Smooth solitary waves for the generalized gKdV-4 equation

We consider a generalization of KdV-type equations with a quartic nonlinearity $u^4$ (gKdV-4), which includes dissipation terms similar to those appearing in the Benjamin-Bona-Mahoney equation as well as in the well-known Camassa–Holm and Degasperis-Procesi equations. Our objective is to construct classical solitary wave solutions (solitons-antisolitons) to this equation.