Solitary Wave Solutions to a Class of Modified Green–Naghdi Systems

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 Duchêne et al. (Stud Appl Math 137:356–415, 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.