Traveling waves for a quasilinear wave equation
(2022) In Nonlinear Analysis, Theory, Methods and Applications 225.- 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=γδ0 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... (More)
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=γδ0 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.
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- author
- Bruell, Gabriele ; Idzik, Piotr and Reichel, Wolfgang
- publishing date
- 2022-12-01
- type
- Contribution to journal
- publication status
- published
- subject
- keywords
- Bifurcation, Nonlinear Maxwell equations, Quasilinear wave equation, Traveling wave
- in
- Nonlinear Analysis, Theory, Methods and Applications
- volume
- 225
- article number
- 113115
- publisher
- Elsevier
- external identifiers
-
- scopus:85138443027
- ISSN
- 0362-546X
- DOI
- 10.1016/j.na.2022.113115
- language
- English
- LU publication?
- no
- id
- ba8efc27-b7b1-4ea2-8b96-bd44add6d054
- date added to LUP
- 2022-12-05 11:30:22
- date last changed
- 2022-12-05 11:30:22
@article{ba8efc27-b7b1-4ea2-8b96-bd44add6d054,
abstract = {{<p>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<sup>∞</sup> being regular and (b) Γ=γδ<sub>0</sub> 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.</p>}},
author = {{Bruell, Gabriele and Idzik, Piotr and Reichel, Wolfgang}},
issn = {{0362-546X}},
keywords = {{Bifurcation; Nonlinear Maxwell equations; Quasilinear wave equation; Traveling wave}},
language = {{eng}},
month = {{12}},
publisher = {{Elsevier}},
series = {{Nonlinear Analysis, Theory, Methods and Applications}},
title = {{Traveling waves for a quasilinear wave equation}},
url = {{http://dx.doi.org/10.1016/j.na.2022.113115}},
doi = {{10.1016/j.na.2022.113115}},
volume = {{225}},
year = {{2022}},
}