Toroidal Lie algebras and Bogoyavlensky's \((2+1)\)-dimensional equation (Q2734541)

In the early 1980s E. Date, M. Kashiwara and others discovered that the Lie algebra \(\mathfrak {gl}(\infty)\) acts on the solutions of the Kadomtsev-Petviashvili (KP) hierarchy. In 1999 Y. Billig, K. Iohara and others derived new hierarchies of Hirota bilinear equations from the 2-toroidal Lie algebra \(\mathfrak g^{tor}\), a central extension of double loop algebras \(\mathfrak g(s^{\pm 1},t^{\pm 1})\) of a simple Lie algebra \(\mathfrak g\). Applying the construction to the case \(\mathfrak g^{tor}=\mathfrak {sl}^{\text{tor}} _l\) the authors obtain an extension of the \(l\)-reduced KP hierarchy of Hirota bilinear form. For simplicity of exposition, the authors consider the case \(\mathfrak g=\mathfrak {sl}_2\). The typical Hirota bilinear equations of lower degree are NEWLINE\[NEWLINE (D^4 _x -4D_x D_t)\tau \cdot \tau=0, \qquad (D_yD_x ^3 +2 D_yD_t -5D_zD_x)\tau \cdot \tau=0. NEWLINE\]NEWLINE The first equation is a famous bilinear form of the KdV equation \(4u_t=6uu_x+u_{xxx}\) with \(u=2(\log \tau)_xx\). The second equation can be written as a Bogoyavlensky (2+1)-dimensional equation that has an equivalent Lax form: NEWLINE\[NEWLINE \frac{\partial P}{\partial z}=[P\partial_y+C,P], NEWLINE\]NEWLINE where \(=\partial_x ^2+u, C=v \partial_x+\frac{3}{4}u_y, v=(\log \tau)_{xy}\). The object of the paper under review is to establish a Lax formalism of the Hirota bilinear equations arising from \(\mathfrak{sl}^{\text{tor}}_l\) for any \(l\geq 2\). For this purpose the \(l\)-Bogoyavlensky hierarchy is introduced and the family of special solutions called breaking solitons is constructed.NEWLINENEWLINEThe structure of the paper is as follows. In Sec. 2 the discussion of the hierarchy from the viewpoint of representation theory and a definition of the algebra \(\mathfrak g ^{\text{tor}}\) are given. It is proved that every vector \(\tau\) in the \(SL^{\text{tor}}_l\)-orbit of the vacuum vector satisfies a hierarchy of Hirota bilinear equations. In Sec. 3 the review of the Lax formalism of the KP hierarchy and the heuristic introduction to the Bogoyavlensky hierarchy are given. Then the \(l\)-Bogoyavlensky hierarchy for each \(l \geq 2\) is considered. In Sec. 4 the residue formula for the Baker-Akhiezer functions of the hierarchy is obtained that lead to the same system of Hirota bilinear equations. In Sec. 5 the special solutions of the hierarchy, i.e. the Wronskian solutions and \(N\)-solitons are derived. In conclusion the table of Hirota equations of low degree are given.