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 $ ^{H{\otimes} B}_{H{\otimes} B} \Bbb {HYD} $. Finally, we obtain a solution of the Hom-Long equation from the Hom-Long dimodules.