KdV solves BKP

Significance

Integrable systems constitute an essential part of modern mathematics and theoretical physics. Interrelations between different integrable systems allow us to uncover unexpected relations between various mathematical and physical problems and eventually, to solve them. This paper provides a simple and surprising relationship between two classical integrable systems—the Korteweg–de Vries (KdV) and type B Kadomtsev–Petviashvili (BKP) hierarchies.

Abstract

In this note, we prove that any $\tau$-function of the Korteweg–de Vries (KdV) hierarchy also solves the type B Kadomtsev–Petviashvili (BKP) hierarchy after a simple rescaling of times.

In this paper, we answer the question about the relation between the Korteweg–de Vries (KdV) and type B Kadomtsev–Petviashvili (BKP) hierarchies raised in ref. 1. Namely, in ref. 1 it was observed that several families of the KdV $\tau$-functions important in enumerative geometry and theoretical physics also solve the BKP hierarchy after a simple rescaling of times. Here, we prove that any $\tau$-function of KdV solves the BKP hierarchy.

In terms of $\tau$-function ${\tau }_{KP}\left(\mathbf{t}\right)$, the Kadomtsev–Petviashvili (KP) hierarchy, introduced in ref. 2, is described by the Hirota bilinear identity

$${\oint }_{\infty }{e}^{\xi \left({\mathbf{t}-\mathbf{t}}^{\prime },z\right)}{\tau }_{KP}\left(\mathbf{t}-\left[z^{-1}\right]\right){\tau }_{KP}\left(\mathbf{t}\prime +\left[z^{-1}\right]\right)dz=0.$$ [1]

This bilinear identity encodes all nonlinear equations of the KP hierarchy. Here, we use the standard shorthand notations

$$\mathbf{t}\pm \left[z^{-1}\right]≔\left\{t_1\pm z^{-1},t_2\pm \frac{1}{2}{z}^{-2},t_3\pm \frac{1}{3}{z}^{-3},\dots \right\}$$ [2]

and

$$\xi \left(\mathbf{t},z\right)=\sum _{k>0}{t}_k{z}^k.$$ [3]

If a $\tau$-function of the KP hierarchy does not depend on even time variables,

$$\frac{\partial }{\partial t_{2k}}{\tau }_{KdV}\left(\mathbf{t}\right)=0\hspace{1em}\forall \hspace{0.17em}k>0,$$ [4]

then it is a $\tau$-function of the KdV hierarchy. Since we still have arbitrary $t_{2k}$ and $t_{2k}^{\prime }$ in the first factor of the integrand in [1], for the KdV hierarchy the Hirota bilinear identity is given by

$${\oint }_{\infty }{e}^{{\xi }^o\left({\mathbf{\text{t-t}}}^{\prime },z\right)}\hspace{0.17em}f\left(z\right)\hspace{0.17em}{\tau }_{KdV}\left(\mathbf{t}-\left[z^{-1}\right]\right){\tau }_{KdV}\left(\mathbf{t}\prime +\left[z^{-1}\right]\right)dz=0$$ [5]

for arbitrary series $f\left(z\right)\in \mathbb{C}\left[\left[z^2\right]\right]$. Here, we introduce

$${\xi }^o\left(\mathbf{t},z\right)=\sum _{k\in {\mathbb{Z}}_{\text{odd}}^{+}}{t}_k{z}^k.$$ [6]

In terms of the Hirota derivatives

$$\begin{array}{ll}\hfill & P\left(D_1,D_2,\dots \right)\hspace{0.17em}f\cdot g\hfill \\ \hfill & \hspace{1em}={\left.P\left(\frac{\partial }{\partial y_1},\frac{\partial }{\partial y_2},\dots \right)f\left(\mathbf{t}+\mathbf{y}\right)g\left(\mathbf{t}-\mathbf{y}\right)\right|}_{\mathbf{y}=\mathbf{0}},\hfill \end{array}$$ [7]

the first few equations of the KdV hierarchy can be represented as

$$\begin{array}{c}\hfill \left(D_1^4-4D_1{D}_3\right){\tau }_{KdV}\cdot {\tau }_{KdV}=0,\\ \hfill \left(D_1^6+4D_1^3{D}_3-32D_3^2\right){\tau }_{KdV}\cdot {\tau }_{KdV}=0,\\ \hfill \left(D_1^6-20D_1^3{D}_3-80D_3^2+144D_1{D}_5\right){\tau }_{KdV}\cdot {\tau }_{KdV}=0.\end{array}$$ [8]

The BKP hierarchy was introduced by Date et al. in ref. 3. It can be represented in terms of $\tau$-function ${\tau }_{BKP}\left(\mathbf{t}\right)$ that depends only on odd time variables by the Hirota bilinear identity similar to ref. 1:

$$\begin{array}{l}\hfill \frac{1}{2\pi i}{\oint }_{\infty }{e}^{{\xi }^o\left(\mathbf{t}-\mathbf{t}\prime ,z\right)}{\tau }_{BKP}\left(\mathbf{t}-2\left[z^{-1}\right]\right){\tau }_{BKP}\left(\mathbf{t}\prime +2\left[z^{-1}\right]\right)\frac{dz}{z}\\ \hfill ={\tau }_{BKP}\left(\mathbf{t}\right){\tau }_{BKP}\left(\mathbf{t}\prime \right).\end{array}$$ [9]

The first equation of the BKP hierarchy in terms of the Hirota derivatives is given by

$$\left(D_1^6-5D_1^3{D}_3-5D_3^2+9D_1{D}_5\right){\tau }_{BKP}\cdot {\tau }_{BKP}=0.$$ [10]

Note that if we map $D_k\mapsto D_k/2$, then this equation coincides with the last equation in [8]. Let us show that this is more than just a coincidence, and all equations of the BKP hierarchy follow from the KdV hierarchy. The main result of this note is Theorem 1.

Theorem 1. For any KdV $\tau$-function,

$${\tau }_{BKP}\left(\mathbf{t}\right)={\tau }_{KdV}\left(\mathbf{t}/2\right)$$ [11]

is a $\tau$-function of the BKP hierarchy.

Proof: Let us consider

$$f\left(z\right)=\frac{e^{{\xi }^o\left(\mathbf{t}-\mathbf{t}\prime ,z\right)}-e^{-{\xi }^o\left(\mathbf{t}-\mathbf{t}\prime ,z\right)}}{z}\in \mathbb{C}\left[\left[z^2\right]\right],$$ [12]

then the Hirota bilinear identity [5] for the KdV hierarchy leads to

$$\begin{array}{l}\hfill \frac{1}{2\pi i}{\oint }_{\infty }{e}^{2{\xi }^0\left({\mathbf{t}-\mathbf{t}}^{\prime },z\right)}{\tau }_{KdV}\left(\mathbf{t}-\left[z^{-1}\right]\right){\tau }_{KdV}\left(\mathbf{t}\prime +\left[z^{-1}\right]\right)\frac{dz}{z}\\ \hfill ={\tau }_{KdV}\left(\mathbf{t}\right){\tau }_{KdV}\left(\mathbf{t}\prime \right).\end{array}$$ [13]

Now, we substitute the times $\mathbf{t}\mapsto \mathbf{t}/2$, $\mathbf{t}\prime \mapsto \mathbf{t}\prime /2$:

$$\begin{array}{l}\hfill \frac{1}{2\pi i}{\oint }_{\infty }{e}^{{\xi }^o\left(\mathbf{t}-\mathbf{t}\prime ,z\right)}{\tau }_{KdV}\left(\mathbf{t}/2-\left[z^{-1}\right]\right){\tau }_{KdV}\left(\mathbf{t}\prime /2+\left[z^{-1}\right]\right)\frac{dz}{z}\\ \hfill ={\tau }_{KdV}\left(\mathbf{t}/2\right){\tau }_{KdV}\left(\mathbf{t}\prime /2\right).\end{array}$$ [14]

This equation coincides with the Hirota bilinear identity for the BKP hierarchy [9] if we identify $\tau$-functions by [11]. This completes the proof.

It is easy to see that the converse statement is false: that is, the KdV hierarchy is a certain reduction of the BKP hierarchy. This reduction, as well as a similar description of the higher Gelfand–Dickey hierarchies, will be discussed elsewhere.

This theorem partially explains the results of refs. 1 and 4. Indeed, in ref. 4 it was noted that the Kontsevich–Witten $\tau$-function—one of the most important solutions of the KdV hierarchy—has a simple expansion in terms of the Schur Q functions. This result was generalized in ref. 1 to a family of KdV $\tau$-functions related to the Brézin–Gross–Witten model. On the basis of these expansions, the author has conjectured (for the Kontsevich–Witten $\tau$-function) and proved (for the Brézin–Gross–Witten $\tau$-function) that these KdV $\tau$-functions also solve the BKP hierarchy. From Theorem 1, it follows that this part of the conjectures in ref. 1 trivially follows from a general relation between the KdV and BKP hierarchies. Moreover, this theorem makes the Schur Q functions a natural basis for expansion of the KdV $\tau$-functions.

There are several different ways to relate KdV and BKP hierarchies. In particular, in ref. 5 the authors describe an identification of the KdV hierarchy with the four reduction of BKP (the remark on p. 1098 in ref. 5). Another reduction from BKP to KdV (so-called one-constrained BKP) was described in refs. 6 and 7. To the best of our understanding, these reductions do not coincide with the one considered in this paper.

Data Availability

There are no data underlying this work.