  We provide the existence and asymptotic description of solitary wave
solutions to a class of modified Green-Naghdi systems, modeling the propagation
of long surface or internal waves. This class was recently proposed by
Duch{\^e}ne, Israwi and Talhouk (Stud. Appl. Math.,137 (2016)) in order to
improve the frequency dispersion of the original Green-Naghdi system while
maintaining the same precision. The solitary waves are constructed from the
solutions of a constrained minimization problem. The main difficulties stem
from the fact that the functional at stake involves low order non-local
operators, intertwining multiplications and convolutions through Fourier
multipliers.
