Unstable simple modes of the nonlinear string
Authors:
Thierry Cazenave and Fred B. Weissler
Journal:
Quart. Appl. Math. 54 (1996), 287-305
MSC:
Primary 35L70; Secondary 45G10, 73D35
DOI:
https://doi.org/10.1090/qam/1388017
MathSciNet review:
MR1388017
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Abstract: We prove instability of high-energy simple modes for the nonlinear vibrating string equation \[ \left \{ {_{u\left | {_{\partial \Omega } = 0,} \right .}^{{u_{tt}} - \left ( {a + b \int _0^\pi {u_x^2 \left ( {t, x} \right ) dx} } \right ) {u_{xx}} = 0,}} \right .\] in $\Omega = \left ( {0, \pi } \right )$, where $a \ge 0, b > 0$.
S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles, Izv. Acad. Nauk. SSSR, Ser. Math. 4, 17–26 (1940)
G. F. Carrier, On the vibration problem of an elastic string, Quart. Appl. Math. 3, 151–165 (1945)
T. Cazenave and F. B. Weissler, Asymptotically periodic solutions for a class of nonlinear coupled oscillators, Portugaliae Math. 52, 109–123 (1995)
R. W. Dickey, Stability of periodic solutions of the nonlinear string equation, Quart. Appl. Math. 38, 253–259 (1980)
P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964
W. Magnus and S. Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics # 20, Interscience Publishers, New York, 1966
L. A. Medeiros and M. Milla Miranda, Solution for the equation of nonlinear vibration in Sobolev spaces of fractionary order, Mat. Aplic. Comp. 6, 257–276 (1987)
R. Narashimha, Nonlinear vibrations of an elastic string, J. Sound Vib. 8, 134–136 (1968)
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1967
S. Bernstein, Sur une classe d’équations fonctionnelles aux dérivées partielles, Izv. Acad. Nauk. SSSR, Ser. Math. 4, 17–26 (1940)
G. F. Carrier, On the vibration problem of an elastic string, Quart. Appl. Math. 3, 151–165 (1945)
T. Cazenave and F. B. Weissler, Asymptotically periodic solutions for a class of nonlinear coupled oscillators, Portugaliae Math. 52, 109–123 (1995)
R. W. Dickey, Stability of periodic solutions of the nonlinear string equation, Quart. Appl. Math. 38, 253–259 (1980)
P. Hartman, Ordinary Differential Equations, John Wiley and Sons, New York, 1964
W. Magnus and S. Winkler, Hill’s equation, Interscience Tracts in Pure and Applied Mathematics # 20, Interscience Publishers, New York, 1966
L. A. Medeiros and M. Milla Miranda, Solution for the equation of nonlinear vibration in Sobolev spaces of fractionary order, Mat. Aplic. Comp. 6, 257–276 (1987)
R. Narashimha, Nonlinear vibrations of an elastic string, J. Sound Vib. 8, 134–136 (1968)
G. Szegö, Orthogonal Polynomials, Amer. Math. Soc., Providence, RI, 1967
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Article copyright:
© Copyright 1996
American Mathematical Society
