Advanced Search
Article Contents
Article Contents

Steady state bifurcations for the Kuramoto-Sivashinsky equation: A computer assisted proof

Abstract Related Papers Cited by
  • We apply the method of self-consistent bounds to prove the existence of multiple steady state bifurcations for Kuramoto-Sivashinski PDE on the line with odd and periodic boundary conditions.
    Mathematics Subject Classification: Primary: 35B32, 37G10, 65G20.


    \begin{equation} \\ \end{equation}
  • [1]

    G. Arioli and H. Koch, Computer-assisted methods for the study of stationary solutions in dissipative systems, applied to the Kuramoto-Sivashinski equation, Arch. Rational Mech. An., 197 (2010), 1033-1051.doi: 10.1007/s00205-010-0309-7.


    L. Cesari, Functional analysis and Galerkin's method, Mich. Math. Jour., 11 (1964), 385-414.doi: 10.1307/mmj/1028999194.


    S.-N. Chow and J. Hale, Methods of Bifurcation Theory, Springer-Verlag, New York, 1982.


    F. Christiansen, P. Cvitanovic and V. Putkaradze, Spatiotemporal chaos in terms of unstable recurrent patterns, Nonlinearity, 10 (1997), 55-70.doi: 10.1088/0951-7715/10/1/004.


    P. Collet, J.-P. Eckmann, H. Epstein and J. Stubbe, Analyticity for the Kuramoto-Sivashinsky equation, Physica D, 67 (1993), 321-326.doi: 10.1016/0167-2789(93)90168-Z.


    E. J. Doedel, AUTO: a program for the bifurcation analysis of autonomous system, Congr. Numer., 30 (1981), 265-284.


    E. J. Doedel and R. C. Paffenroth, The AUTO2000: command line user interface, in Proceedings of the 9-th Python Conference, 2001, 233-241.


    C. Foias, B. Nicolaenko, G. Sell and R. Temam, Inertial manifolds for the Kuramoto-Sivashinsky equation and an estimate of their lowest dimension, J. Math. Pures Appl., 67 (1988), 197-226.


    J. Hyman and B. Nicolaenko, The Kuramoto-Sivashinsky equation; A bridge between PDEs and dynamical systems, Physica D, 18 (1986), 113-126.doi: 10.1016/0167-2789(86)90166-1.


    J. S. Il'yashenko, Global Analysis of the Phase Portrait for the Kuramoto-Sivashinsky equation, J. Dyn. Diff. Eq., 4 (1992), 585-615.doi: 10.1007/BF01048261.


    M. Jolly, I. Kevrekidis and E. Titi, Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations, Physica D, 44 (1990), 38-60.doi: 10.1016/0167-2789(90)90046-R.


    M. Jolly, R. Rosa and R. Temam, Evaluating the dimension of an inertial manifold for the Kuramoto-Sivashinsky Equation, Adv. Differential Equations, 5 (2000), 31-66.


    M. Jolly, R. Rosa and R. Temam, Acurate computations on inertial manifolds, SIAM J. Sci. Compt., 22 (2000), 2216-2238.doi: 10.1137/S1064827599351738.


    I. Kevrekidis, B. Nicolaenko and C. Scovel, Back in saddle again: a computer assisted study of the Kuramoto-Sivashinsky equation, SIAM J. Appl. Math., 50 (1990), 760-790.doi: 10.1137/0150045.


    Y. Kuramoto and T. Tsuzuki, Persistent propagation of concentration waves in dissipative media far from thermal equilibrium, Prog. Theor. Phys., 55 (1976), 356-369.doi: 10.1143/PTP.55.356.


    S. Maier-Paape, U. Miller, K. Mischaikow and T. Wanner, Rigorous numerics for the Cahn-Hilliard equation on the unit square, Revista Matematica Complutense, 21 (2008), 351-426.doi: 10.5209/rev_REMA.2008.v21.n2.16380.


    R. E. Moore, Interval Analysis, Prentice Hall, Englewood Cliffs, N.J., 1966.


    A. Neumeier, Interval Methods for Systems of Equations, Cambrigde University Press, 1990.


    A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, Texts in Applied Mathematics, 37. Springer-Verlag, New York, 2000.


    G. I. Sivashinsky, Nonlinear analysis of hydrodynamical instability in laminar flames - 1. Derivation of basic equations, Acta Astron, 4 (1977), 1177-1206.doi: 10.1016/0094-5765(77)90096-0.


    C. Scovel, I. Kevrekidis and B. Nicolaenko, Scaling laws and the prediction of bifurcations in systems modeling pattern formation, Physics Letters A, 130 (1988), 73-80.doi: 10.1016/0375-9601(88)90242-3.


    P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: The Kuramoto-Sivashinsky equation, Foundations of Computational Mathematics, 1 (2001), 255-288.doi: 10.1007/s002080010010.


    P. Zgliczyński, Trapping regions and an ODE-type proof of existence and uniqueness for Navier-Stokes equations with periodic boundary conditions on the plane, Univ. Iag. Acta Math., 41 (2003), 89-113.


    P. Zgliczyński, On smooth dependence on initial conditions for dissipative PDEs, an ODE-type approach, J. Diff. Eq., 195 (2003), 271-283.doi: 10.1016/j.jde.2003.07.009.


    P. Zgliczyński, Attracting fixed points for the Kuramoto-Sivashinsky equation - a computer assisted proof, SIAM Journal on Applied Dynamical Systems, 1 (2002), 215-235.doi: 10.1137/S111111110240176X.

  • 加载中

Article Metrics

HTML views() PDF downloads(184) Cited by(0)

Access History

Other Articles By Authors



    DownLoad:  Full-Size Img  PowerPoint