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The Gross-Pitaewskii equation with time and space dependent coefficients

Federico Lai

TL;DR

This work analyzes a generalized, nonautonomous Gross-Pitaewskii system with time- and space-dependent coefficients, proving the existence of weak solutions via an elliptic regularization framework. A variational approach with a family of functionals $\mathcal{F}_\varepsilon$ yields minimizers whose limits provide a weak solution to the original parabolic system, together with uniform estimates and regularity results. The authors also establish a segregation phenomenon in the long-time limit when the inter-species coupling grows unbounded, showing that the limit configurations become mutually exclusive across species. Overall, the paper provides a rigorous variational and asymptotic toolkit for nonautonomous multi-species GP-type dynamics with potential coefficient blow-up or vanishing, contributing to the mathematical understanding of time-delayed segregation in reaction-diffusion systems.

Abstract

We study the existence of weak solutions of a generalized Gross-Pitaewskii equation, with time and space dependent coefficients that could blow up or vanish asymptotically in time, with initial data not necessarily segregated. We also study the asymptotic behavior in time of these solutions, including cases in which the segregation phenomenon appears.

The Gross-Pitaewskii equation with time and space dependent coefficients

TL;DR

This work analyzes a generalized, nonautonomous Gross-Pitaewskii system with time- and space-dependent coefficients, proving the existence of weak solutions via an elliptic regularization framework. A variational approach with a family of functionals yields minimizers whose limits provide a weak solution to the original parabolic system, together with uniform estimates and regularity results. The authors also establish a segregation phenomenon in the long-time limit when the inter-species coupling grows unbounded, showing that the limit configurations become mutually exclusive across species. Overall, the paper provides a rigorous variational and asymptotic toolkit for nonautonomous multi-species GP-type dynamics with potential coefficient blow-up or vanishing, contributing to the mathematical understanding of time-delayed segregation in reaction-diffusion systems.

Abstract

We study the existence of weak solutions of a generalized Gross-Pitaewskii equation, with time and space dependent coefficients that could blow up or vanish asymptotically in time, with initial data not necessarily segregated. We also study the asymptotic behavior in time of these solutions, including cases in which the segregation phenomenon appears.

Paper Structure

This paper contains 8 sections, 9 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Functional setting and main results
  3. Hypotheses on the coefficients
  4. Notations
  5. Functional setting
  6. Main results
  7. Existence of weak solutions
  8. Asymptotic properties and segregation condition

Key Result

Theorem 2.1

Under the conditions of hyponcoeff, and ip13, ip11, ip12, considering ip9, ip10, there exists $w=(w_1,\dots,w_k) \in \mathcal{U}$, that is a weak solution, in the sense of $H^{-1}(\Omega \times (0,T))^k$ for all $T>0$ (and, in particular, in the sense of distribution of $\Omega \times \mathbb{R}^+$) if $\mathcal{U}=\mathcal{U}_{v_0,g}$, or of if $\mathcal{U}=\mathcal{U}_{v_0}$.

Theorems & Definitions (18)

  • Theorem 2.1: Existence theorem
  • Theorem 2.2: Limit segregation principle
  • Theorem 3.1: Existence of minimizers
  • proof
  • Corollary 3.2
  • proof
  • Corollary 3.3
  • proof
  • Theorem 3.4: Uniform estimates
  • proof
  • ...and 8 more