 
Summary: ON THE EXACT SOLUTIONS OF THE
INTERMEDIATE LONGWAVE EQUATION
J. P. Albert J. F. Toland
Department of Mathematics School of Mathematical Sciences
University of Oklahoma University of Bath
Norman, Oklahoma 73019, U. S. A. Claverton Down, Bath BA2 7AY, U. K.
To the memory of Peter Hess
1. Introduction. The intermediate longwave equation was introduced by R. I.
Joseph [4] as a mathematical model of nonlinear dispersive waves on the interface
between two fluids of different positive densities contained at rest in a long channel
with a horizontal top and bottom, the lighter fluid forming a horizontal layer above
a layer of the same depth of the heavier fluid. When variables have been rescaled
it is the pseudodifferential operator equation (see [5])
t + 2x  (NH)x + (1/H)x = 0, (1)
where H > 0 and the Fourier multiplier operator NH is given by
NH(k) = (k coth kH)(k).
In common with the classical KdV and BenjaminOno equations, between which
it was intended to form a modeltheoretical bridge [4], equation (1) was found to
have a family of exact solitarywave solutions: namely,
(x, t) = C,H (x  Ct),
