Boundary Value Problems for Nonlinear Dispersive Wave Equations

PDE Seminar
Tuesday, October 20, 2009 - 15:05
1.5 hours (actually 80 minutes)
Skiles 255
University of Memphis
Under the classical small-amplitude, long wave-length assumptions in which the Stokes number is of order one, so featuring a balance between nonlinear and dispersive effects, the KdV-equation u_t+ u_x + uu_x + u_xxx = 0 (1) and the regularized long wave equation, or BBM-equation u_t + u_x + uu_x-u_xxt = 0 (2) are formal reductions of the full, two-dimensional Euler equations for free surface flow. This talk is concerned with the two-point boundary value problem for (1) and (2) wherein the wave motion is specified at both ends of a finite stretch of length L of the media of propagation. After ascertaining natural boundary specifications that constitute well posed problems, it is shown that the solution of the two-point boundary value problem, posed on the interval [0;L], say, converges as L converges to infinity, to the solution of the quarter-plane boundary value problem in which a semi-infinite stretch [0;1) of the medium is disturbed at its finite end (the so-called wavemaker problem). In addition to its intrinsic interest, our results provide justification for the use of the two-point boundary-value problem in numerical studies of the quarter plane problem for both the KdV-equation and the BBM-equation.