REDUCTION OF PDES ON UNBOUNDED-DOMAINS - APPLICATION - UNSTEADY WATER-WAVES PROBLEM

For a certain class of partial differential equations in cylindrical d omains, we show that all small time-dependent solutions are described by a reduced system of equations on the real line, which contains nonl ocal terms. As an application, we investigate the system describing no nlinear water waves travelling on the free surface of an inviscid flui d. Two-dimensional gravity waves are characterized by the parameter la mbda, the inverse square of the Froude number. For lambda close to the critical value lambda(0) = 1, we obtain a reduced system of four nonl ocal equations. We show that the terms of lowest order in mu = lambda - 1 lead to the Korteweg-de Vries equation for the lowest-order approx imation of the free surface.