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Bifurcation of Harmonic Solutions of an Integro-Differential Equation Modelling Resonant Sloshing

D. W. Reynolds
SIAM Journal on Applied Mathematics
Vol. 49, No. 2 (Apr., 1989), pp. 362-372
Stable URL: http://www.jstor.org/stable/2102078
Page Count: 11
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Bifurcation of Harmonic Solutions of an Integro-Differential Equation Modelling Resonant Sloshing
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Abstract

This paper proves the existence of multiple solutions (μ, u) of $u'''(t) - \lambda u'(t) - 2u(t)u'(t) = \mu_1 (\mathbf{D} u')(t) + \mu_2 \sin t,$ such that u has period 2 π and mean zero. D is a singular integral operator. The equation models steady gravity waves on the surface of a shallow tank, oscillating near the primary resonance frequency. It is shown here that for each $\lambda > -1$, the equation has one, two, or three solutions, depending on the position of μ = (μ1, μ2) in a neighbourhood of zero in R2. This extends previous work, which required that λ be close to -1. The method depends on the existence of cnoidal solutions when $\lambda > -1$ and μ = 0. The Lyapunov-Schmidt procedure is used to reduce the problem to a single bifurcation equation, which is analysed using some results of Hale and Taboas.

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