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Solutions of differential equations in Freud-weighted Sobolev spaces

Maxime Breden, Hugo Chu

TL;DR

The paper develops a rigorous framework for solving differential equations in Freud-weighted Sobolev spaces, uniting Freud-type orthogonal polynomials, Poincaré inequalities, and discrete Painlevé dynamics. It constructs Sobolev orthogonal bases and leverages a bidiagonal differentiation structure to obtain quantitative embedding and compactness results in the non-classical Freud setting, including a sharp bound for the quartic potential. The authors then apply these tools to the Gross–Pitaevskii equation with sextic potential, proving existence of standing-wave solutions via computer-assisted proofs that combine spectral methods with Newton–Kantorovich-type fixed-point arguments. Overall, the work provides a rigorous, numerically effective approach for analysis and computation of PDEs in Freud-weighted spaces and demonstrates CAPs on unbounded domains using spectral bases not diagonalizing the differential operator.

Abstract

We lay some mathematically rigorous foundations for the resolution of differential equations with respect to semi-classical bases and topologies, namely Freud-Sobolev polynomials and spaces. In this quest, we uncover an elegant theory melding various topics in Numerical and Functional Analysis: Poincaré inequalities, Sobolev orthogonal polynomials, Painlevé equations and more. Brought together, these ingredients allow us to quantify the compactness of Sobolev embeddings on Freud-weighted spaces and finally resolve some differential equations in this topology. As an application, we rigorously and tightly enclose solutions of the Gross-Pitaevskii equation with sextic potential.

Solutions of differential equations in Freud-weighted Sobolev spaces

TL;DR

The paper develops a rigorous framework for solving differential equations in Freud-weighted Sobolev spaces, uniting Freud-type orthogonal polynomials, Poincaré inequalities, and discrete Painlevé dynamics. It constructs Sobolev orthogonal bases and leverages a bidiagonal differentiation structure to obtain quantitative embedding and compactness results in the non-classical Freud setting, including a sharp bound for the quartic potential. The authors then apply these tools to the Gross–Pitaevskii equation with sextic potential, proving existence of standing-wave solutions via computer-assisted proofs that combine spectral methods with Newton–Kantorovich-type fixed-point arguments. Overall, the work provides a rigorous, numerically effective approach for analysis and computation of PDEs in Freud-weighted spaces and demonstrates CAPs on unbounded domains using spectral bases not diagonalizing the differential operator.

Abstract

We lay some mathematically rigorous foundations for the resolution of differential equations with respect to semi-classical bases and topologies, namely Freud-Sobolev polynomials and spaces. In this quest, we uncover an elegant theory melding various topics in Numerical and Functional Analysis: Poincaré inequalities, Sobolev orthogonal polynomials, Painlevé equations and more. Brought together, these ingredients allow us to quantify the compactness of Sobolev embeddings on Freud-weighted spaces and finally resolve some differential equations in this topology. As an application, we rigorously and tightly enclose solutions of the Gross-Pitaevskii equation with sextic potential.
Paper Structure (30 sections, 21 theorems, 204 equations, 3 figures)

This paper contains 30 sections, 21 theorems, 204 equations, 3 figures.

Table of Contents

  1. Introduction
  2. A natural Sobolev Hilbert structure
  3. A Numerical-Functional Analysis of Freud-weighted Sobolev spaces
  4. A differential counterpart to Freud's conjecture
  5. Weighted Sobolev spaces
  6. The Functional Analytic point-of-view
  7. Sobolev orthogonal polynomials
  8. Some examples outside of our setting
  9. The differentiation operator
  10. Functional and Numerical Analysis on Freud-weighted Sobolev spaces
  11. Freud-type orthogonal polynomials
  12. A precise analysis of the discrete Painlevé I equation
  13. Compactness estimates
  14. Proof of Theorem \ref{['thm:quant-compact']}
  15. Proof of Theorem \ref{['thm:quant-poinc']}
  16. ...and 15 more sections

Key Result

Proposition 1

For all $n\geq 1$, the polynomials $\mathcal{Q} =\{q_n\}_{n\in\mathbb{N}}$ orthonormal with respect to $(\cdot,\cdot)$ are uniquely defined by the properties

Figures (3)

  • Figure 1: Plots of orthonormal polynomials with respect to $\nu(\mathrm{d} x) = e^{-V(x)}\mathrm{d} x/Z$ with $V(x) = x^4/4-\kappa x^2/2$ with $\kappa = 4$.
  • Figure 2: Plots of the numerical approximations $\bar{\varphi}_1$ and $\bar{\varphi}_2$ in Theorems \ref{['thm:GP1']} and \ref{['thm:GP2']}.
  • Figure 3: Bounds on $b_n$ for $k = 4$, $c^-=0.987$ and $c^+=1.025$.

Theorems & Definitions (59)

  • Proposition 1
  • proof
  • Example 2
  • Theorem 3
  • Theorem 4
  • Remark 5
  • Theorem 6
  • Theorem 7
  • Lemma 8
  • proof
  • ...and 49 more