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The Journal of Fourier Analysis and Applications Volume 2, Number 5, 1996

Summary: The Journal of Fourier Analysis and Applications
Volume 2, Number 5, 1996
On Strongly Interacting
Internal Solitary Waves
J. Marshall Ash, Jonathan Cohen, and Gang Wang
ABSTRACT. The Cauchy problem and global well-posedness for a mathematical model of the
strong interaction of two-dimensional, long, internal gravity waves propogating on neighboring
pycnoclines in a stratified fluid have been studied by Bona, Ponce, Saut, Tom, and others. We
show that global well-posedness occurs even when the initial data is rough.
1. Introduction
This paper is concerned with the initial-value problem
u, + Uxxx + a3Vxxx + UUx + al vVx + a2(uv)x = 0
u(x,O) = uo(x)
v(x,O) = voex),
where ab a2, a3, bl, b2, and r are real constants with bi, b: positive; U = u(x, r), v = vex, t)
are real-valued functions of the two real variables x and t; and subscripts adorning u and v denote
partial differentiation. This system has the structure of a pair ofKorteweg-de Vries equations coupled
through both dispersive and nonlinear effects. It was derived by Gear and Grimshaw [GG] as a model
to describe the strong interaction of weakly nonlinear, long waves. System (1) was studied extensively


Source: Ash, J. Marshall - Department of Mathematical Sciences, DePaul University


Collections: Mathematics