Geodesic
Global Bifurcation for the Whitham Equation
M. Ehrnström1 ; H. Kalisch2
1 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
2 Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway
Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 13-30

Voir la notice de l'article provenant de la source EDP Sciences

We prove the existence of a global bifurcation branch of 2π-periodic, smooth, traveling-wave solutions of the Whitham equation. It is shown that any subset of solutions in the global branch contains a sequence which converges uniformly to some solution of Hölder class Cα, α  1/2. Bifurcation formulas are given, as well as some properties along the global bifurcation branch. In addition, a spectral scheme for computing approximations to those waves is put forward, and several numerical results along the global bifurcation branch are presented, including the presence of a turning point and a ‘highest’, cusped wave. Both analytic and numerical results are compared to traveling-wave solutions of the KdV equation.
DOI : 10.1051/mmnp/20138502

M. Ehrnström 1 ; H. Kalisch 2

1 Department of Mathematical Sciences, Norwegian University of Science and Technology, 7491 Trondheim, Norway
2 Department of Mathematics, University of Bergen Postbox 7800, 5020 Bergen, Norway 
@article{10_1051_mmnp_20138502,
     author = {M. Ehrnstr\"om and H. Kalisch},
     title = {Global {Bifurcation} for the {Whitham} {Equation}},
     journal = {Mathematical modelling of natural phenomena},
     pages = {13--30},
     publisher = {mathdoc},
     volume = {8},
     number = {5},
     year = {2013},
     doi = {10.1051/mmnp/20138502},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138502/}
}
TY  - JOUR
AU  - M. Ehrnström
AU  - H. Kalisch
TI  - Global Bifurcation for the Whitham Equation
JO  - Mathematical modelling of natural phenomena
PY  - 2013
SP  - 13
EP  - 30
VL  - 8
IS  - 5
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138502/
DO  - 10.1051/mmnp/20138502
LA  - en
ID  - 10_1051_mmnp_20138502
ER  - 
%0 Journal Article
%A M. Ehrnström
%A H. Kalisch
%T Global Bifurcation for the Whitham Equation
%J Mathematical modelling of natural phenomena
%D 2013
%P 13-30
%V 8
%N 5
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.1051/mmnp/20138502/
%R 10.1051/mmnp/20138502
%G en
%F 10_1051_mmnp_20138502
M. Ehrnström; H. Kalisch. Global Bifurcation for the Whitham Equation. Mathematical modelling of natural phenomena, Tome 8 (2013) no. 5, pp. 13-30. doi: 10.1051/mmnp/20138502

[1] H. Amann Math. Nachr. 1997 5 56

[2] J. L. Bona, T. Colin, D. Lannes Arch. Ration. Mech. Anal. 2005 373 410

[3] J. Boyd. Chebyshev and Fourier spectral methods. Dover Publications Inc., Mineola, 2001.

[4] J. P. Boyd J. Comput. Phys. 2003 98 110

[5] B. Buffoni Arch. Ration. Mech. Anal. 2004 25 68

[6] B. Buffoni, J. F. Toland. Analytic theory of global bifurcation. Princeton Series in Applied Mathematics, Princeton University Press, Princeton, 2003.

[7] C. Canuto, M. Y. Hussaini, A. Quarteroni, T.A. Zang Springer Series in Computational Physics. Springer

[8] W. Craig Comm. Partial Differential Equations 1985 787 1003

[9] M. Ehrnström, M. D. Groves, E. Wahlén Nonlinearity 2012 2903 2936

[10] M. Ehrnström, H. Kalisch Differential Integral Equations 2009 1193 1210

[11] T. Gneiting Proc. Amer. Math. Soc. 2001 2309 2318

[12] M. D. Groves, E. Wahlén J. Math. Fluid Mech. 593 627 2011

[13] Y. Katznelson. An introduction to harmonic analysis. Cambridge Mathematical Library, Cambridge University Press, Cambridge, 2004.

[14] H. Kielhöfer. Bifurcation theory. Applied Mathematical Sciences, vol. 156, Springer, New York, 2004.

[15] P. I. Naumkin, I. A. Shishmarev. Nonlinear nonlocal equations in the theory of waves. Translations of Mathematical Monographs. vol. 133. American Mathematical Society, Providence, 1994.

[16] G. Schneider, C. E. Wayne Comm. Pure Appl. Math. 2000 1475 1535

[17] H. Triebel. Theory of function spaces. Monographs in Mathematics, vol. 78, Birkhäuser, Basel, 1983.

[18] G. B. Whitham Proc. R. Soc. Lond., A 1967 6 25

[19] G. B. Whitham. Linear and nonlinear waves. Pure and Applied Mathematics, John Wiley Sons, New York, 1974.

Cité par Sources :