Geodesic
Bloch–Floquet band gaps for water waves over a periodic bottom
Christophe Lacave1 orcid ; Matthieu Ménard2 orcid ; Catherine Sulem3 orcid
1Université Savoie Mont Blanc, Chambéry, France
2Université Libre de Bruxelles, Brussels, Belgium
3University of Toronto, Toronto, Canada
EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 243-288

Voir la notice de l'article provenant de la source EMS Press

DOI

A central object in the analysis of the water wave problem is the Dirichlet–Neumann operator. This paper is devoted to the study of its spectrum in the context of the water wave system linearized near equilibrium in a domain with a variable bottom, assumed to be a C2 periodic function. We use the analyticity of the Dirichlet–Neumann operator with respect to the bottom variation and combine it with general properties of elliptic systems and spectral theory for self-adjoint operators to develop a Bloch–Floquet theory and describe the structure of its spectrum. We find that, under some conditions on the bottom variations, the spectrum is composed of bands separated by gaps, with explicit formulas for their sizes and locations.
DOI : 10.4171/emss/93
Classification : 76B15, 35P15
Mots-clés : water wave problem, periodic bottom, spectral gaps

Christophe Lacave  1   ; Matthieu Ménard  2   ; Catherine Sulem  3

1 Université Savoie Mont Blanc, Chambéry, France
2 Université Libre de Bruxelles, Brussels, Belgium
3 University of Toronto, Toronto, Canada 
Christophe Lacave; Matthieu Ménard; Catherine Sulem. Bloch–Floquet band gaps for water waves over a periodic bottom. EMS surveys in mathematical sciences, Tome 12 (2025) no. 1, pp. 243-288. doi: 10.4171/emss/93
@article{10_4171_emss_93,
     author = {Christophe Lacave and Matthieu M\'enard and Catherine Sulem},
     title = {Bloch{\textendash}Floquet band gaps for water waves over a periodic bottom},
     journal = {EMS surveys in mathematical sciences},
     pages = {243--288},
     year = {2025},
     volume = {12},
     number = {1},
     doi = {10.4171/emss/93},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/emss/93/}
}
TY  - JOUR
AU  - Christophe Lacave
AU  - Matthieu Ménard
AU  - Catherine Sulem
TI  - Bloch–Floquet band gaps for water waves over a periodic bottom
JO  - EMS surveys in mathematical sciences
PY  - 2025
SP  - 243
EP  - 288
VL  - 12
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4171/emss/93/
DO  - 10.4171/emss/93
ID  - 10_4171_emss_93
ER  - 
%0 Journal Article
%A Christophe Lacave
%A Matthieu Ménard
%A Catherine Sulem
%T Bloch–Floquet band gaps for water waves over a periodic bottom
%J EMS surveys in mathematical sciences
%D 2025
%P 243-288
%V 12
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4171/emss/93/
%R 10.4171/emss/93
%F 10_4171_emss_93

Cité par Sources :