Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

Jian Li; Chuanzhong Li

doi:10.1088/1572-9494/ac01e5

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Communications in Theoretical Physics ›› 2021, Vol. 73 ›› Issue (8) : 85002. DOI: 10.1088/1572-9494/ac01e5

Mathematical Physics

Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation

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Abstract

In this paper, we construct Hamiltonian systems for 2N particles whose force depends on the distances between the particles. We obtain the generalized finite nonperiodic Toda equations via a symmetric group transformation. The solutions of the generalized Toda equations are derived using the tau functions. The relationship between the generalized nonperiodic Toda lattices and Lie algebras is then be discussed and the generalized Kac-van Moerbeke hierarchy is split into generalized Toda lattices, whose integrability and Darboux transformation are studied.

Key words

Hamiltonian systems / Toda lattices / Darboux transformation / Lie algebra / Kac-van Moerbeke hierarchy

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Jian Li, Chuanzhong Li. Generalizations of the finite nonperiodic Toda lattice and its Darboux transformation[J]. Communications in Theoretical Physics, 2021, 73(8): 85002 https://doi.org/10.1088/1572-9494/ac01e5

1. Introduction

Many mathematicians have made significant contributions to the Toda equations and have achieved many meaningful results. For example, in 1974, the complete integrability of the Toda lattice was demonstrated by Henon [1]. A few years later, the nonlinear interactions in chains were well known [2]. Also, in 1974, according to Flaschka [3, 4], the Lax form was equivalent to the periodic Toda lattice:

\begin{array}{rcl} \frac{d L}{d t} = [B, L], \end{array}

(1.1)

where L is a tridiagonal matrix,

B = {(L)}_{> 0} - {(L)}_{< 0}

. Further, Moser showed the complete integrability of nonperiodic Toda lattices [5]. With the development of integrable systems, the relationship was established between nonperiodic Toda lattices and their geometry and topology. Then, by considering the full Kostant–Toda hierarchy with real variables [6–8], it was shown that the regular solutions for the full Kostant–Toda lattices could be split into the algebra sl_n(R). In fact, in recent years, Kodama has made many outstanding achievements with nonperiodic Toda lattices [9–13]. According to [14], any finite dimensional simple Lie algebra or Kac–Moody algebra can be split into generalized Toda lattices, and all these equations are integrable systems. There are many applications of two-dimensional Toda equations, including the inverse scattering method, the Hirota direct method, Darboux transformations [15, 16], and so on [17–24].

This paper is arranged as follows. In section 2, we give the generalized Toda systems for

2 N

particles, and the properties and the solutions of the generalized Toda equations are both discussed. In section 3, the relationship between the generalized nonperiodic Toda lattices and Lie algebras is studied. From the example given in this article, the generalized Kac-van Moerbeke hierarchy can be split into two Toda lattices. In section 4, Darboux transformations of the two-dimensional generalized Toda equations are obtained.

2. Coupled Toda lattice

In this section, we consider

2 n

particles, whose force depends on the distances between the particles. For those particles, the coupled Hamiltonians

(H, \hat{H})

are given by

\begin{array}{rcl} {\begin{cases} H = \frac{1}{2} \sum_{k = 1}^{n} (p_{k}^{2} + ε {\hat{p}}_{k}^{2}) + \sum_{k = 1}^{n - 1} e^{- (q_{k + 1} - q_{k})} \cosh [- \sqrt{ε} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})], \\ \hat{H} = \sum_{k = 1}^{n} p_{k} {\hat{p}}_{k} + \frac{1}{\sqrt{ε}} \sum_{k = 1}^{n - 1} e^{- (q_{k + 1} - q_{k})} \sinh [- \sqrt{ε} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})], \end{cases} \end{array}

(2.1)

where

\frac{d q_{k}}{d t} = \frac{\partial H}{\partial p_{k}}, \frac{d {\hat{q}}_{k}}{d t} = \frac{\partial \hat{H}}{\partial p_{k}}, \frac{d p_{k}}{d t} = - \frac{\partial H}{\partial q_{k}}, \frac{d {\hat{p}}_{k}}{d t} = - \frac{\partial \hat{H}}{\partial q_{k}}

. The Hamiltonian systems then represent a generalized finite nonperiodic lattice described by:

\begin{array}{rcl} {\begin{cases} \frac{d q_{k}}{d t} = p_{k}, \\ \frac{d {\hat{q}}_{k}}{d t} = {\hat{p}}_{k}, \\ \frac{d p_{k}}{d t} = - e^{- (q_{k + 1} - q_{k})} \cosh [- \sqrt{ε} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})] + e^{- (q_{k} - q_{k - 1})} \cosh [- \sqrt{ε} ({\hat{q}}_{k} - {\hat{q}}_{k - 1})], \\ \frac{d {\hat{p}}_{k}}{d t} = - \frac{1}{\sqrt{ε}} e^{- (q_{k + 1} - q_{k})} \sinh [- \sqrt{ε} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})] + \frac{1}{\sqrt{ε}} e^{- (q_{k} - q_{k - 1})} \sinh [- \sqrt{ε} ({\hat{q}}_{k} - {\hat{q}}_{k - 1})], \end{cases} \end{array}

(2.2)

where the formal boundary conditions are given by

e^{- (q_{1} - q_{0})} = e^{- ({\hat{q}}_{1} - {\hat{q}}_{0})} = e^{- (q_{n + 1} - q_{n})} = e^{- ({\hat{q}}_{n + 1} - {\hat{q}}_{n})} = 0

q_{0} = {\hat{q}}_{0} = 0

and

q_{n + 1} = {\hat{q}}_{n + 1} = \infty

. In fact, the equations (2.2) can be expressed in the Lax form, and we introduce a new set of variables, as follows:

\begin{array}{rcl} {\begin{cases} a_{k} = \frac{1}{2} e^{- \frac{1}{2} (q_{k + 1} - q_{k})} \cosh [\frac{- \sqrt{ε}}{2} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})], \\ {\hat{a}}_{k} = \frac{1}{2 \sqrt{ε}} e^{- \frac{1}{2} (q_{k + 1} - q_{k})} \sinh [\frac{- \sqrt{ε}}{2} ({\hat{q}}_{k + 1} - {\hat{q}}_{k})], \\ b_{k} = - \frac{1}{2} p_{k}, \\ {\hat{b}}_{k} = - \frac{1}{2} {\hat{p}}_{k} . \end{cases} \end{array}

(2.3)

Based on the above transformation (2.3), the generalized Toda equations (2.2) become

\begin{array}{rcl} {\begin{cases} \frac{d a_{k}}{d t} = a_{k} (b_{k + 1} - b_{k}) - ε {\hat{a}}_{k} ({\hat{b}}_{k + 1} - {\hat{b}}_{k}), \\ \frac{d {\hat{a}}_{k}}{d t} = {\hat{a}}_{k} (b_{k + 1} - b_{k}) - a_{k} ({\hat{b}}_{k + 1} - {\hat{b}}_{k}), \\ \frac{d b_{k}}{d t} = 2 (a_{k}^{2} + ε {\hat{a}}_{k}^{2} - a_{k - 1}^{2} - ε {\hat{a}}_{k - 1}^{2}), \\ \frac{d {\hat{b}}_{k}}{d t} = 4 (a_{k} {\hat{a}}_{k} - a_{k - 1} {\hat{a}}_{k - 1}), \end{cases} \end{array}

(2.4)

where

a_{0} = {\hat{a}}_{0} = 0

and

a_{n} = {\hat{a}}_{n} = 0

. From the two extended symmetric matrices

L (\hat{L})

, and

a_{k}, b_{k} ({\hat{a}}_{k}, {\hat{b}}_{k})

we can obtain the elements of

L (\hat{L})

, using the expressions given by

\begin{array}{rcl} L = (\begin{array}{ccccc} b_{1} & a_{1} \\ a_{1} & b_{2} & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & b_{n - 1} & a_{n - 1} \\ a_{n - 1} & b_{n} \end{array}), \end{array}

(2.5)

\begin{array}{rcl} \hat{L} = (\begin{array}{ccccc} {\hat{b}}_{1} & {\hat{a}}_{1} \\ {\hat{a}}_{1} & {\hat{b}}_{2} & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & {\hat{b}}_{n - 1} & {\hat{a}}_{n - 1} \\ {\hat{a}}_{n - 1} & {\hat{b}}_{n} \end{array}) . \end{array}

(2.6)

Thus, the systems (2.5) and (2.6) can be written as

\begin{array}{rcl} {\begin{cases} \frac{d}{d t} L (t) = [B, L] + ε [\hat{B}, \hat{L}], \\ \frac{d}{d t} \hat{L} (t) = [\hat{B}, L] + [B, \hat{L}], \end{cases} \end{array}

(2.7)

where

{\begin{cases} B = {(L)}_{> 0} - {(L)}_{< 0}, \\ \hat{B} = {(\hat{L})}_{> 0} - {(\hat{L})}_{< 0}, \end{cases}

{(L)}_{> 0} {((\hat{L})}_{> 0})

is the upper triangular matrix of

L (\hat{L})

, and

{(L)}_{< 0} {((\hat{L})}_{< 0})

is the lower triangular matrix of

L (\hat{L})

. According to [3], we find that the functions

t r (L^{k}), t r ({\hat{L}}^{k})

are constant for any k, and

\begin{array}{rcl} {\begin{cases} t r ([B, L^{k} + 2 ε \sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j} L^{k - 2 j} {\hat{L}}^{2 j}] + ε [\hat{B}, \sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j - 1} L^{k - 2 j + 1} {\hat{L}}^{2 j - 1}]) = \\ \frac{d}{d t} t r (L^{k} + 2 ε \sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j} L^{k - 2 j} {\hat{L}}^{2 j}) = 0, \\ t r ([\hat{B}, L^{k} + 2 ε \sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j} L^{k - 2 j} {\hat{L}}^{2 j}] + [B, \sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j - 1} L^{k - 2 j + 1} {\hat{L}}^{2 j - 1}]) = \\ \frac{d}{d t} t r (\sum_{j = 1}^{[\frac{k}{2}]} C_{k}^{2 j - 1} L^{k - 2 j + 1} {\hat{L}}^{2 j - 1}) = 0. \end{cases} \end{array}

(2.8)

For the

2 n - 2

independent invariant functions,

\begin{array}{rcl} {\begin{cases} H_{k} (L, \hat{L}) = \frac{1}{k + 1} t r (L^{k + 1} + 2 ε \sum_{j = 1}^{[\frac{k + 1}{2}]} C_{k + 1}^{2 j} L^{k + 1 - 2 j} {\hat{L}}^{2 j}), \\ {\hat{H}}_{k} (L, \hat{L}) = \frac{1}{k + 1} t r (\sum_{j = 1}^{[\frac{k + 1}{2}]} C_{k + 1}^{2 j - 1} L^{k - 2 j + 2} {\hat{L}}^{2 j - 1}), \end{cases} \end{array}

(2.9)

where

[]

represents an integer operation and

C_{k + 1}^{2 j - 1}

represents a binomial coefficient. From the calculation and transformation shown in (2.2), we obtain the relationship between the independent invariant functions and Hamiltonian

(H, \hat{H})

\begin{array}{rcl} {\begin{cases} H (L, \hat{L}) = \frac{1}{2} t r (L^{2} + ε {\hat{L}}^{2}), \\ \hat{H} (L, \hat{L}) = \frac{1}{2} t r (\hat{L} L + L \hat{L}) . \end{cases} \end{array}

(2.10)

According to the properties of tridiagonal symmetric matrices [5], this has real and distinct eigenvalues. In fact,

{\hat{q}}_{k + 1} - {\hat{q}}_{k}

and

q_{k + 1} - q_{k}

tend to

\infty

when

t \to \pm \infty

. Furthermore, it is easy to see that

{\hat{a}}_{k}

and a_k tend to zero, so

L (\hat{L})

degenerates into a diagonal matrix

d i a g (λ_{1}, λ_{2}, \dots, λ_{n})

(

d i a g ({\hat{λ}}_{1}, {\hat{λ}}_{2}, \dots, {\hat{λ}}_{n})

\begin{array}{rcl} {\begin{cases} L (t) \to d i a g (λ_{1}, λ_{2}, \dots, λ_{n}), \\ \hat{L} (t) \to d i a g ({\hat{λ}}_{1}, {\hat{λ}}_{2}, \dots, {\hat{λ}}_{n}) . \end{cases} \end{array}

(2.11)

There are many methods to solve the the Toda lattice, such as QR-factorization [25], Gram-Schmidt orthogonalization [26], and so on. Based on the methods provided above and the generalized initial matrices

(L (0), \hat{L} (0))

, the factorization of the generalized initial matrices can be expressed as

\begin{array}{rcl} {\begin{cases} e^{t L (0)} \cosh (ε t \hat{L} (0)) = k (t) r (t) + ε \hat{k} (t) \hat{r} (t), \\ \frac{1}{\sqrt{ε}} e^{t L (0)} \sinh (\sqrt{ε} t \hat{L} (0)) = \hat{k} (t) r (t) + k (t) \hat{r} (t), \end{cases} \end{array}

(2.12)

where

k (t), \hat{k} (t) \in {S O}_{n}

and

r (t), \hat{r} (t)

are upper triangular matrices. In order to solve the generalized Toda equations (2.4), we introduce the τ-functions [27]. We now define two moment matrices,

\begin{array}{rcl} {\begin{cases} M (t) := e^{2 t L (0)} \cosh (2 \sqrt{ε} t \hat{L} (0)) = r^{T} (t) r (t) + ε {\hat{r}}^{T} (t) \hat{r} (t), \\ \hat{M} (t) := \frac{1}{\sqrt{ε}} e^{2 t L (0)} \sinh (2 \sqrt{ε} t \hat{L} (0)) = r^{T} (t) \hat{r} (t) + {\hat{r}}^{T} (t) r (t), \end{cases} \end{array}

(2.13)

where

r^{T} (t) ({\hat{r}}^{T} (t))

is the transpose of

r (t) (\hat{r} (t))

. From the decomposition of (2.13), the matrices r(t) and

\hat{r} (t)

can be obtained by the factorization of (2.12).

Definition 1.The tau functions $τ_{j}, {\hat{τ}}_{j}$ ( $j = 1, \dots, n - 1$ ) are defined by,

\begin{array}{rcl} {\begin{cases} τ_{j} := det (M_{j} (t)) = \prod_{i = 1}^{j} {\tilde{r}}_{i} + ε \sum_{\sum_{i = 1}^{j} k_{i} = 0 m o d 2}^{j} ({\tilde{r}}_{1}^{[k_{1}]} {\tilde{r}}_{2}^{[k_{2}]} \dots {\tilde{r}}_{j}^{[k_{j}]}) + ε^{2} \prod_{i = 1}^{j} {\hat{\tilde{r}}}_{i}, \\ {\hat{τ}}_{j} := det ({\hat{M}}_{j} (t)) = \sum_{m = 1}^{j} {\tilde{r}}_{1} \dots {\hat{\tilde{r}}}_{m} \dots {\tilde{r}}_{j} + ε \sum_{\sum_{i = 1}^{j} k_{i} = 1 m o d 2}^{j} ({\tilde{r}}_{1}^{[k_{1}]} {\tilde{r}}_{2}^{[k_{2}]} \dots {\tilde{r}}_{j}^{[k_{j}]}), \end{cases} \end{array}

(2.14)

where M_j (

{\hat{M}}_{j}

) are the j × j upper-left sub-matrix of M(t) (

\hat{M} (t)

), and

{\tilde{r}}_{p}^{[k_{p}]} = {\begin{cases} {\tilde{r}}_{p}, k_{p} = 0, \\ {\hat{\tilde{r}}}_{p}, k_{p} = 1. \end{cases}

Let diag (r(t))=diag

(r_{1} (t), \dots, r_{n} (t))

and

d i a g (\hat{r} (t)) = d i a g ({\hat{r}}_{1} (t), \dots, {\hat{r}}_{n} (t))

; note that

{\tilde{r}}_{i} = r_{i}^{2} + ε {\hat{r}}_{i}^{2}

{\hat{\tilde{r}}}_{i} = r_{i} {\hat{r}}_{i} + {\hat{r}}_{i} r_{i}

From the gauge transformation and the Gram-Schmidt method of orthonormalization, we then have

\begin{array}{rcl} {\begin{cases} a_{j} (t) = a_{j} (0) \frac{r_{k + 1} r_{k} + ε {\hat{r}}_{k + 1} {\hat{r}}_{k}}{r_{k}^{2} - ε {\hat{r}}_{k}^{2}} + ε {\hat{a}}_{j} (0) \frac{{\hat{r}}_{k + 1} r_{k} + r_{k + 1} {\hat{r}}_{k}}{r_{k}^{2} - ε {\hat{r}}_{k}^{2}}, \\ {\hat{a}}_{j} (t) = {\hat{a}}_{j} (0) \frac{r_{k + 1} r_{k} + ε {\hat{r}}_{k + 1} {\hat{r}}_{k}}{r_{k}^{2} - ε {\hat{r}}_{k}^{2}} + a_{j} (0) \frac{{\hat{r}}_{k + 1} r_{k} + r_{k + 1} {\hat{r}}_{k}}{r_{k}^{2} - ε {\hat{r}}_{k}^{2}} . \end{cases} \end{array}

(2.15)

Therefore, with (2.14) and (2.4), we obtain the solutions

(a_{j} (t), {\hat{a}}_{j} (t), b_{j} (t), {\hat{b}}_{j} (t))

\begin{array}{rcl} {\begin{cases} a_{j} (t) = a_{j} (0) \frac{x τ_{j} (t) - ε y {\hat{τ}}_{j} (t)}{τ_{j}^{2} (t) - ε {\hat{τ}}_{j}^{2} (t)} + ε {\hat{a}}_{j} (0) \frac{y τ_{j} (t) - x {\hat{τ}}_{j} (t)}{τ_{j}^{2} (t) - ε {\hat{τ}}_{j}^{2} (t)}, \\ {\hat{a}}_{j} (t) = {\hat{a}}_{j} (0) \frac{x τ_{j} (t) - ε y {\hat{τ}}_{j} (t)}{τ_{j}^{2} (t) - ε {\hat{τ}}_{j}^{2} (t)} + a_{j} (0) \frac{y τ_{j} (t) - x {\hat{τ}}_{j} (t)}{τ_{j}^{2} (t) - ε {\hat{τ}}_{j}^{2} (t)}, \end{cases} \end{array}

(2.16)

where

x = {(\frac{a_{1} \mp \sqrt{a_{1}^{2} - ε a_{2}^{2}}}{2})}^{\frac{1}{2}}

y = {(\frac{a_{1} \mp \sqrt{a_{1}^{2} - ε a_{2}^{2}}}{2 ε})}^{\frac{1}{2}}

and

a_{1} = τ_{j + 1} (t) τ_{j - 1} (t) + ε {\hat{τ}}_{j + 1} (t) {\hat{τ}}_{j - 1} (t)

a_{2} = {\hat{τ}}_{j + 1} (t) τ_{j - 1} (t) + τ_{j + 1} (t) {\hat{τ}}_{j - 1} (t)

. Furthermore, we have

\begin{array}{rcl} {\begin{cases} b_{j} (t) = \frac{1}{4} \frac{d}{d t} [l o g (c + \sqrt{ε} \hat{c}) + l o g (c - \sqrt{ε} \hat{c})], \\ {\hat{b}}_{j} (t) = \frac{1}{4 \sqrt{ε}} \frac{d}{d t} [l o g (c + \sqrt{ε} \hat{c}) - l o g (c - \sqrt{ε} \hat{c})], \end{cases} \end{array}

(2.17)

where

c = \frac{τ_{j} (t) τ_{j - 1} (t) - ε {\hat{τ}}_{j} (t) {\hat{τ}}_{j - 1} (t)}{τ_{j - 1}^{2} (t) - ε {\hat{τ}}_{j - 1}^{2} (t)}

\hat{c} = \frac{{\hat{τ}}_{j} (t) τ_{j - 1} (t) - τ_{j} (t) {\hat{τ}}_{j - 1} (t)}{τ_{j - 1}^{2} (t) - ε {\hat{τ}}_{j - 1}^{2} (t)}

3. Generalized Toda lattice on Lie algebras

We first give the Lax equations of the generalized nonperiodic Toda lattice, which are connected with the Lie algebra g. The forms are as follows:

\begin{array}{rcl} {\begin{cases} \frac{d}{d t} P = [P, N] + ε [\hat{P}, \hat{N}] \\ \frac{d}{d t} \hat{P} = [P, \hat{N}] + [\hat{P}, N], \end{cases} \end{array}

(3.1)

where

P (t), \hat{P} (t)

are the elements of g and

N (t) (\hat{N} (t))

are the projections of

P (t) (\hat{P} (t))

, which are given by

\begin{array}{rcl} {\begin{cases} P (t) = \sum_{i = 1}^{l} f_{i} (t) h_{α_{i}} + \sum_{i = 1}^{l} g_{i} (t) (e_{- α_{i}} + e_{α_{i}}), \\ \hat{P} (t) = \sum_{i = 1}^{l} {\hat{f}}_{i} (t) h_{α_{i}} + \sum_{i = 1}^{l} {\hat{g}}_{i} (t) (e_{- α_{i}} + e_{α_{i}}), \\ N (t) = - \sum_{i = 1}^{l} g_{i} (t) h_{- α_{i}}, \\ \hat{N} (t) = - \sum_{i = 1}^{l} {\hat{g}}_{i} (t) h_{- α_{i}}, \end{cases} \end{array}

(3.2)

where

h_{α_{i}}, e_{\pm α_{i}} (i = 1, 2, \dots, l)

are the Chevalley basis of the algebra,

[h_{α_{i}}, h_{α_{j}}] = 0

[h_{α_{i}}, e_{\pm α_{i}}] = \pm C_{i j} e_{\pm α_{i}}

[e_{α_{i}}, e_{- α_{j}}] = δ_{i j} h_{α_{j}}

, and

C_{i j} = α_{i} (h_{α_{i}})

is the Cartan matrix. In fact, Chevalley proved the complete integrability of the algebra, and Kostant discussed the geometry of the iso-spectral variety for the single-component system [28]. According to (3.1), this then gives

\begin{array}{rcl} {\begin{cases} \frac{d f_{i}}{d t} = g_{i}, \frac{d {\hat{f}}_{i}}{d t} = {\hat{g}}_{i}, \\ \frac{d g_{i}}{d t} = - (\sum_{j = 1}^{l} f_{j}) g_{i} - ε (\sum_{j = 1}^{l} {\hat{f}}_{j}) {\hat{g}}_{i}, \\ \frac{d {\hat{g}}_{i}}{d t} = - (\sum_{j = 1}^{l} {\hat{f}}_{j}) g_{i} - (\sum_{j = 1}^{l} f_{j}) {\hat{g}}_{i}, \end{cases} \end{array}

(3.3)

and the relations of

f_{i}, {\hat{f}}_{i}, g_{i}, {\hat{g}}_{i}

with τ-functions are given by

\begin{array}{rcl} {\begin{cases} f_{i} (t) = \frac{d}{d t} \frac{l o g [τ_{i} (t) + \sqrt{ε} {\hat{τ}}_{i} (t)] + l o g [τ_{i} (t) - \sqrt{ε} {\hat{τ}}_{i} (t)]}{2}, \\ {\hat{f}}_{i} (t) = \frac{d}{d t} \frac{l o g [τ_{i} (t) + \sqrt{ε} {\hat{τ}}_{i} (t)] - l o g [τ_{i} (t) - \sqrt{ε} {\hat{τ}}_{i} (t)]}{2 \sqrt{ε}}, \end{cases} \end{array}

(3.4)

\begin{array}{rcl} {\begin{cases} g_{i} (t) = g_{i} (0) \prod_{j = 1}^{l} [{(τ_{j} (t))}^{- C_{i j}} + 2 ε \sum_{m = 1}^{[\frac{- C_{i j}}{2}]} (C_{- C_{i j}}^{2 m} {(τ_{j} (t))}^{- C_{i j} - 2 m} {({\hat{τ}}_{j} (t))}^{2 m})] \\ + ε {\hat{g}}_{i} (0) \prod_{j = 1}^{l} [\sum_{m = 1}^{[\frac{- C_{i j}}{2}]} (C_{- C_{i j}}^{2 m - 1} {(τ_{j} (t))}^{- C_{i j} - 2 m + 1} {({\hat{τ}}_{j} (t))}^{2 m - 1})], \\ {\hat{g}}_{i} (t) = {\hat{g}}_{i} (0) \prod_{j = 1}^{l} [{(τ_{j} (t))}^{- C_{i j}} + 2 ε \sum_{m = 1}^{[\frac{- C_{i j}}{2}]} (C_{- C_{i j}}^{2 m} {(τ_{j} (t))}^{- C_{i j} - 2 m} {({\hat{τ}}_{j} (t))}^{2 m})] \\ + g_{i} (0) \prod_{j = 1}^{l} [\sum_{m = 1}^{[\frac{- C_{i j}}{2}]} (C_{- C_{i j}}^{2 m - 1} {(τ_{j} (t))}^{- C_{i j} - 2 m + 1} {({\hat{τ}}_{j} (t))}^{2 m - 1})] . \end{cases} \end{array}

(3.5)

The integrability of the system can be ensured, because we give a specific example to illustrate it below.

Definition 2.Let $g_{1} = {S O}_{2 n}$ , $X, \hat{X} \in {S O}_{2 n}$ , if $X^{2 k - 1} + 2 ε \sum_{j = 1}^{[\frac{2 k - 1}{2}]} C_{2 k - 1}^{2 j} X^{2 k - 1 - 2 j} {\hat{X}}^{2 j}, \sum_{j = 1}^{[\frac{2 k - 1}{2}]} C_{2 k - 1}^{2 j - 1} X^{2 k - 2 j} {\hat{X}}^{2 j - 1} \in {S O}_{2 n}$ , the specific forms of $X, \hat{X}$ are given by the tridiagonal matrix,

\begin{array}{rcl} X = (\begin{array}{ccccc} 0 & α_{1} \\ - α_{1} & 0 & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & 0 & α_{2 n - 1} \\ - α_{2 n - 1} & 0 \end{array}), \end{array}

(3.6)

\begin{array}{rcl} \hat{X} = (\begin{array}{ccccc} 0 & {\hat{α}}_{1} \\ - {\hat{α}}_{1} & 0 & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & 0 & {\hat{α}}_{2 n - 1} \\ - {\hat{α}}_{2 n - 1} & 0 \end{array}) . \end{array}

(3.7)

In this way, we obtain all the even flows, which are called the generalized Kac-van Moerbeke hierarchy:

\begin{array}{rcl} {\begin{cases} \frac{\partial X}{\partial t_{2 j}} = [\prod_{s o} X^{2 j} + 2 ε \sum_{i = 1}^{j} C_{2 j}^{2 i} X^{2 j - 2 i} {\hat{X}}^{2 i}, X] + ε [\prod_{s o} \sum_{i = 1}^{j} C_{2 j}^{2 i - 1} X^{2 j - 2 i + 1} {\hat{X}}^{2 i - 1}, \hat{X}], \\ \frac{\partial \hat{X}}{\partial t_{2 j}} = [\prod_{s o} \sum_{i = 1}^{j} C_{2 j}^{2 i - 1} X^{2 j - 2 i + 1} {\hat{X}}^{2 i - 1}, X] + [\prod_{s o} X^{2 j} + 2 ε \sum_{i = 1}^{j} C_{2 j}^{2 i} X^{2 j - 2 i} {\hat{X}}^{2 i}, \hat{X}] . \end{cases} \end{array}

(3.8)

Consider the number of t₂ flows when j = 1,

\begin{array}{rcl} {\begin{cases} \frac{\partial α_{k}}{\partial t_{2}} = α_{k} [(α_{k - 1}^{2} - α_{k + 1}^{2}) + ε α_{k} ({\hat{α}}_{k - 1}^{2} - {\hat{α}}_{k + 1}^{2})] \\ + 2 ε {\hat{α}}_{k} ({\hat{α}}_{k - 1} α_{k - 1} - {\hat{α}}_{k + 1} α_{k + 1}), \\ \frac{\partial {\hat{α}}_{k}}{\partial t_{2}} = {\hat{α}}_{k} [(α_{k - 1}^{2} - α_{k + 1}^{2}) + ε α_{k} ({\hat{α}}_{k - 1}^{2} - {\hat{α}}_{k + 1}^{2})] \\ + 2 α_{k} ({\hat{α}}_{k - 1} α_{k - 1} - {\hat{α}}_{k + 1} α_{k + 1}), \end{cases} \end{array}

(3.9)

with

α_{0} = {\hat{α}}_{0} = {\hat{α}}_{2 n} = α_{2 n} = 0

and

k = 1, \dots, 2 n - 1

. From the t₂ flow, we find that the symmetric Toda lattice is equivalent to this system , so X and

\hat{X}

can be expressed by the generalized matrices

\begin{array}{rcl} {\begin{cases} X^{2} + ε {\hat{X}}^{2} = T^{(1)} \otimes (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) + T^{(2)} \otimes (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}), \\ X \hat{X} + \hat{X} X = {\hat{T}}^{(1)} \otimes (\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}) + {\hat{T}}^{(2)} \otimes (\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}), \end{cases} \end{array}

(3.10)

where

T^{(i)}

and

{\hat{T}}^{(i)}

(for i = 1, 2) are symmetric tridiagonal matrices, the specific forms of which are given as follows:

\begin{array}{rcl} T^{(i)} = (\begin{array}{ccccc} b_{1}^{(i)} & a_{1}^{(i)} \\ a_{2}^{(i)} & b_{2}^{(i)} & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & b_{n - 1}^{(i)} & a_{n - 1}^{(i)} \\ a_{n - 1}^{(i)} & b_{n}^{(i)} \end{array}), \end{array}

(3.11)

\begin{array}{rcl} {\hat{T}}^{(i)} = (\begin{array}{ccccc} {\hat{b}}_{1}^{(i)} & {\hat{a}}_{1}^{(i)} \\ {\hat{a}}_{2}^{(i)} & {\hat{b}}_{2}^{(i)} & ⋱ \\ ⋱ & ⋱ & ⋱ \\ ⋱ & {\hat{b}}_{n - 1}^{(i)} & {\hat{a}}_{n - 1}^{(i)} \\ {\hat{a}}_{n - 1}^{(i)} & {\hat{b}}_{n}^{(i)} \end{array}) . \end{array}

(3.12)

According to [29], we then have

\begin{array}{rcl} {\begin{cases} a_{k}^{(1)} = α_{2 k - 1} α_{2 k} + ε {\hat{α}}_{2 k - 1} {\hat{α}}_{2 k}, \\ {\hat{a}}_{k}^{(1)} = {\hat{α}}_{2 k - 1} α_{2 k} + α_{2 k - 1} {\hat{α}}_{2 k}, \end{cases} {\begin{cases} b_{k}^{(1)} = - α_{2 k - 2}^{2} - α_{2 k - 1}^{2} - ε ({\hat{α}}_{2 k - 2}^{2} + {\hat{α}}_{2 k - 1}^{2}), \\ {\hat{b}}_{k}^{(1)} = - 2 ({\hat{α}}_{2 k - 2} α_{2 k - 2} - {\hat{α}}_{2 k - 1} α_{2 k - 1}), \end{cases} \end{array}

(3.13)

\begin{array}{rcl} {\begin{cases} a_{k}^{(2)} = α_{2 k} α_{2 k + 1} + ε {\hat{α}}_{2 k} {\hat{α}}_{2 k + 1}, \\ {\hat{a}}_{k}^{(2)} = {\hat{α}}_{2 k} α_{2 k + 1} + α_{2 k} {\hat{α}}_{2 k + 1}, \end{cases} {\begin{cases} b_{k}^{(2)} = - α_{2 k - 1}^{2} - α_{2 k}^{2} - ε ({\hat{α}}_{2 k - 1}^{2} + {\hat{α}}_{2 k}^{2}), \\ {\hat{b}}_{k}^{(2)} = - 2 ({\hat{α}}_{2 k - 1} α_{2 k - 1} - {\hat{α}}_{2 k} α_{2 k}), \end{cases} \end{array}

(3.14)

from the structures of

T^{(i)}

and

{\hat{T}}^{(i)}

(i = 1, 2), the generalized Kac-van Moerbeke hierarchy for X²,

{\hat{X}}^{2}

can be split into generalized Toda lattices, and all of these equations are integrable systems.

4. Darboux transformation for the generalized Toda equations

We give two generalized affine Kac–Moody algebras

(g_{1}, {\hat{g}}_{1})

, and the corresponding Toda equations are given as follows:

\begin{array}{rcl} {\begin{cases} w_{j, x y} = e^{\sum_{i = 1}^{n} C_{j i} w_{i}} \cosh (\sqrt{ε} \sum_{i = 1}^{n} C_{j i} {\hat{w}}_{i}) - v_{j} e^{\sum_{i = 1}^{n} C_{0 i} w_{i}} \cosh (\sqrt{ε} \sum_{i = 1}^{n} C_{0 i} {\hat{w}}_{i}), \\ {\hat{w}}_{j, x y} = \frac{1}{\sqrt{ε}} e^{\sum_{i = 1}^{n} C_{j i} w_{i}} \sinh (\sqrt{ε} \sum_{i = 1}^{n} C_{j i} {\hat{w}}_{i}) - \frac{v_{j}}{\sqrt{ε}} e^{\sum_{i = 1}^{n} C_{0 i} w_{i}} \sinh (\sqrt{ε} \sum_{i = 1}^{n} C_{0 i} {\hat{w}}_{i}), \end{cases} j = 1, \dots, n, \end{array}

(4.1)

where C_ij are generalized Cartan matrices mentioned, and

v_{j} (j = 0, 1, 2 \dots, n)

satisfy

C (v_{0}, v_{1}

, \dots, v_{n})^{T} = 0

. When

g_{1} = {\hat{g}}_{1} = A_{n}^{(1)}

, the equations (4.1) will be changed into the two-dimensional generalized Toda equations

\begin{array}{rcl} {\begin{cases} u_{j, x y} = e^{u_{j} - u_{j - 1}} \cosh [\sqrt{ε} ({\hat{u}}_{j} - {\hat{u}}_{j - 1})] - e^{u_{j + 1} - u_{j}} \cosh [\sqrt{ε} ({\hat{u}}_{j + 1} - {\hat{u}}_{j})], \\ {\hat{u}}_{j, x y} = \frac{1}{\sqrt{ε}} e^{u_{j} - u_{j - 1}} \sinh [\sqrt{ε} ({\hat{u}}_{j} - {\hat{u}}_{j - 1})] - \frac{1}{\sqrt{ε}} e^{u_{j + 1} - u_{j}} \sinh [\sqrt{ε} ({\hat{u}}_{j + 1} - {\hat{u}}_{j})] . \end{cases} \end{array}

(4.2)

According to [30], any equation in (4.1) is integrable, and their Lax pairs can also be obtained. We consider the Lax pairs, based on the compatibility condition of the system (4.1), to be expressed as:

\begin{array}{rcl} {\begin{cases} Φ_{x} = (λ J + P) Φ + ε \hat{P} \hat{Φ}, \\ {\hat{Φ}}_{x} = (λ J + P) \hat{Φ} + \hat{P} Φ, \\ Φ_{y} = \frac{1}{λ} (Q Φ + ε \hat{Q} \hat{Φ}), \\ {\hat{Φ}}_{y} = \frac{1}{λ} (\hat{Q} Φ + Q \hat{Φ}), \end{cases} \end{array}

(4.3)

where

\begin{array}{rcl} {\begin{cases} J = \sum_{j = 0}^{n} e_{i}, \\ P = \sum_{i = 1}^{n} P_{i} h_{i}, \\ \hat{P} = \sum_{i = 1}^{n} {\hat{P}}_{i} h_{i}, \\ Q = \sum_{i = 0}^{n} Q_{i} f_{i}, \\ \hat{Q} = \sum_{i = 0}^{n} {\hat{Q}}_{i} f_{i}, \end{cases} \end{array}

(4.4)

and

e_{i} = E_{i, i + 1}

f_{i} = E_{i + 1, i}

h_{i} = E_{i, i} - E_{i + 1, i + 1}

, (

i = 0, 1, \dots, n

), are the basis of the algebra. The compatibility condition implies that

P_{j} = {(w_{j})}_{x}

{\hat{P}}_{j} = {({\hat{w}}_{j})}_{x}

Q_{j} = e^{\sum_{i = 1}^{n} C_{j i} w_{i}}

and

{\hat{Q}}_{j} = e^{\sum_{i = 1}^{n} C_{j i} {\hat{w}}_{i}}

. We let

\begin{array}{rcl} {\begin{cases} P = \frac{V_{x} V}{V^{2} - ε {\hat{V}}^{2}} - \frac{ε {\hat{V}}_{x} \hat{V}}{V^{2} - ε {\hat{V}}^{2}}, \\ \hat{P} = \frac{{\hat{V}}_{x} V}{V^{2} - ε {\hat{V}}^{2}} - \frac{V_{x} \hat{V}}{V^{2} - ε {\hat{V}}^{2}}, \end{cases} {\begin{cases} Q = \frac{{V J}^{T} V}{V^{2} - ε {\hat{V}}^{2}} - \frac{ε \hat{V} J^{T} \hat{V}}{V^{2} - ε {\hat{V}}^{2}}, \\ \hat{Q} = \frac{\hat{V} J^{T} V}{V^{2} - ε {\hat{V}}^{2}} - \frac{{V J}^{T} \hat{V}}{V^{2} - ε {\hat{V}}^{2}}, \end{cases} \end{array}

(4.5)

where

V = {(V_{j k})}_{1 ⩽ j, k ⩽ N}

and

\hat{V} = {({\hat{V}}_{j k})}_{1 ⩽ j, k ⩽ N}

are N × N block diagonal matrices. From (4.3), the integrability conditions of Lax pairs are then expressed as

\begin{array}{rcl} {\begin{cases} P_{t} + [J, Q] = 0, \\ {\hat{P}}_{t} + [J, \hat{Q}] = 0. \end{cases} \end{array}

(4.6)

J, V, \hat{V}

satisfy certain symmetry conditions, there are two real symmetric matrices K,

\hat{K}

that satisfy

\begin{array}{rcl} {\begin{cases} K^{2} + ε {\hat{K}}^{2} = I, Ω K Ω = ω^{2 - m} K, \\ \hat{K} K + K \hat{K} = 0, Ω \hat{K} Ω = ω^{2 - m} \hat{K}, \end{cases} \end{array}

(4.7)

and

\begin{array}{rcl} {\begin{cases} \bar{J} = J, Ω J Ω^{- 1} = ω J, K J K + ε \hat{K} \hat{J} \hat{K} = J^{T}, \\ \hat{K} J K + K J \hat{K} = 0, \end{cases} \end{array}

(4.8)

\begin{array}{rcl} {\begin{cases} \bar{V} = V, Ω V Ω^{- 1} = \pm V, V^{T} K V + ε ({\hat{V}}^{T} \hat{K} V + {\hat{V}}^{T} K \hat{V} + V^{T} \hat{K} \hat{V}) = K, \\ \bar{\hat{V}} = \hat{V}, Ω \hat{V} Ω^{- 1} = \pm \hat{V}, {\hat{V}}^{T} K V + V^{T} \hat{K} V + V^{T} K \hat{V} + ε {\hat{V}}^{T} \hat{K} \hat{V} = \hat{K}, \end{cases} \end{array}

(4.9)

where

ω = e^{\frac{2 π i}{N}}

Ω = d i a g (N, C) (I_{r_{1}}, ω^{- 1} I_{r_{2}}, \dots, ω^{- N + 1} I_{r_{N}})

. It is obvious that

Ω^{N} = I

, so we let

θ = e^{\frac{π i}{N}}

Θ = d i a g (N, C) (I_{r_{1}}, θ^{- 1} I_{r_{2}}, \dots, θ^{- N + 1} I_{r_{N}})

, and

θ^{2} = ω, Θ^{2} = Ω

. For the symmetry above, the selection of positive and negative signs does not effect the symmetry of Lax pairs as long as m is an integer. Except for

A_{n}^{(1)}

, the coefficient matrices of the Lax pairs for the two-dimensional generalized Toda equations produced by an infinite series of affine Kac–Moody algebras satisfy the symmetry above.

Lemma 1.If $V (x, y)$ and $\hat{V} (x, y)$ satisfy

\begin{array}{rcl} {\begin{cases} \bar{V} = V, Ω V Ω^{- 1} = \pm V, V^{T} K V + ε {\hat{V}}^{T} \hat{K} V + ε ({\hat{V}}^{T} K \hat{V} + V^{T} \hat{K} \hat{V}) = K, \\ \bar{\hat{V}} = \hat{V}, Ω \hat{V} Ω^{- 1} = \pm \hat{V}, {\hat{V}}^{T} K V + V^{T} \hat{K} V + V^{T} K \hat{V} + ε {\hat{V}}^{T} \hat{K} \hat{V} = \hat{K}, \end{cases} \end{array}

(4.10)

then

P, \hat{P}

Q, \hat{Q}

satisfy

\begin{array}{rcl} {\begin{cases} \bar{P} = P, Ω P Ω^{- 1} = P, K P K + ε \hat{K} \hat{P} K + ε (K \hat{P} \hat{K} + \hat{K} P K) = - P^{T}, \\ \bar{\hat{P}} = \hat{P}, Ω \hat{P} Ω^{- 1} = \hat{P}, \hat{K} P K + K \hat{P} K + ε (\hat{K} \hat{P} \hat{K} + K P \hat{K}) = - ε {\hat{P}}^{T}, \end{cases} \end{array}

(4.11)

\begin{array}{rcl} {\begin{cases} \bar{Q} = Q, Ω Q Ω^{- 1} = Q, K P K + ε \hat{K} \hat{Q} K + ε (K \hat{Q} \hat{K} + \hat{K} Q K) = Q^{T}, \\ \bar{\hat{Q}} = \hat{Q}, Ω \hat{Q} Ω^{- 1} = \hat{Q}, \hat{K} Q K + K \hat{Q} K + ε (\hat{K} \hat{Q} \hat{K} + K Q \hat{K}) = ε {\hat{Q}}^{T} . \end{cases} \end{array}

(4.12)

Lemma 2.

\begin{array}{rcl} {\begin{cases} K^{'} = θ^{m - 2} Θ K Θ, J^{'} = θ^{- 1} Θ^{- 1} J Θ, Q^{'} = θ Θ^{- 1} Q Θ, \\ {\hat{K}}^{'} = θ^{m - 2} Θ \hat{K} Θ, {\hat{Q}}^{'} = θ Θ^{- 1} \hat{Q} Θ, \end{cases} \end{array}

(4.13)

are real matrices, and

K^{'}

{\hat{K}}^{'}

are symmetric matrices.

Proof.By calculating the symmetry of the previous assumption, we can obtain lemma 1. According to (4.7), $ω^{- (j - 1)} K_{j k} ω^{- (k - 1)} = ω^{2 - m} K_{j k}$ , $ω^{- (j - 1)} {\hat{K}}_{j k} ω^{- (k - 1)} = ω^{2 - m} {\hat{K}}_{j k}$ , so $K_{j k} \neq 0$ , ${\hat{K}}_{j k} \neq 0$ ; we have $j + k \equiv m m o d N$ because of $θ^{N} = - 1$ . When $K_{j k} \neq 0$ , ${\hat{K}}_{j k} \neq 0$ , $ω^{- (j - 1)} {\hat{K}}_{j k} ω^{- (k - 1)} = \pm ω^{2 - (j + k)} {\hat{K}}_{j k} = \pm ω^{2 - m} {\hat{K}}_{j k}$ , so ${\hat{K}}_{j k}^{'} = \pm {\hat{K}}_{j k}$ ; therefore, ${\hat{K}}^{'}$ is a real symmetry matrix, and the expression of $K^{'}$ tells us that $K^{'}$ is also a real symmetry matrix. From (4.8), $ω^{- (j - 1)} {\hat{J}}_{j k} ω^{k - 1} = ω {\hat{J}}_{j k}$ , we obtain $K \equiv j + 1 m o d N$ as ${\hat{J}}_{j k} \neq 0$ . When ${\hat{J}}_{j k} \neq 0$ , $ω^{- (j - 1)} {\hat{J}}_{j k} ω^{k - 1} = \pm θ {\hat{J}}_{j k}$ then ${\hat{J}}^{'} = θ^{- 1} Θ^{- 1} \hat{J} Θ$ is a real matrix. $Q^{'}, {\hat{Q}}^{'}, J^{'}$ can be proved similarly.

Lemma 3.Assuming V, $\hat{V}$ satisfy (4.9), then Φ, $\hat{Φ}$ are the solutions of equation (4.3); when $λ = λ_{0}$ , we can give the following conclusions:

	(1) $Ω Φ, Ω \hat{Φ}$ are the solutions of (4.3) when $λ = λ_{0}$ ;
	(2) $\bar{Φ}, \bar{\hat{Φ}}$ are the solutions of (4.3) when $λ = {\bar{λ}}_{0}$ ;
	(3) ${\begin{cases} Ψ = K Φ + ε \hat{K} \hat{Φ}, \\ \hat{Ψ} = \hat{K} Φ + K \hat{Φ}, \end{cases}$ are the solutions of the conjugated Lax pairs when $λ = - λ_{0}$ :

\begin{array}{rcl} {\begin{cases} Ψ_{x} = (- λ_{0} J^{T} + P^{T}) Ψ + (- λ_{0} J^{T} + {\hat{P}}^{T}) \hat{Ψ}, \\ {\hat{Ψ}}_{x} = (- λ_{0} J^{T} + ε {\hat{P}}^{T}) Ψ + (- λ_{0} J^{T} + P^{T}) \hat{Ψ}, \\ Ψ_{y} = - \frac{1}{λ_{0}} (Q^{T} Ψ + {\hat{Q}}^{T} \hat{Ψ}), \\ {\hat{Ψ}}_{y} = - \frac{1}{λ_{0}} (Q^{T} \hat{Ψ} + ε {\hat{Q}}^{T} Ψ); \end{cases} \end{array}

(4.14)

(1) ${\begin{cases} Φ^{'} = Θ^{- 1} Φ, \\ {\hat{Φ}}^{'} = Θ^{- 1} \hat{Φ}, \end{cases}$ are the solutions of

\begin{array}{rcl} {\begin{cases} Φ_{x}^{'} = (λ J^{'} + P) Φ^{'} + ε (λ J^{'} + \hat{P}) {\hat{Φ}}^{'}, \\ {\hat{Φ}}_{x}^{'} = (λ J^{'} + \hat{P}) Φ^{'} + (λ J^{'} + P) {\hat{Φ}}^{'}, \\ Φ_{y}^{'} = \frac{1}{λ} (Q^{'} Φ^{'} + ε {\hat{Q}}^{'} {\hat{Φ}}^{'}), \\ {\hat{Φ}}_{y} = \frac{1}{λ} ({\hat{Q}}^{'} Φ^{'} + Q^{'} {\hat{Φ}}^{'}), \end{cases} a s λ = θ^{- 1} λ_{0}, \end{array}

(4.15)

where

J^{'}

Q^{'}

and

{\hat{Q}}^{'}

are given by lemma 2; when

λ_{0}

is real,

Φ^{'}

{\hat{Φ}}^{'}

are real solutions.

Proof.From

\begin{array}{rcl} {\begin{cases} Ω Φ_{x} = Ω (λ_{0} J + P) Φ + Ω (λ_{0} J + ε \hat{P}) \hat{Φ} = (ω λ_{0} J + P) Ω Φ + (ω λ_{0} J + ε \hat{P}) Ω \hat{Φ}, \\ Ω {\hat{Φ}}_{x} = Ω (λ_{0} J + \hat{P}) Φ + Ω (λ_{0} J + P) \hat{Φ} = (ω λ_{0} J + \hat{P}) Ω Φ + (ω λ_{0} J + P) Ω \hat{Φ}, \end{cases} \end{array}

(4.16)

and

\begin{array}{rcl} {\begin{cases} {(Ω Φ)}_{x} = \frac{1}{λ_{0}} (Q Φ + ε \hat{Q} \hat{Φ}) = \frac{1}{ω λ_{0}} (Q Ω Φ + ε \hat{Q} Ω \hat{Φ}), \\ {(Ω \hat{Φ})}_{x} = \frac{1}{λ_{0}} (Q Φ + ε \hat{Q} \hat{Φ}) = \frac{1}{ω λ_{0}} (\hat{Q} Ω Φ + Q Ω \hat{Φ}), \end{cases} \end{array}

(4.17)

we can prove conclusions (1), (2). From

\begin{array}{rcl} {\begin{cases} K Φ_{x} + ε \hat{K} {\hat{Φ}}_{x} = K (λ_{0} J + P) Φ + ε \hat{K} (λ_{0} J + \hat{P}) Φ + \hat{K} (λ_{0} J + P) \hat{Φ} + ε K (λ_{0} J + \hat{P}) \hat{Φ} \\ \hat{K} Φ_{x} + K {\hat{Φ}}_{x} = \hat{K} (λ_{0} J + P) Φ + K (λ_{0} J + \hat{P}) Φ + K (λ_{0} J + P) \hat{Φ} + ε \hat{K} (λ_{0} J + \hat{P}) \hat{Φ} \\ = (λ_{0} J^{T} - P^{T}) K Φ + (λ_{0} J^{T} - P^{T}) \hat{K} Φ + (λ_{0} J^{T} - {\hat{P}}^{T}) K \hat{Φ} + ε (λ_{0} J^{T} - P^{T}) \hat{K} \hat{Φ} \\ = (λ_{0} J^{T} - ε {\hat{P}}^{T}) K Φ + (λ_{0} J^{T} - P^{T}) \hat{K} Φ + ε (λ_{0} J^{T} - ε {\hat{P}}^{T}) \hat{K} \hat{Φ} + (λ_{0} J^{T} - P^{T}) K \hat{Φ}, \end{cases} \end{array}

(4.18)

and

\begin{array}{rcl} {\begin{cases} K Φ_{y} + ε \hat{K} {\hat{Φ}}_{y} = \frac{1}{λ_{0}} (K Q Φ + ε \hat{K} \hat{Q} Φ + \hat{K} Q \hat{Φ} + ε K \hat{Q} \hat{Φ}), \\ \hat{K} Φ_{y} + K {\hat{Φ}}_{y} = \frac{1}{λ_{0}} (\hat{K} Q Φ + K \hat{Q} Φ + K Q \hat{Φ} + ε \hat{K} \hat{Q} \hat{Φ}), \end{cases} \end{array}

(4.19)

we have proved conclusion (3). According to the expressions of

J^{'}, {\hat{J}}^{'}, Q^{'}

and

{\hat{Q}}^{'}

, (4) is obvious.

Assume

G (x, y, λ), \hat{G} (x, y, λ)

are M × M matrices given by:

\begin{array}{rcl} {\begin{cases} G (x, y, λ) = \sum_{j = 0}^{L} G (x, y) λ^{L - j}, \\ \hat{G} (x, y, λ) = \sum_{j = 0}^{L} \hat{G} (x, y) λ^{L - j}, \end{cases} \end{array}

(4.20)

where

G_{i}, {\hat{G}}_{i}

(i = 1, 2, \dots, L)

are independent of λ and

G_{0} = I

{\hat{G}}_{0} = 0

. Thus,

V (x, y)

\hat{V} (x, y)

have a similar symmetry satisfying (4.10):

\begin{array}{rcl} {\begin{cases} \bar{\tilde{V}} = \tilde{V}, Ω \tilde{V} Ω^{- 1} = \pm \tilde{V}, {\tilde{V}}^{T} K \tilde{V} + ε ({\hat{\tilde{V}}}^{T} \hat{K} \tilde{V} + {\hat{\tilde{V}}}^{T} K \hat{V} + {\tilde{V}}^{T} \hat{K} \hat{\tilde{V}}) = K, \\ \bar{\hat{\tilde{V}}} = \hat{\tilde{V}}, Ω \hat{\tilde{V}} Ω^{- 1} = \pm \hat{\tilde{V}}, {\hat{\tilde{V}}}^{T} K \tilde{V} + {\tilde{V}}^{T} \hat{K} \tilde{V} + V^{T} K \hat{\tilde{V}} + ε {\hat{\tilde{V}}}^{T} \hat{K} \hat{\tilde{V}} = \hat{K}, \end{cases} \end{array}

(4.21)

where

\tilde{V} (x, y), \hat{\tilde{V}} (x, y)

are partition diagonal matrices.

Theorem 1.Let Φ, $\hat{Φ}$ be the solutions of (4.3), so $\tilde{Φ} = G Φ + ε \hat{G} \hat{Φ}$ , $\tilde{\hat{Φ}} = \hat{G} Φ + G \hat{Φ}$ satisfy

\begin{array}{rcl} {\begin{cases} {\tilde{Φ}}_{x} = (λ J + \tilde{P}) \tilde{Φ} + ε (λ J + \hat{\tilde{P}}) \hat{\tilde{Φ}}, \\ {\hat{\tilde{Φ}}}_{x} = (λ J + \hat{\tilde{P}}) \tilde{Φ} + (λ J + \tilde{P}) \hat{\tilde{Φ}}, \\ {\tilde{Φ}}_{y} = \frac{1}{λ} (\tilde{Q} \tilde{Φ} + ε \hat{\tilde{Q}} \hat{\tilde{Φ}}), \\ {\hat{\tilde{Φ}}}_{y} = \frac{1}{λ} (\hat{\tilde{Q}} \tilde{Φ} + \tilde{Q} \hat{\tilde{Φ}}), \end{cases} \end{array}

(4.22)

where

\begin{array}{rcl} {\begin{cases} \tilde{P} = \frac{{\tilde{V}}_{x} \tilde{V}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} - \frac{ε {\hat{\tilde{V}}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \\ \hat{\tilde{P}} = \frac{{\hat{\tilde{V}}}_{x} \tilde{V}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} - \frac{{\tilde{V}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \end{cases} {\begin{cases} \tilde{Q} = \frac{\tilde{V} {\tilde{J}}^{T} \tilde{V}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} - \frac{ε \hat{\tilde{V}} J^{T} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \\ \hat{\tilde{Q}} = \frac{\hat{\tilde{V}} J^{T} \tilde{V}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} - \frac{\tilde{V} J^{T} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} . \end{cases} \end{array}

(4.23)

Proof.An L-order two-dimensional Darboux transformation can be obtained by generalizing the method of [31]. Let $s (1 ⩽ s ⩽ M - 1)$ be a positive integer, $(λ_{1}, λ_{1}, \dots, λ_{n})$ are different complex numbers and ${\bar{λ}}_{i} + λ_{k} \neq 0 (j, k = 1, 2, \dots, L$ ), H_j, ${\hat{H}}_{j}$ are the solutions of Lax pairs when $λ = λ_{j}$ ; we now have

\begin{array}{rcl} {\begin{cases} G (λ) = \prod_{l = 1}^{L} (λ + {\bar{λ}}_{l}) [E - \sum_{j, k = 1}^{L} \frac{1}{(λ + {\bar{λ}}_{k}) (T^{2} - ε {\hat{T}}^{2})} (H_{j} {T H}_{k}^{*} K - ε {\hat{H}}_{j} \hat{T} H_{k}^{*} K + ε^{2} {\hat{H}}_{j} {T H}_{k}^{*} K \\ - ε^{2} H_{j} \hat{T} H_{k}^{*} K + H_{j} T {\hat{H}}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + ε {\hat{H}}_{j} {T H}_{k}^{*} \hat{K} - ε H_{j} \hat{T} H_{k}^{*} \hat{K})], \\ \hat{G} (λ) = \prod_{l = 1}^{L} (λ + {\bar{λ}}_{l}) \sum_{j, k = 1}^{L} \frac{- 1}{(λ + {\bar{λ}}_{k}) (T^{2} - ε {\hat{T}}^{2})} ({\hat{H}}_{j} {T H}_{k}^{*} K - H_{j} \hat{T} H_{k}^{*} K + ε H_{j} T {\hat{H}}_{k}^{*} K - \\ ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} K + {\hat{H}}_{j} T {\hat{H}}_{k}^{*} \hat{K} + H_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + H_{j} {T H}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K}), \end{cases} \end{array}

(4.24)

where

\begin{array}{rcl} {\begin{cases} T_{j k} = \frac{1}{{\bar{λ}}_{j} + λ_{k}} (H_{j}^{*} {K H}_{k} + {\hat{H}}_{j}^{*} \hat{K} H_{k} + {\hat{H}}_{j}^{*} K {\hat{H}}_{k} + ε H_{j}^{*} \hat{K} {\hat{H}}_{k}), \\ {\hat{T}}_{j k} = \frac{1}{{\bar{λ}}_{j} + λ_{k}} (H_{j}^{*} \hat{K} H_{k} + ε {\hat{H}}_{j}^{*} {K H}_{k} + H_{j}^{*} K {\hat{H}}_{k} + ε^{2} {\hat{H}}_{j}^{*} \hat{K} {\hat{H}}_{k}), \end{cases} j, k = 1, \dots, L . \end{array}

(4.25)

Since

G (λ), \hat{G} (λ)

are the L-degree polynomials of λ, this can be also written as:

\begin{array}{rcl} {\begin{cases} G (λ) = λ^{L} E + λ^{L - 1} G_{1} + λ^{L - 2} G_{2} + \dots + λ G_{L - 1} + G_{L}, \\ \hat{G} (λ) = λ^{L - 1} {\hat{G}}_{1} + λ^{L - 2} {\hat{G}}_{2} + \dots + λ {\hat{G}}_{L - 1} + {\hat{G}}_{L} . \end{cases} \end{array}

(4.26)

From (4.24), we have

\begin{array}{rcl} {\begin{cases} G_{1} = \sum_{l = 1}^{L} {\bar{λ}}_{l} - \sum_{j, k = 1}^{L} \frac{1}{T^{2} - ε {\hat{T}}^{2}} (H_{j} {T H}_{k}^{*} K - ε {\hat{H}}_{j} \hat{T} H_{k}^{*} K + ε^{2} {\hat{H}}_{j} {T H}_{k}^{*} K - ε^{2} H_{j} \hat{T} H_{k}^{*} K + \\ H_{j} T {\hat{H}}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + ε {\hat{H}}_{j} {T H}_{k}^{*} \hat{K} - ε H_{j} \hat{T} H_{k}^{*} \hat{K}), \\ {\hat{G}}_{1} = \sum_{j, k = 1}^{L} \frac{- 1}{T^{2} - ε {\hat{T}}^{2}} ({\hat{H}}_{j} {T H}_{k}^{*} K - H_{j} \hat{T} H_{k}^{*} K + ε H_{j} T {\hat{H}}_{k}^{*} K - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} K + \\ {\hat{H}}_{j} T {\hat{H}}_{k}^{*} \hat{K} + H_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + H_{j} {T H}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K}), \\ G_{L} = (\prod_{l = 1}^{L} {\bar{λ}}_{l}) [E - \sum_{j, k = 1}^{L} \frac{1}{{\bar{λ}}_{k} (T^{2} - ε {\hat{T}}^{2})} (H_{j} {T H}_{k}^{*} K - ε {\hat{H}}_{j} \hat{T} H_{k}^{*} K + ε^{2} {\hat{H}}_{j} {T H}_{k}^{*} K - ε^{2} H_{j} \hat{T} H_{k}^{*} K + \\ H_{j} T {\hat{H}}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + ε {\hat{H}}_{j} {T H}_{k}^{*} \hat{K} - ε H_{j} \hat{T} H_{k}^{*} \hat{K})], \\ {\hat{G}}_{L} = (\prod_{l = 1}^{L} {\bar{λ}}_{l}) \sum_{j, k = 1}^{L} \frac{- 1}{{\bar{λ}}_{k} (T^{2} - ε {\hat{T}}^{2})} ({\hat{H}}_{j} {T H}_{k}^{*} K - H_{j} \hat{T} H_{k}^{*} K + ε H_{j} T {\hat{H}}_{k}^{*} K - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} K + \\ {\hat{H}}_{j} T {\hat{H}}_{k}^{*} \hat{K} + H_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K} + H_{j} {T H}_{k}^{*} \hat{K} - ε {\hat{H}}_{j} \hat{T} {\hat{H}}_{k}^{*} \hat{K}) . \end{cases} \end{array}

(4.27)

Assuming

G (λ)

\hat{G} (λ)

are given by (4.24), then theorem 1 is established. According to [24], we have

\begin{array}{rcl} {\begin{cases} \tilde{P} = P - [J, G_{1}], \\ \hat{\tilde{P}} = \hat{P} - [J, {\hat{G}}_{1}], \end{cases} {\begin{cases} \tilde{Q} = \frac{G_{L} {Q G}_{L} - ε ({\hat{G}}_{L} \hat{Q} G_{L} + G_{L} \hat{Q} {\hat{G}}_{L} + {\hat{G}}_{L} Q {\hat{G}}_{L})}{G_{L}^{2} - ε {\hat{G}}_{L}^{2}}, \\ \hat{\tilde{Q}} = \frac{{\hat{G}}_{L} {Q G}_{L} + G_{L} \hat{Q} G_{L} - G_{L} Q {\hat{G}}_{L} - ε {\hat{G}}_{L} \hat{Q} {\hat{G}}_{L}}{G_{L}^{2} - ε {\hat{G}}_{L}^{2}}, \end{cases} \end{array}

(4.28)

and if

\tilde{V}, \hat{\tilde{V}}

satisfy

\begin{array}{rcl} {\begin{cases} \tilde{V} = G_{L} V + ε {\hat{G}}_{L} \hat{V}, \\ \hat{\tilde{V}} = {\hat{G}}_{L} V + G_{L} \hat{V}, \end{cases} \end{array}

(4.29)

then

\begin{array}{rcl} {\begin{cases} \tilde{P} = \frac{{\tilde{V}}_{x} \tilde{V} - ε {\hat{\tilde{V}}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \\ \hat{\tilde{P}} = \frac{{\hat{\tilde{V}}}_{x} \tilde{V} - {\tilde{V}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \end{cases} {\begin{cases} \tilde{Q} = \frac{\tilde{V} J^{T} \tilde{V} - ε \hat{\tilde{V}} J^{T} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \\ \hat{\tilde{Q}} = \frac{\hat{\tilde{V}} J^{T} \tilde{V} - \tilde{V} J^{T} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \end{cases} \end{array}

(4.30)

satisfies (4.28). From (4.25), we obtain

\begin{array}{rcl} {\begin{cases} T_{j k, x} = H_{j}^{*} {K J H}_{k} + {\hat{H}}_{j}^{*} \hat{K} {J H}_{k} + {\hat{H}}_{j}^{*} K J {\hat{H}}_{k} + ε H_{j}^{*} \hat{K} J {\hat{H}}_{k}, \\ {\hat{T}}_{j k, x} = H_{j}^{*} \hat{K} {J H}_{k} + H_{j}^{*} K J {\hat{H}}_{k} + ε {\hat{H}}_{j}^{*} {K J H}_{k} + ε^{2} {\hat{H}}_{j}^{*} \hat{K} J {\hat{H}}_{k}, \end{cases} \end{array}

(4.31)

\begin{array}{rcl} {\begin{cases} T_{j k, y} = \frac{1}{{\bar{λ}}_{j} λ_{k}} [H_{j}^{*} {K Q H}_{k} + {\hat{H}}_{j}^{*} \hat{K} {Q H}_{k} + {\hat{H}}_{j}^{*} K \hat{Q} H_{k} + {\hat{H}}_{j}^{*} K Q {\hat{H}}_{k} + \\ ε (H_{j}^{*} \hat{K} \hat{Q} H_{k} + H_{j}^{*} K \hat{Q} {\hat{H}}_{k} + {\hat{H}}_{j}^{*} \hat{K} \hat{Q} {\hat{H}}_{k} + H_{j}^{*} \hat{K} Q {\hat{H}}_{k})], \\ {\hat{T}}_{j k, y} = \frac{1}{{\bar{λ}}_{j} λ_{k}} [H_{j}^{*} \hat{K} {Q H}_{k} + H_{j}^{*} K \hat{Q} H_{k} + H_{j}^{*} K Q {\hat{H}}_{k} + ε ({\hat{H}}_{j}^{*} {K Q H}_{k} + H_{j}^{*} \hat{K} \hat{Q} {\hat{H}}_{k}) \\ + ε^{2} ({\hat{H}}_{j}^{*} \hat{K} \hat{Q} H_{k} + H_{j}^{*} \hat{K} Q {\hat{H}}_{k} + {\hat{H}}_{j}^{*} K \hat{Q} {\hat{H}}_{k})] . \end{cases} \end{array}

(4.32)

According to (4.28), the following expressions are established for all λ:

\begin{array}{rcl} {\begin{cases} (λ J + \tilde{P}) G + ε \hat{\tilde{P}} \hat{G} - G (λ J + P) - ε \hat{G} \hat{P} - G_{x} = 0, \\ (λ J + \tilde{P}) \hat{G} + \hat{\tilde{P}} G - \hat{G} (λ J + P) - G \hat{P} - {\hat{G}}_{x} = 0, \\ λ^{- 1} (\tilde{Q} G + ε \hat{\tilde{Q}} \hat{G}) - λ^{- 1} (G Q + ε \hat{G} \hat{Q}) - G_{y} = 0, \\ λ^{- 1} (\hat{\tilde{Q}} G + \tilde{Q} \hat{G}) - λ^{- 1} (\hat{G} Q + G \hat{Q}) - {\hat{G}}_{y} = 0, \end{cases} \end{array}

(4.33)

and the conclusion of theorem 1 is valid. In fact,

\begin{array}{rcl} {\begin{cases} \frac{\tilde{V} J \tilde{V} - ε \hat{\tilde{V}} J \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} = \frac{G_{L} {Q G}_{L} + ε {\hat{G}}_{L} \hat{Q} G_{L} - ε (G_{L} \hat{Q} {\hat{G}}_{L} + {\hat{G}}_{L} Q {\hat{G}}_{L})}{G_{L}^{2} - ε {\hat{G}}_{L}^{2}}, \\ \frac{\hat{\tilde{V}} J \tilde{V} - \tilde{V} J \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} = \frac{{\hat{G}}_{L} {Q G}_{L} + G_{L} \hat{Q} G_{L} - G_{L} Q {\hat{G}}_{L} - ε {\hat{G}}_{L} \hat{Q} {\hat{G}}_{L}}{G_{L}^{2} - ε {\hat{G}}_{L}^{2}}, \end{cases} \end{array}

(4.34)

are obvious; what we need to prove is that

\begin{array}{rcl} {\begin{cases} \frac{{\tilde{V}}_{x} \tilde{V} - ε {\hat{\tilde{V}}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} = P - [J, G_{1}], \\ \frac{{\hat{\tilde{V}}}_{x} \tilde{V} - {\tilde{V}}_{x} \hat{\tilde{V}}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}} = \hat{P} - [J, {\hat{G}}_{1}] . \end{cases} \end{array}

(4.35)

From the first two equations of (4.34), the following equations are true,

\begin{array}{rcl} {\begin{cases} G_{x} + G (λ J + P) + ε \hat{G} \hat{P} = (λ J + P - [J, G_{1}]) G + ε (\hat{P} - [J, {\hat{G}}_{1}]) \hat{G}, \\ {\hat{G}}_{x} + \hat{G} (λ J + P) + G \hat{P} = (λ J + P - [J, G_{1}]) \hat{G} + (\hat{P} - [J, {\hat{G}}_{1}]) G, \end{cases} \end{array}

(4.36)

when

λ = 0

. On the other hand, from (4.30), we get

\begin{array}{rcl} {\begin{cases} G_{L, x} + G_{L} P + ε {\hat{G}}_{L} \hat{P} = \frac{{\tilde{V}}_{x} \tilde{V} G_{L} - ε ({\hat{\tilde{V}}}_{x} \hat{\tilde{V}} G_{L} + {\tilde{V}}_{x} \hat{\tilde{V}} {\hat{G}}_{L} - {\hat{\tilde{V}}}_{x} \tilde{V} {\hat{G}}_{L})}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \\ {\hat{G}}_{L, x} + {\hat{G}}_{L} P + G_{L} \hat{P} = \frac{{\hat{\tilde{V}}}_{x} \tilde{V} G_{L} - {\tilde{V}}_{x} \hat{\tilde{V}} G_{L} + {\tilde{V}}_{x} \tilde{V} {\hat{G}}_{L} - ε {\hat{\tilde{V}}}_{x} \hat{\tilde{V}} {\hat{G}}_{L}}{{\tilde{V}}^{2} - ε {\hat{\tilde{V}}}^{2}}, \end{cases} \end{array}

(4.37)

and the proof is complete.

Remarks:

	1.The equations and systems involved in this article are weakly coupled when $ε = 0;$
	2.The equations and systems involved in this article are strongly coupled when $ε = 1$ .

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Acknowledgments

Chuanzhong Li is supported by the National Natural Science Foundation of China under Grant No. 12 071 237 and by the K C Wong Magna Fund in Ningbo University.