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Stability analysis of the $(1+1)$-dimensional Nambu-Goto action gas models

Alfred M. Grundland, Javier de Lucas, Bartosz M. Zawora

TL;DR

The paper addresses nonlinear stability of (1+1)-dimensional gas models derived from the Nambu-Goto action, focusing on Chaplygin gas and Born–Infeld equations. It develops and applies the energy-Casimir method within an infinite-dimensional Poisson framework, deriving Hamiltonian structures, identifying Casimir functionals, and formulating an explicit stability algorithm. By locating equilibria such as $p\rho=\sqrt{2\lambda}$ and $p\rho=1$ (under appropriate limits), and verifying Lyapunov stability for carefully constructed Casimirs and quadratic forms, the authors establish rigorous nonlinear stability for both models and illustrate the approach with concrete exact solutions. This work provides a geometric, variational route to stability in brane-inspired fluid models and showcases the utility of the energy-Casimir method in relativistic-like and nonlinear wave contexts.

Abstract

The aim of this paper is to perform a nonlinear stability analysis of the $(1+1)$-dimensional Nambu-Goto action gas models. The energy-Casimir method is employed to discuss in detail the Lyapunov stability of the Chaplygin and Born-Infeld models. Particular solutions are considered and their stability is studied in order to illustrate the application of our results.

Stability analysis of the $(1+1)$-dimensional Nambu-Goto action gas models

TL;DR

The paper addresses nonlinear stability of (1+1)-dimensional gas models derived from the Nambu-Goto action, focusing on Chaplygin gas and Born–Infeld equations. It develops and applies the energy-Casimir method within an infinite-dimensional Poisson framework, deriving Hamiltonian structures, identifying Casimir functionals, and formulating an explicit stability algorithm. By locating equilibria such as and (under appropriate limits), and verifying Lyapunov stability for carefully constructed Casimirs and quadratic forms, the authors establish rigorous nonlinear stability for both models and illustrate the approach with concrete exact solutions. This work provides a geometric, variational route to stability in brane-inspired fluid models and showcases the utility of the energy-Casimir method in relativistic-like and nonlinear wave contexts.

Abstract

The aim of this paper is to perform a nonlinear stability analysis of the -dimensional Nambu-Goto action gas models. The energy-Casimir method is employed to discuss in detail the Lyapunov stability of the Chaplygin and Born-Infeld models. Particular solutions are considered and their stability is studied in order to illustrate the application of our results.

Paper Structure

This paper contains 16 sections, 2 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Poisson structures on infinite-dimensional spaces
  3. Nonlinear stability
  4. Lyapunov stability
  5. The energy-Casimir method
  6. Chaplygin gas model
  7. Lagrangian formulation of the Chaplygin gas model
  8. Hamiltonian formulation of the Chaplygin gas model
  9. Stability of the --dimensional Chaplygin gas model
  10. Born-Infeld model
  11. Lagrangian formulation of the Born-Infeld model
  12. Hamiltonian formulation of the Born-Infeld model
  13. Stability algorithm for the Born-Infeld model
  14. Stability analysis of particular solutions
  15. Chaplygin gas model
  16. ...and 1 more sections

Key Result

Theorem 3.1

Let $(\mathcal{A},\{\cdot,\cdot\})$ be a Poisson Banach space and let $H\in\mathcal{A}$ be a Hamiltonian functional such that $\dot{H}(u)=0$ for any solution $u\in V$ of Eq::NonAutDyn. Additionally, let $C\in\mathcal{A}$ denote a Casimir functional, i.e. $\{C,\,\cdot\,\}=0$ and let $H_C:=H+C\in\math and Then, indeed $\|\cdot\|$ is a norm and $u_e\in V$ is a stable equilibrium point of Eq::NonAutD

Theorems & Definitions (2)

  • Theorem 3.1
  • Lemma 3.2