Reduction of PDEs on Unbounded Domains. Application: Unsteady Water-Waves Problem |
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Authors: | M Hărăguş |
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Institution: | Institut Non-Linéaire de Nice, UMR CNRS 129, Université de Nice, 1361 Route des Lucioles, Sophia Antipolis, 06560 Valbonne, France Institute of Mathematics of the Romanian Academy, P.O. BOX 1-764, 70700 Bucharest, Romania, FR
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Abstract: | Summary. For a certain class of partial differential equations in cylindrical domains, we show that all small time-dependent solutions
are described by a reduced system of equations on the real line, which contains nonlocal terms. As an application, we investigate
the system describing nonlinear water waves travelling on the free surface of an inviscid fluid. Two-dimensional gravity waves
are characterized by the parameter λ , the inverse square of the Froude number. For λ close to the critical value λ
0
=1 , we obtain a reduced system of four nonlocal equations. We show that the terms of lowest order in μ=λ-1 lead to the Korteweg—de Vries equation for the lowest-order approximation of the free surface.
Received February 23, 1994; final revision received October 13, 1997; accepted for publication October 16, 1997. |
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Keywords: | , reduction, partial differential equations, unbounded domain, water waves, Korteweg—,de Vries equation |
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