Two approaches to instability analysis of the viscous Burgers' equation

Messoud Efendiev; Taylan Sengul; Burhan Tiryakioglu

doi:10.3934/dcdss.2023202

Article Contents

2024, Volume 17, Issue 4: 1621-1638. Doi: 10.3934/dcdss.2023202

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Two approaches to instability analysis of the viscous Burgers' equation

1.
Department of Mathematics, Marmara University, 34722, Istanbul, Turkey
2.
Helmholtz Zentrum München, Institut für Computational Biology, Ingolstädter Landstrasse 1, Neuherberg, 85764, Germany

^*Corresponding author: Burhan Tiryakioglu
^*Corresponding author: Burhan Tiryakioglu

Dedicated to Yihong Du on the occasion of his 60th birthday.

Received: January 2023

Revised: October 2023

Early access: November 2023

Published: April 2024

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Abstract

The 1D Burger's equation with Dirichlet boundary conditions exhibits a first transition from the trivial steady state to a sinusoidal patterned steady state as the parameter $ \lambda $ which controls the linear term exceeds 1. The main goal of this paper is to present two different approaches regarding the transition of this patterned steady state. We believe that these approaches can be extended to study the dynamics of more interesting models. As a first approach, we consider an external forcing on the equation which supports a sinusoidal solution as a stable steady state which loses its stability at a critical threshold. We use the method of continued fractions to rigorously analyze the associated linear problem. In particular, we find that the system exhibits a mixed type transition with two distinct basins for initial conditions one of which leads to a local steady state and the other leaves a small neighborhood of the origin. As a second approach, we consider the dynamics on the center-unstable manifold of the first two modes of the unforced system. In this approach, the secondary transition produces two branches of steady state solutions. On one of these branches there is another transition which indicates a symmetry breaking phenomena.

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Mathematics Subject Classification: Primary: 37L10, 70K50; Secondary: 30B70.

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Figure 1. The first eigenvalue of (9)

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Figure 2. The first three eigenvalues of (9)

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Figure 3. he 3D bifurcation diagram of the system (19) in the $ (u_1, u_2, \lambda) $ space with $ \alpha = 1 $. The black, blue and red steady states are respectively the trivial, mixed and pure steady state solutions

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Figure 4. Bifurcation diagram for (18) when $ \alpha = 1 $

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Table 1. The coordinates of the nontrivial branching point which connects the "mixed" solution branches with "pure" solution branches

	Third order truncation	Fourth order truncation
Nontrivial branching point	$ (u_1, u_2, \lambda)=(0, 5.81, 6.35) $	$ (u_1, u_2, \lambda)=(0, 5.50, 6.14) $

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