Abstract
This article investigates the Hirota–Satsuma–Ito equation with variable coefficient using the Hirota bilinear method and the long wave limit method. The equation is proved to be Painlevé integrable by Painlevé analysis. On the basis of the bilinear form, the forms of two-soliton solutions, three-soliton solutions, and four-soliton solutions are studied specifically. The appropriate parameter values are chosen and the corresponding figures are presented. The breather waves solutions, lump solutions, periodic solutions and the interaction of breather waves solutions and soliton solutions, etc. are given. In addition, we also analyze the different effects of the parameters on the figures. The figures of the same set of parameters in different planes are presented to describe the dynamical behavior of solutions. These are important for describing water waves in nature.
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