Summary: ON THE EXACT SOLUTIONS OF THE
INTERMEDIATE LONG-WAVE 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 long-wave equation was introduced by R. I.
Joseph  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 re-scaled
it is the pseudo-differential operator equation (see )
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 Benjamin-Ono equations, between which
it was intended to form a model-theoretical bridge , equation (1) was found to
have a family of exact solitary-wave solutions: namely,
(x, t) = C,H (x - Ct),