Abstract
A family of periodic travelling wave solutions parameterized by the wavenumber is shown to bifurcate from the trivial solution in a perturbed KdV equation. Studying linearized eigenvalue problem about each periodic travelling wave solution, all of them are shown to be unstable immediately after the bifurcation in contrast to the Eckhaus stability/instability. Analysis from a dynamical viewpoint suggests that “modulated periodic waves” are obtained by a secondary bifurcation from periodic travelling waves as a super critical Hopf bifurcation.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
N.J. Balmforth, G.R. Ierley and R. Worthing, Pulse dynamics in an unstable medium. SIAM J. Appl. Math.,57 (1997), 205–251.
P.W. Bates, X. Chen and T.Ogawa, in preparation.
D.J. Benney, Long waves on liquid films. J. Math. Phys.,45 (1966), 150–155.
J. Carr, Applications of Center Manifold Theory. Springer-Verlag, 1981.
P. Collet and J.-P. Eckmann, Instabilities and Fronts in Extended Systems. Princeton University Press, 1990.
S.-N. Chow, J.K. Hale and J. Mallet-Paret, An example of bifurcation to homoclinic orbits. J. Diff. Eqns.,37 (1980), 351–373.
G. Derks and S.van Gils, On the uniqueness of traveling waves in perturbed Korteweg-de Vries equations. Japan J. Indust. Appl. Math.,10 (1993), 413–430.
P.G. Drazin and R.S. Johnson, Solitons: An Introduction. Cambridge University Press, 1989.
N.M. Ercolani, D.W. McLaughlin and H. Roitner, Attractors and transients for a perturbed KdV equation: a nonlinear spectral analysis. J. Nonlinear Sci.,3 (1993), 477–539.
N. Fenichel, Persistence and smoothness of invariant manifolds for flows. Indiana Univ. Math. J.,21 (1971), 193–226.
R.A. Gardner, On the struncure of the spectra of periodic travelling waves. J. Math. Pures Appl.,72 (1993), 415–439.
C.K.R.T. Jones, Geometric singular perturbation theory. C.I.M.E. Lectures, Session on Dynamical Systems, 1994.
T. Kawahara and S. Toh, Nonlinear dispersive periodic waves in the presence of instability and damping. Phys. Fluids,28, No. 6 (1985), 1636–1638.
H. McKean and E. Trubowitz, Hill’s operator and hyperbolic function theory in the presence of infinitely many branch points. Comm. Pure Appl. Math.,29 (1976), 143–226.
T. Ogawa, Travelling wave solutions to a perturbed Korteweg-de Vries equation. Hiroshima Math. J.,24 (1994), 401–422.
T. Ogawa, Wave patterns in nearly-integrable systems. Proceedings of the International Conference on Asymptotics in Nonlinear Diffusive Systems, Tohoku Math. Publ.8, 1998, 139–148
T. Ogawa and H. Suzuki, On the spectra of pulses in nearly integrable system. SIAM J. Appl. Math.,57 (1997), 485–500.
K. Sakamoto, Invariant manifolds in singular perturbation problems for ordinary differential equations. Proc. Roy. Soc. Edinb.,116A (1990), 45–78.
J. Topper and T. Kawahara, Approximate equations for long nonlinear waves on a viscous fluid. J. Phys. Soc. Japan,44 (1978), 663–666.
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Ogawa, T. Periodic travelling waves and their modulation. Japan J. Indust. Appl. Math. 18, 521–542 (2001). https://doi.org/10.1007/BF03168589
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF03168589