Stationary shock-like transition fronts in dispersive systems. (22nd September 2020)

• Record Type: Journal Article
• Authors: Gavrilyuk, Sergey; Nkonga, Boniface; Shyue, Keh-Ming; Truskinovsky, Lev
• Is Part Of: Nonlinearity. Volume 33:Number 10(2020)
• Page Start: 5477
• Page End: 5509
• Publication Date: 2020-09-22
• Subjects: dispersive regularization -- expansion shocks -- zero dispersion limit.; 35L40 -- 35Q35 -- 35Q74; Nonlinear theories -- Periodicals; Mathematical analysis -- Periodicals; Mathematical analysis; Nonlinear theories; Periodicals; 515
• DOI: 10.1088/1361-6544/ab95ac
• Languages: English
• ISSNs: 0951-7715

Abstract

Abstract: We show that, contrary to popular belief, lower order dispersive regularization of hyperbolic systems does not exclude the development of the localized shock-like transition fronts. To guide the numerical search of such solutions, we generalize Rankine–Hugoniot relations to cover the case of higher order dispersive discontinuities and study their properties in an idealized case of a transition between two periodic wave trains with different wave lengths. We present evidence that smoothed stationary fronts of this type are numerically stable in the case when regularization is temporal and one of the adjacent states is homogeneous. In the zero dispersion limit such shock-like transition fronts, that are not travelling waves and apparently require for their description more complex anzats, evolve into travelling wave type jump discontinuities.