Bogoyavlensky–modified KdV hierarchy and toroidal Lie algebra \(\textrm{sl}^\textrm{tor}_{2}\)

By principal representation of toroidal Lie algebra \(\mathrm{sl^{tor}_2}\), we construct an integrable system: Bogoyavlensky–modified KdV (B–mKdV) hierarchy, which is \((2+1)\)-dimensional generalization of modified KdV hierarchy. Firstly, bilinear equations of B–mKdV hierarchy are obtained by fermionic representation of \(\mathrm{sl^{tor}_2}\) and boson–fermion correspondence, which are rewritten into Hirota bilinear forms. Also Fay-like identities of B–mKdV hierarchy are derived. Then from B–mKdV bilinear equations, we investigate Lax structure, which is another equivalent formulation of B–mKdV hierarchy. Conversely, we also derive B–mKdV bilinear equations from Lax structure. Other equivalent formulations of wave functions and dressing operator are needed when discussing bilinear equations and Lax structure. After that, Miura links between Bogoyavlensky–KdV hierarchy and B–mKdV hierarchy are discussed. Finally, we construct soliton solutions of B–mKdV hierarchy.