Floquet’s Theorem and Stability of Periodic Solitary Waves
Tóm tắt
This paper is concerned with the spectrum the Hill operator L(y) = −y′′ + Q(x) y in
$${L^{2}_{\rm per}[0, \pi]}$$
. We show that the eigenvalues of L can be characterized by knowing one of its eigenfunctions. Applications are given to nonlinear stability of a class of periodic problems.
Tài liệu tham khảo
Akhiezer, N. I.: Elements of theory of elliptic functions, translation of mathematical monographs, vol. 79. AMS (1990)
Angulo J., Natali F.: Positivity properties of the Fourier transform and the stability of periodic travelling-waves solutions. SIAM J. Math. Anal. 40(3), 1123–1151 (2008)
Angulo J., Bona J., Scialom M.: Stability of Cnoidal waves. Adv. Differ. Equ. 11(12), 1321–1374 (2006)
Byrd P.F., Friedman M.D.: A Handbook of Elliptic Integrals for Engineers and Physicists. Springer, Berlin (1954)
Gallay T., Hărăgus M.: Orbital stability of periodic waves for the nonlinear Schrödinger equation. J. Dynam. Differ. Equ. 19(4), 825–865 (2007)
Grillakis M., Shatah J., Straus W.: Stability theory of solitary waves in the presence of symmetry I. J. Funct. Anal. 74, 160–197 (1987)
Grillakis M., Shatah J., Straus W.: Stability theory of solitary waves in the presence of symmetry II. J. Funct. Anal. 94, 308–348 (1990)
Haupt, O.: Über eine methode zum Beweis von Oszillationstheoremen. Math. Ann. 76, 67–104 (1914)
Lopes O.: A linearized instability result for solitary waves. Discrete Contin. Dyn. Syst. 8, 115–119 (2002)
Magnus W., Winkler S.: Hill’s Equation, Intersience Tracts in Pure and Applied Mathematics, vol. 20. Wiley, New York (1966)
Neves A.: Isoinertial family of operators and convergence of KdV cnoidal waves to solitons. J. Differ. Equ. 244(4), 875–886 (2008)