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...

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Vydáno v:	Journal of nonlinear science Ročník 33; číslo 5; s. 82
Hlavní autoři:	Lizarraga, Ian, Marangell, Robert
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	New York Springer US 01.10.2023 Springer Nature B.V
Témata:	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
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Popis
Shrnutí:	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.
Bibliografie:	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