Soliton Solutions of the Generalized Dullin-Gottwald-Holm Equation with Parabolic Law Nonlinearity

In this paper, soliton solutions of the generalized Dullin-Gottwald-Holm (gDGH) equation with parabolic law nonlinearity are investigated. The gDGH describes the behavior of waves in shallow water with surface tension. There are only a few studies in the literature regarding gDGH equation with parabolic law nonlinearity, and to our best knowledge, the unified Riccati equation expansion method (UREEM) has not been applied to this equation before. Many soliton solutions of the considered gDGH equation are successfully attained using the UREEM, which is a powerful technique for solving nonlinear partial differential equations. We verify that the obtained analytical solutions satisfy the gDGH equation using Mathematica. Furthermore, some plots of the acquired solitons are demonstrated with the aid of Matlab to examine the properties of the soliton solutions. The obtained results show that the considered gDGH equation admits dark, bright, singular, and periodic solutions. This study may contribute to a comprehensive investigation of the soliton solutions of the gDGH equation, which has practical applications in fields such as oceanography and nonlinear optics.