The Hom-Long dimodule category and nonlinear equations

In this paper, we construct a kind of new braided monoidal category over two Hom-Hopf algerbas $(H,alpha)$ and $(B,beta)$ and associate it with two nonlinear equations. We first introduce the notion of an $(H,B)$-Hom-Long dimodule and show that the Hom-Long dimodule category $^{B}_{H} Bbb L$ is an autonomous category. Second, we prove that the category $^{B}_{H} Bbb L$ is a braided monoidal category if $(H,alpha)$ is quasitriangular and $(B,beta)$ is coquasitriangular and get a solution of the quantum Yang-Baxter equation. Also, we show that the category $^{B}_{H} Bbb L$ can be viewed as a subcategory of the Hom-Yetter-Drinfeld category $^{Ho B}_{Ho B} Bbb {HYD}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.