Spectral Stability of Shock-fronted Travelling Waves Under Viscous Relaxation

Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Li et al. (Physica D 423:132916, 2021) has demonstrated the existence of such waves for two classes of regularizations, including viscous relaxation (see Li et al. in Physic...

Ausführliche Beschreibung

Gespeichert in:
Bibliographische Detailangaben
Veröffentlicht in:Journal of nonlinear science Jg. 33; H. 5; S. 82
Hauptverfasser: Lizarraga, Ian, Marangell, Robert
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York Springer US 01.10.2023
Springer Nature B.V
Schlagworte:
Analysis
Classical Mechanics
Differential equations
Economic Theory/Quantitative Economics/Mathematical Methods
Eigenvalues
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Partial differential equations
Perturbation theory
Regularization
Singular perturbation
Stability
Theoretical
Traveling waves
Waves
Spectral stability
34D15
Shock waves
37C75
35B35
74J40
Geometric singular perturbation theory
Reaction-nonlinear diffusion PDEs
ISSN:0938-8974, 1432-1467
Online-Zugang:Volltext
Tags: Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
Beschreibung
Zusammenfassung:Reaction-nonlinear diffusion partial differential equations can exhibit shock-fronted travelling wave solutions. Prior work by Li et al. (Physica D 423:132916, 2021) has demonstrated the existence of such waves for two classes of regularizations, including viscous relaxation (see Li et al. in Physica D 423:132916, 2021). Their analysis uses geometric singular perturbation theory: for sufficiently small values of a parameter ε > 0 characterizing the ‘strength’ of the regularization, the waves are constructed as perturbations of a singular heteroclinic orbit. Here we show rigorously that these waves are spectrally stable for the case of viscous relaxation. Our approach is to show that for sufficiently small ε > 0 , the ‘full’ eigenvalue problem of the regularized system is controlled by a reduced slow eigenvalue problem defined for ε = 0 . In the course of our proof, we examine the ways in which this geometric construction complements and differs from constructions of other reduced eigenvalue problems that are known in the wave stability literature.
Bibliographie:ObjectType-Article-1
SourceType-Scholarly Journals-1
ObjectType-Feature-2
content type line 14
ISSN:0938-8974
1432-1467
DOI:10.1007/s00332-023-09941-x