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 ${}_H^B\mathbb{L}$ is an autonomous category. Second, we prove that the category ${}_H^B\mathbb{L}$ 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 ${}_H^B\mathbb{L}$ can be viewed as a subcategory of the Hom-Yetter-Drinfeld category ${}_{H\otimes B}^{H\otimes B}\mathbb{H}\mathbb{Y}\mathbb{D}$. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.