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By principal representation of toroidal Lie algebra sl 2 tor , 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 sl 2 tor 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. [ABSTRACT FROM AUTHOR]

The modified Kadomtsev-Petviashvili hierarchy of B type (mBKP hierarchy) and its constrained cases are investigated from the aspect of the Lax formulation. Starting from the Lax equation of the mBKP hierarchy, we firstly derive the bilinear equations and show the existence of the tau functions. Then the additional symmetries are constructed, and the corresponding generator is showed to be the squared eigenfunction symmetry. After that, the actions of the additional symmetries on the tau functions are given. At last, the Lax formulation of the constrained mBKP hierarchy is investigated and the corresponding bilinear equations are also discussed. [ABSTRACT FROM AUTHOR]

A class of non-local integrable equations is proposed as a new generalization of the constrained 1st modified KP hierarchy (i.e. Kupershmidt–Kiso version), which is called the generalized k-modified intermediate long wave ( gILW k for short) hierarchy. Then the bilinear formulations for the gMILW k hierarchy are constucted. Based on the bilinear formulations, rational and soliton solutions are obtained by using the boson-fermion correspondence. [ABSTRACT FROM AUTHOR]

In a companion paper (arXiv:1910.03150) to this one, we proved that the Gromov–Witten theory of a Fano orbifold line of type D is governed by a system of Hirota Bilinear Equations. The goal of this paper is to prove that every solution to the Hirota Bilinear Equations determines a solution to a new integrable hierarchy of Lax equations. We suggest the name extended D-Toda hierarchy for this new system of Lax equations, because it should be viewed as the analogue of Carlet's extended bi-graded Toda hierarchy, which is known to govern the Gromov–Witten theory of Fano orbifold lines of type A. [ABSTRACT FROM AUTHOR]

In this paper, two kinds of gauge transformations for the constrained q-deformed modified KP hierarchy and their relations with additional symmetries are discussed. The corresponding gauge transformations are required to keep both the Lax equation and the Lax operator, which can be obtained by choosing the special generating functions. Then the successive gauge transformations for the constrained q-modified KP hierarchy are investigated. Finally, we find that additional symmetry flows commute with the gauge transformation for the q-mKP hierarchy. [ABSTRACT FROM AUTHOR]

In this paper, with the help of the S function and ghost symmetry for the discrete KP hierarchy which is a semi-discrete version of the KP hierarchy, the ghost flow on its eigenfunction (adjoint eigenfunction) and the spectral representation of its Baker-Akhiezer function and adjoint Baker-Akhiezer function are derived. From these observations above, some important distinctions between the discrete KP hierarchy and KP hierarchy are shown. Also we give the ghost flow on the tau function and another kind of proof of the ASvM formula of the discrete KP hierarchy. [ABSTRACT FROM AUTHOR]

The additional symmetries of the constrained CKP (cCKP) and BKP (cBKP) hierarchies are given by their actions on the Lax operators, and their actions on the eigenfunction and adjoint eigenfunction {Φ, Ψ} are presented explicitly. Furthermore, we show that acting on the space of the wave operator, ∂* forms new centerless W and W W-subalgebra of centerless W respectively. In order to define above symmetry flows ∂* of the cCKP and cBKP hierarchies, two vital operators Y are introduced to revise the additional symmetry flows of the CKP and BKP hierarchies. [ABSTRACT FROM AUTHOR]