331 equations, 2 figures)

This paper contains 21 sections, 43 theorems, 331 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. The Liouville problem
  3. The magnetic interpolation problem
  4. An equivalent optimization problem and stability in quantum mechanics
  5. Generalized Liouville equation
  6. Preliminary observations
  7. Proof of Theorem \ref{['thm:liouville']}
  8. Sufficient form of solutions
  9. Necessary form of solutions
  10. The symmetry space of the solutions
  11. Some special solutions of interest
  12. Magnetic stability
  13. Preliminaries on magnetic inequalities
  14. Variational equation
  15. Supersymmetric magnetic identity
  16. ...and 6 more sections

Key Result

Theorem 1

Let $f\colon \mathbb{C} \to \mathbb{C}$ be a nonzero polynomial. All the weak solutions $\psi \in L^1_\mathrm{loc}(\mathbb{R}^2;\mathbb{R})$ of the equation eq:Liouville-psi, such that $\int_{\mathbb{R}^2} |f|^2 e^{\psi} < \infty$, are of the form where $P,Q$ are two coprime complex polynomials which satisfy $f= P' Q -P Q'$. Moreover, and if $(P,Q)$ and $(\tilde{P},\tilde{Q})$ are pairs of polyn

Figures (2)

  • Figure 1: Our current upper and lower bounds for $\gamma_*(\beta)$; see \ref{['rmk:bermuda']}. The upper bound from the left (green) is given by Townes soliton $u_0$ and the one from the right (blue) by the lowest "vortex ring" $u_1$. The lower bounds are $C_{\rm LGN}$ (blue), $2\pi\beta$ (red) and \ref{['eq:gamma-bounds-refined']} (yellow). Horiz./vert. axis for $\beta$ / $\gamma$.
  • Figure 2: Construction of squares.

Theorems & Definitions (106)

  • Theorem 1: Generalized Liouville equation
  • Definition 1.1: Self-generated magnetic field and energy functional
  • Definition 1.2: Self-magnetic minimization problem
  • Theorem 2: Magnetic stability
  • Remark 1.3: Supersymmetry
  • Remark 1.4: Vorticity and the degree of zeros of the minimizers
  • Remark 1.5: Vortex quantization
  • Remark 1.6: Soliton manifold completeness
  • Remark 1.7: Improved bounds
  • Remark 1.8: Extended interpolation
  • ...and 96 more