Dispersionless BKP Research Articles

The constrained dispersionless BKP hierarchy and its additional symmetries

In this paper, the constrained dispersionless BKP hierarchy are given by restricting the formal Laurent series [Formula: see text] of the dispersionless BKP hierarchy to [Formula: see text]. Its additional symmetries are given by introducing vital formal Laurent series [Formula: see text]. The additional flows form the nonabelian [Formula: see text]-grading Lie algebra. Furthermore, the additional flows acting on [Formula: see text] and [Formula: see text] are presented.

Decompositions and linear superpositions of B‐type Kadomtsev‐Petviashvili equations

The existence of decomposition solutions of the well‐known nonlinear BKP hierarchy is explored. It is shown that these decompositions provide simple and interesting relationships between classical integrable systems and the BKP hierarchy. Further, some special decomposition solutions display a rare property: they can be linearly superposed. With the emphasis on the case of the fifth BKP equation, the structure characteristic having linear superposition solutions is analyzed. Finally, we obtain similar superposed solutions in the dispersionless BKP hierarchy.

Dispersionless BKP Equation, the Manakov–Santini System and Einstein–Weyl Structures

We construct a map from solutions of the dispersionless BKP (dBKP) equation to solutions of the Manakov–Santini (MS) system. This map defines an Einstein–Weyl structure corresponding to the dBKP equation through the general Lorentzian Einstein–Weyl structure corresponding to the MS system. We give a spectral characterisation of reduction in the MS system, which singles out the image of the dBKP equation solution, and also consider more general reductions of this class. We define the BMS system and extend the map defined above to the map (Miura transformation) of solutions of the BMS system to solutions of the MS system, thus obtaining an Einstein–Weyl structure for the BMS system.

Integrable hierarchies associated to infinite families of Frobenius manifolds

We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to A N , D N and B N singularities. In the case of A N Frobenius manifolds our hierarchy turns out to coincide with the dispersionless KP hierarchy; for B N Frobenius manifolds it coincides with the dispersionless BKP hierarchy; and for D N hierarchy it is a certain reduction of the dispersionless 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain coefficients of A N , D N and B N Frobenius potentials.

Dispersionless and multicomponent BKP hierarchies with quantum torus symmetries

In this article, we will construct the additional perturbative quantum torus symmetry of the dispersionless BKP hierarchy basing on the $W_{\infty}$ infinite dimensional Lie symmetry. These results show that the complete quantum torus symmetry is broken from the BKP hierarchy to its dispersionless hierarchy. Further a series of additional flows of the multicomponent BKP hierarchy will be defined and these flows constitute an $N$-folds direct product of the positive half of the quantum torus symmetries.

Dispersionless BKP Hierarchy and Quadrant Löwner Equation

We show that $N$-variable reduction of the dispersionless BKP hierarchy is described by a L\"owner type equation for the quadrant.

On kernel formulas and dispersionless Hirota equations of the extended dispersionless BKP hierarchy

We derive dispersionless Hirota equations of the extended dispersionless BKP(EdBKP) hierarchy proposed by Takasaki from the method of kernel formula provided by Carroll and Kodama. Moreover, we verify associativity equations (WDVV equations) in the EdBKP hierarchy from dispersionless Hirota equations and give a realization of associative algebra with structure constants expressed in terms of the residue formula.

On Dispersionless BKP Hierarchy and its Reductions

Integrable dispersionless Kadomtsev-Petviashvili (KP) hierarchy of B type is considered. Addition formula for the τ-function and conformally invariant equations for the dispersionless BKP (dBKP) hierarchy are derived. Symmetry constraints for the dBKP hierarchy are studied.

Quasi-classical limit of BKP hierarchy andW-infinity symmetries

Previous results on quasi-classical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the Baker-Akhiezer function and the tau function are reformulated so as to fit into the analysis of quasi-classical limit. Two subalgebras $\WB_{1+\infty}$ and $\wB_{1+\infty}$ of the W-infinity algebras $W_{1+\infty}$ and $w_{1+\infty}$ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasi-classical limit, the dispersionless BKP hierarchy. The quantum W-infinity algebra $\WB_{1+\infty}$ emerges in symmetries of the BKP hierarchy. In quasi-classical limit, these $\WB_{1+\infty}$ symmetries are shown to be contracted into $\wB_{1+\infty}$ symmetries of the dispersionless BKP hierarchy.