Traveling waves for a quasilinear wave equation. (December 2022)

• Record Type: Journal Article
• Authors: Bruell, Gabriele; Idzik, Piotr; Reichel, Wolfgang
• Is Part Of: Nonlinear analysis. Volume 225(2022)
• Publication Date: 2022-12
• Subjects: primary 35B32 35L72 -- secondary 35C07 35Q61; Nonlinear Maxwell equations -- Quasilinear wave equation -- Traveling wave -- Bifurcation; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2022.113115
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: We consider a 2+1 dimensional wave equation appearing in the context of polarized waves for the nonlinear Maxwell equations. The equation is quasilinear in the time derivatives and involves two material functions V and Γ . We prove the existence of traveling waves which are periodic in the direction of propagation and localized in the direction orthogonal to the propagation direction. Depending on the nature of the nonlinearity coefficient Γ we distinguish between two cases: (a) Γ ∈ L ∞ being regular and (b) Γ = γ δ 0 being a multiple of the delta potential at zero. For both cases we use bifurcation theory to prove the existence of nontrivial small-amplitude solutions. One can regard our results as a persistence result which shows that guided modes known for linear wave-guide geometries survive in the presence of a nonlinear constitutive law. Our main theorems are derived under a set of conditions on the linear wave operator. They are subsidized by explicit examples for the coefficients V in front of the (linear) second time derivative for which our results hold.