2-BKP Darboux Transformations

• 2-BKP Darboux Transformations are explicit gauge symmetries that generate new tau-functions for multicomponent integrable hierarchies via operator factorizations.
• They utilize pseudo-differential Lax operators and squared eigenfunction potentials to maintain bilinear identities and reduction constraints.
• Iterated transformations reveal Pfaffian identities and additional symmetry flows, uniting gauge techniques with fermionic structures in integrable models.

The 2-component BKP (2-BKP) Darboux transformations are algebraic symmetries of the 2-BKP integrable hierarchy, a generalization of the classical BKP hierarchy associated with the infinite-dimensional $b_\infty$ and $d_\infty$ Lie algebras. The 2-BKP hierarchy governs tau-functions that encode solutions to multicomponent nonlinear systems, including the Novikov-Veselov equation and the total descendent potential of $D$-type singularities. The Darboux transformations provide a mechanism to generate new tau-functions and solutions from existing ones by applying explicit operator factorizations, leading to a unified structure encapsulating gauge symmetries, Pfaffian identities, and addition formulae (Chen et al., 19 Nov 2025, Zabrodin, 8 Jun 2025, Yang et al., 2021).

1. The 2-Component BKP Hierarchy and Tau Function Structure

The 2-BKP hierarchy is formulated in terms of a tau-function $\tau(\mathbf t)$, where the time variables are vector-valued: $$\mathbf t = \bigl(t^{(1)}, t^{(2)}\bigr),\quad t^{(a)} = \bigl(t^{(a)}_1 = x_a,\, t^{(a)}_3,\, t^{(a)}_5, \dots\bigr),\quad a = 1,2.$$ The tau-function satisfies a bilinear Hirota identity,

$\Res_{z} z^{-1} \, \tau(\mathbf t - 2[z^{-1}]_1)\, \tau(\mathbf t' + 2[z^{-1}]_1)\, e^{\xi(t^{(1)}-t^{(1)'},z)} = \Res_{z} z^{-1} \, \tau(\mathbf t - 2[z^{-1}]_2)\, \tau(\mathbf t' + 2[z^{-1}]_2)\, e^{\xi(t^{(2)}-t^{(2)'},z)},$

where $[z^{-1}]_a$ is a formal shift in the $a$-th set of time variables and $\xi(t^{(a)},z)=\sum_{i\ge1} t^{(a)}_{2i-1}z^{2i-1}$.

The Shiota Lax formalism introduces operators $L_a$ and $H$: $$L_a = \partial_a + \sum_{i\geq1} u_{a,i}\, \partial_a^{-i}, \quad H = \partial_1\partial_2 + \rho,\quad \rho = 2\,\partial_1\partial_2\log\tau,$$ where these satisfy integrability constraints and generate the hierarchy's time flows. The operators $L_a$ are scalar pseudo-differential operators in $\partial_a$ alone (after projection), and $H$ encodes the interaction between the two sectors (Chen et al., 19 Nov 2025).

2. Definition of the 2-BKP Darboux Transformations

Given a 2-BKP eigenfunction $q(\mathbf t)$ satisfying $\partial_{a,n}q = B^{(a)}_n(q)$ and $H(q) = 0$, the basic Darboux (gauge) transformation is presented as: $$T_a[q] = 1 - 2\,q^{-1} \partial_a^{-1} q_{x_a},\qquad a=1,2.$$ Applying $T_a[q]$ constructs a new tau-function,

$$\tau^{[1]}(\mathbf t) = q(\mathbf t)\, \tau(\mathbf t),$$

which continues to satisfy the 2-BKP bilinear equations. The transformed Lax operators are

$$L_a^{[1]} = T_a[q]\, L_a\, T_a[q]^{-1},\qquad H^{[1]} = \partial_1\, T_1[q]\, \partial_1^{-1}\, H\, T_1[q]^{-1} = \partial_2\, T_2[q]\, \partial_2^{-1}\, H\, T_2[q]^{-1},$$

demonstrating covariance of the Lax structure under Darboux transformations (Chen et al., 19 Nov 2025).

On Baker–Akhiezer functions $\psi_a(\mathbf t, z) = W_a \exp \xi(t^{(a)}, z)$, the transformation acts simply as $$\psi_a^{[1]}(\mathbf t, z) = T_a[q](\psi_a(\mathbf t, z))$$.

3. Squared Eigenfunction Potentials and Alternative Transformations

The transformation theory incorporates the use of "squared-eigenfunction potentials" (SEPs): $$\Omega(p, r) = \partial_1^{-1}\bigl(p\, r_{x_1} - r\, p_{x_1}\bigr) = -\partial_2^{-1}\bigl(p\, r_{x_2} - r\, p_{x_2}\bigr).$$ Given another eigenfunction $\widetilde q$, its Darboux transform is

$$\widetilde q^{[1]} = q^{-1} \Omega(\widetilde q, q),$$

and the transformation

$\tau \longmapsto \Omega(\widetilde q, q)\, \tau$

provides a further class of 2-BKP Darboux-type symmetries. These constructions generalize the classical squared-eigenfunction mechanism for generating new solutions from pairs of eigenfunctions, and clarify the algebraic interplay between tau-functions and their eigenfunction symmetries (Chen et al., 19 Nov 2025).

4. Darboux Transformations in Reductions and Lax Structures

The 2-BKP hierarchy admits reductions via constraints on the Lax operators. For integers $(M_1, M_2)$, define

$\mathcal L = L_1^{M_1} + L_2^{M_2}.$

If an eigenfunction $q$ satisfies $\mathcal L(q) = c\, q$, then the Darboux transformation $q \mapsto q\, \tau$ preserves the reduction constraint. The transformed reduced Lax operator becomes

$$\mathcal L^{[1]} = \Bigl(T_1[q]\, \pi_1(\mathcal L)\, T_1[q]^{-1}\Bigr)_{\geq 1} + \Bigl(T_2[q]\, \pi_2(\mathcal L)\, T_2[q]^{-1}\Bigr)_{\geq 0},$$

where $\pi_a$ is the projection onto the $\partial_a$-subalgebra, and negative parts are eliminated to ensure the reduced structure. This ensures compatibility of the Darboux formalism with reduction schemes such as rational or finite-gap solutions (Chen et al., 19 Nov 2025).

5. Iterated Darboux Transformations and Pfaffian Identities

Successive Darboux transformations applied to a sequence of eigenfunctions $\{q_1, ..., q_m\}$ lead to structured identities for the resultant tau-function: $$\tau^{[0]} = \tau,\quad \tau^{[k]} = q_k^{[k-1]}\, \tau^{[k-1]},\quad q_j^{[j-1]} = T[q_{j-1}^{[j-2]}] \cdots T[q_1](q_j).$$ After $m$ applications,

$$\tau^{[m]}(\mathbf t) = \mathrm{Pf}(Q_m(q_m,...,q_1))\,\tau(\mathbf t),$$

where $Q_m$ is an antisymmetric matrix with entries $Q_{ij} = \Omega(q_{m+1-i}, q_{m+1-j})$, and, for odd $m$, an extra column is adjoined.

A central result is the 2-BKP Pfaffian addition formula obtained by specializing the $q_k$ to Baker–Akhiezer wavefunctions,

$$\frac{\tau\Bigl(\mathbf t - 2\sum_{q=1}^{N_1} [\lambda_q^{-1}]_1 - 2\sum_{q=1}^{N_2} [\mu_q^{-1}]_2 \Bigr)}{\tau(\mathbf t)}\,\prod_{i<j}\frac{\lambda_i - \lambda_j}{\lambda_i + \lambda_j}\prod_{s<\ell}\frac{\mu_\ell - \mu_s}{\mu_\ell + \mu_s} = \mathrm{Pf}\begin{pmatrix}E & F \ -F^{T} & G\end{pmatrix},$$

with $E$, $F$, $G$ constructed from pairings of the wavefunctions indexed by spectral parameters $\lambda_i$ and $\mu_j$ (Chen et al., 19 Nov 2025).

These Pfaffian relations are reminiscent of the addition formulae for classical BKP and KP hierarchies and serve as higher analogues of Bäcklund transformation superposition principles (Zabrodin, 8 Jun 2025, Yang et al., 2021).

6. Additional Symmetries as Special Darboux Transformations

The 2-BKP Darboux transformations naturally include the so-called "additional symmetries." In particular, the action of the BKP-vertex operators

$$X_{ab}(\lambda, \mu) = \varepsilon_{ab}(\lambda, \mu)\, e^{\xi(t^{(a)}, \mu) - \xi(t^{(b)}, \lambda)}\, e^{-2\xi(\tilde\partial_{t^{(a)}},\mu^{-1}) + 2\xi(\tilde\partial_{t^{(b)}}, \lambda^{-1})}$$

on the tau-function translates to

$$X_{ab}(\lambda, \mu)\, \tau = (-1)^a\, \Omega\bigl( \psi_b(-\lambda), \psi_a(\mu) \bigr)\, \tau,$$

and so $\tau \mapsto \tau + C X_{ab}(\lambda,\mu)\, \tau$ yields another tau-function. The infinitesimal case matches the Adler–Shiota–van Moerbeke additional symmetry flow. Thus, the operator formalism $T_a[q]$ subsumes both Darboux and additional-flow symmetries within a universal gauge-theoretic framework (Chen et al., 19 Nov 2025).

7. Significance and Broader Connections

The 2-BKP Darboux framework unites several aspects of multicomponent integrable hierarchies: explicit tau-function generation, gauge symmetries in the Lax representation, and the algebraic structure of addition formulae via Pfaffian determinants. The same algebraic machinery supports connections to free-fermion constructions for the (single-component) BKP hierarchy, with deep analogues in the operator insertion and bilinear-identity frameworks developed by the Kyoto school (Zabrodin, 8 Jun 2025, Yang et al., 2021).

A plausible implication is that the full scope of 2-BKP Darboux transformations and their associated Pfaffian and fermionic structure offers systematic tools for exploring novel families of solutions, exploring reduction to discrete or difference analogues, and relating integrable hierarchies to moduli of curves and singularity theory via tau-function geometry.