Analytical solutions and molecule states of the (3+1)-dimensional variable coecient Date-Jimbo-Kashiwara-Miwa equation

Abstract

In the article, the (3+1)-dimensional variable coefficient Date-Jimbo-Kashiwara-Miwa (vcDJKM) equation is researched systematically. The Hirota bilinear method is utilized to construct $N$-soliton solutions and by imposing appropriate condition, the resonant $Y$-type solitons and the mixed solutions formed from resonant $Y$-type solitons are obtained. In addition, the positive quadratic function is exploited to search for lump solutions as well as the new degenerating breather method is employed to derive lump solutions. More importantly, with the help of velocity resonant principle, soliton molecules, breather molecules and lump molecules can be derived. Considering various forms of variable coefficients, these obtained solutions with all kinds of shapes, including $S$-type, parabolic-type and periodic-type are demonstrated by three-dimensional graphics, density and contour plots. And the results of this research can further advance the investigation of nonlinear partial differential equations with variable coefficients.