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A class of reductions of the two-component KP hierarchy and the Hirota–Ohta system

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Abstract

We introduce a class of reductions of the two-component KP hierarchy that includes the Hirota–Ohta system hierarchy. The description of the reduced hierarchies is based on the Hirota bilinear identity and an extra bilinear relation characterizing the reduction. We derive the reduction conditions in terms of the Lax operator and higher linear operators of the hierarchy, as well as in terms of the basic two-component KP system of equations.

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References

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Funding

The reported study was funded by the RFBR and NSFC (project No. 21-51-53017).

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Authors and Affiliations

  1. Landau Institute for Theoretical Physics, Russian Academy of Sciences, Chernogolovka, Moscow Region, Russia

    L. V. Bogdanov

  2. Department of Mathematics, Faculty of Science, Ningbo University, Ningbo, China

    Lingling Xue

Authors
  1. L. V. Bogdanov
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  2. Lingling Xue
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Correspondence to L. V. Bogdanov.

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Translated from Teoreticheskaya i Matematicheskaya Fizika, 2022, Vol. 211, pp. 37–47 https://doi.org/10.4213/tmf10243.

Appendix Reductions in terms of the Lax–Sato equations

Here, we briefly describe the the two-component KP hierarchy with times (12) [2] in terms of the Lax–Sato equations and discuss the class of reductions corresponding to bilinear relation (13). In the scalar case, reductions of this type were described in [3].

The Lax–Sato equations define the dynamics of pseudodifferential operators

$$\begin{aligned} \, &L=\partial + U_1\partial^{-1} + U_2\partial^{-2} + \cdots, \\ &M=\sigma_3 + V_1\partial^{-1} + V_2\partial^{-2} + \cdots, \end{aligned}$$
where \(U_n\) and \(V_n\) are \(2\times 2\) matrices, \(\partial=\partial_x\), \(\sigma_3= \bigl(\begin{smallmatrix} 1&\hphantom{-}0\\ 0&-1 \end{smallmatrix}\bigr)\), with the characteristic properties
$$[L,M]=0, \qquad M^2=1.$$
For odd times, we have
$$ \frac{\partial L}{\partial t_{2n+1}}=[(L^{2n+1})_+, L], \qquad \frac{\partial M}{\partial t_{2n+1}}=[(L^{2n+1})_+, M],$$
(A.1)
and for even times,
$$ \frac{\partial L}{\partial t_{2n}}=[(L^{2n}M)_+, L], \qquad \frac{\partial M}{\partial t_{2n}}=[(L^{2n}M)_+, M].$$
(A.2)
The Gelfand–Dickey reductions for this hierarchy are defined by the following conditions: for odd flows,
$$(L^{2n+1})_-=0,\qquad (L^{2n+1})_+= \mathbf{D}^{(2n+1)},$$
where \(D^{(2n+1)}\) is a differential operator of order \(2n+1\) with matrix coefficients, and for even flows,
$$(L^{2n}M)_-=0,\qquad (L^{2n}M)_+= \mathbf{D}^{(2n)}.$$
Introducing a formal pseudodifferential dressing operator (related to the operator \((1+\hat K_+)\) [2])
$$P=I + W_1\partial^{-1} + W_2\partial^{-2} + \cdots,$$
allows expressing the operators \(L\) and \(M\) as
$$L=P\partial P^{-1}, \qquad M=P \sigma_3 P^{-1}.$$
The operators \(L\) and \(M\) defined this way manifestly have the necessary characteristic properties. The dynamics of the dressing operator is governed by the Sato equations [7]
$$ \begin{aligned} \, &\frac{\partial P}{\partial t_{2n+1}}=- (P\partial^{2n+1} P^{-1})_- P, \\ &\frac{\partial P}{\partial t_{2n}}=-(P\partial^{2n}\sigma_3 P^{-1})_-P, \end{aligned}$$
(A.3)
which implies Eqs. (A.1) and (A.2). To find a dressing operator starting from \(L\) (or \(M\)), we must solve the factorization problem \(LP=P\partial\), \(MP=P\sigma_3\). The reduction to the Hirota–Ohta system hierarchy is described by the conditions
$$\begin{aligned} \, &L^*=JLJ,\\ &M^*=JMJ,\\ &P^*=-JP^{-1}J. \end{aligned}$$
A class of reductions corresponding to bilinear relation (13) is defined by the following conditions: for \({A=I}\),
$$(P\partial^n J P ^*)_-=0,$$
and for \(A=\sigma_3\),
$$(P\sigma_3 J\partial^n P ^*)_-=0.$$
Introducing the differential operators \(\mathbf{A}_k=P\partial^{k} P^\dagger\) (for \(A=I\)) or \(\mathbf{A}_k=P\sigma_3\partial^{k} P^\dagger\) (for \(A=\sigma_3\)), where we use the notation \(P^\dagger=J P^*J^{-1}\), we obtain the relations
$$\begin{aligned} \, &L \mathbf{A}_k=\mathbf{A}_k L^{\dagger},\\ &M\mathbf{A}_k=\mathbf{A}_k M^{\dagger}, \end{aligned}$$
and also relations of form (29).

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Bogdanov, L.V., Xue, L. A class of reductions of the two-component KP hierarchy and the Hirota–Ohta system. Theor Math Phys 211, 473–482 (2022). https://doi.org/10.1134/S0040577922040031

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