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:where L is a tridiagonal matrix, $B={(L)}_{\gt 0}-{(L)}_{\lt 0}$. 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 $2N$ 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.

In this section, we consider $2n$ particles, whose force depends on the distances between the particles. For those particles, the coupled Hamiltonians $(H,\widehat{H})$ are given bywhere $\tfrac{{\rm{d}}{q}_{k}}{{\rm{d}}t}=\tfrac{\partial H}{\partial {p}_{k}},\tfrac{{\rm{d}}{\hat{q}}_{k}}{{\rm{d}}t}=\tfrac{\partial \widehat{H}}{\partial {p}_{k}},\tfrac{{\rm{d}}{p}_{k}}{{\rm{d}}t}=-\tfrac{\partial H}{\partial {q}_{k}},\tfrac{{\rm{d}}{\hat{p}}_{k}}{{\rm{d}}t}=-\tfrac{\partial \widehat{H}}{\partial {q}_{k}}$. The Hamiltonian systems then represent a generalized finite nonperiodic lattice described by:where the formal boundary conditions are given by ${{\rm{e}}}^{-({q}_{1}-{q}_{0})}={{\rm{e}}}^{-({\hat{q}}_{1}-{\hat{q}}_{0})}={{\rm{e}}}^{-({q}_{n+1}-{q}_{n})}={{\rm{e}}}^{-({\hat{q}}_{n+1}-{\hat{q}}_{n})}=0$ at ${q}_{0}={\hat{q}}_{0}=0$ 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:Based on the above transformation (2.3), the generalized Toda equations (2.2) becomewhere ${a}_{0}={\hat{a}}_{0}=0$ and ${a}_{n}={\hat{a}}_{n}=0$. From the two extended symmetric matrices $L(\widehat{L})$, and ${a}_{k},{b}_{k}({\widehat{a}}_{k},{\widehat{b}}_{k})$ we can obtain the elements of $L(\widehat{L})$, using the expressions given by Thus, the systems (2.5) and (2.6) can be written aswhere $\left\{\begin{array}{l}B={(L)}_{\gt 0}-{(L)}_{\lt 0},\\ \widehat{B}={(\hat{L})}_{\gt 0}-{(\hat{L})}_{\lt 0},\end{array}\right.$ ${(L)}_{\gt 0}{((\hat{L})}_{\gt 0})$ is the upper triangular matrix of $L(\hat{L})$, and ${(L)}_{\lt 0}{((\hat{L})}_{\lt 0})$ is the lower triangular matrix of $L(\hat{L})$. According to [3], we find that the functions ${tr}({L}^{k}),{tr}({\widehat{L}}^{k})$ are constant for any k, andFor the $2{\rm{n}}-2$ independent invariant functions,where $[\,]$ represents an integer operation and ${C}_{k+1}^{2j-1}$ represents a binomial coefficient. From the calculation and transformation shown in (2.2), we obtain the relationship between the independent invariant functions and Hamiltonian $(H,\widehat{H})$ ,According to the properties of tridiagonal symmetric matrices [5], this has real and distinct eigenvalues. In fact, ${\widehat{q}}_{k+1}-{\widehat{q}}_{k}$ and ${q}_{k+1}-{q}_{k}$ tend to $\infty $ when ${\rm{t}}\to \pm \infty $. Furthermore, it is easy to see that ${\hat{a}}_{k}$ and a_k tend to zero, so $L(\widehat{L})$ degenerates into a diagonal matrix $\mathrm{diag}({\lambda }_{1},{\lambda }_{2},\cdots ,{\lambda }_{n})$ ($\mathrm{diag}({\widehat{\lambda }}_{1},{\widehat{\lambda }}_{2},\cdots ,{\widehat{\lambda }}_{n})$),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 aswhere $k(t),\hat{k}(t)\in {{SO}}_{n}$ 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,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).

From the gauge transformation and the Gram-Schmidt method of orthonormalization, we then haveTherefore, with (2.14) and (2.4), we obtain the solutions $({a}_{j}(t),{\hat{a}}_{j}(t),{b}_{j}(t),{\hat{b}}_{j}(t))$,where $x={\left(\tfrac{{a}_{1}\mp \sqrt{{a}_{1}^{2}-\varepsilon {a}_{2}^{2}}}{2}\right)}^{\tfrac{1}{2}}$, $y={\left(\tfrac{{a}_{1}\mp \sqrt{{a}_{1}^{2}-\varepsilon {a}_{2}^{2}}}{2\varepsilon }\right)}^{\tfrac{1}{2}}$ and ${a}_{1}={\tau }_{j+1}(t){\tau }_{j-1}(t)+\varepsilon {\hat{\tau }}_{j+1}(t){\hat{\tau }}_{j-1}(t)$, ${a}_{2}={\hat{\tau }}_{j+1}(t){\tau }_{j-1}(t)\,+{\tau }_{j+1}(t){\hat{\tau }}_{j-1}(t)$. Furthermore, we havewhere $c=\tfrac{{\tau }_{j}(t){\tau }_{j-1}(t)-\varepsilon {\hat{\tau }}_{j}(t){\hat{\tau }}_{j-1}(t)}{{\tau }_{j-1}^{2}(t)-\varepsilon {\hat{\tau }}_{j-1}^{2}(t)}$, $\hat{c}=\tfrac{{\hat{\tau }}_{j}(t){\tau }_{j-1}(t)-{\tau }_{j}(t){\hat{\tau }}_{j-1}(t)}{{\tau }_{j-1}^{2}(t)-\varepsilon {\hat{\tau }}_{j-1}^{2}(t)}$.

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:where $P(t),\widehat{P}(t)$ are the elements of g and $N(t)(\widehat{N}(t))$ are the projections of $P(t)(\widehat{P}(t))$, which are given bywhere ${h}_{{\alpha }_{i}},{e}_{\pm {\alpha }_{i}}(i=1,2,\ldots ,l)$ are the Chevalley basis of the algebra, $[{h}_{{\alpha }_{i}},{h}_{{\alpha }_{j}}]=0$, $[{h}_{{\alpha }_{i}},{e}_{\pm {\alpha }_{i}}]=\pm {C}_{{ij}}{e}_{\pm {\alpha }_{i}}$, $[{e}_{{\alpha }_{i}},{e}_{-{\alpha }_{j}}]={\delta }_{{ij}}{h}_{{\alpha }_{j}}$, and ${C}_{{ij}}={\alpha }_{i}({h}_{{\alpha }_{i}})$ 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 givesand the relations of ${f}_{i},{\hat{f}}_{i},{g}_{i},{\hat{g}}_{i}$ with τ-functions are given by The integrability of the system can be ensured, because we give a specific example to illustrate it below.

In this way, we obtain all the even flows, which are called the generalized Kac-van Moerbeke hierarchy:Consider the number of t₂ flows when j = 1,with ${\alpha }_{0}={\widehat{\alpha }}_{0}={\widehat{\alpha }}_{2n}={\alpha }_{2n}=0$ and $k=1,\cdots ,2n-1$. From the t₂ flow, we find that the symmetric Toda lattice is equivalent to this system , so X and $\widehat{X}$ can be expressed by the generalized matriceswhere ${T}^{(i)}$ and ${\widehat{T}}^{(i)}$ (for i = 1, 2) are symmetric tridiagonal matrices, the specific forms of which are given as follows: According to [29], we then have from the structures of ${T}^{(i)}$ and ${\widehat{T}}^{(i)}$ (i = 1, 2), the generalized Kac-van Moerbeke hierarchy for X², ${\widehat{X}}^{2}$ can be split into generalized Toda lattices, and all of these equations are integrable systems.

We give two generalized affine Kac–Moody algebras $({g}_{1},{\widehat{g}}_{1})$, and the corresponding Toda equations are given as follows:where C_ij are generalized Cartan matrices mentioned, and ${v}_{j}(j=0,1,2\ldots ,n)$ satisfy $C({v}_{0},{v}_{1}$ $,\ldots ,{v}_{n}{)}^{T}=0$. When ${g}_{1}={\widehat{g}}_{1}={A}_{n}^{(1)}$, the equations (4.1) will be changed into the two-dimensional generalized Toda equationsAccording 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:whereand ${e}_{i}={E}_{i,i+1}$, ${f}_{i}={E}_{i+1,i}$, ${h}_{i}={E}_{i,i}-{E}_{i+1,i+1}$, ($i=0,1,\ldots ,n$), are the basis of the algebra. The compatibility condition implies that ${P}_{j}={({w}_{j})}_{x}$, ${\widehat{P}}_{j}={({\widehat{w}}_{j})}_{x}$, ${Q}_{j}={{\rm{e}}}^{{\sum }_{i=1}^{n}{C}_{{ji}}{w}_{i}}$ and ${\widehat{Q}}_{j}={{\rm{e}}}^{{\sum }_{i=1}^{n}{C}_{{ji}}{\widehat{w}}_{i}}$. We letwhere $V={({V}_{{jk}})}_{1\leqslant j,k\leqslant N}$ and $\widehat{V}={({\widehat{V}}_{{jk}})}_{1\leqslant j,k\leqslant N}$ are N × N block diagonal matrices. From (4.3), the integrability conditions of Lax pairs are then expressed asAs $J,\,V,\,\widehat{V}$ satisfy certain symmetry conditions, there are two real symmetric matrices K, $\widehat{K}$ that satisfyand where $\omega ={e}^{\tfrac{2\pi i}{N}}$, ${\rm{\Omega }}=\mathrm{diag}({\rm{N}},{\mathbb{C}})({{\rm{I}}}_{{r}_{1}},{\omega }^{-1}{{\rm{I}}}_{{r}_{2}},\ldots ,{\omega }^{-N+1}{{\rm{I}}}_{{r}_{N}})$. It is obvious that ${{\rm{\Omega }}}^{N}=I$, so we let $\theta ={e}^{\tfrac{\pi {\rm{i}}}{N}}$, ${\rm{\Theta }}=\mathrm{diag}({\rm{N}},{\mathbb{C}})({{\rm{I}}}_{{r}_{1}},{\theta }^{-1}{{\rm{I}}}_{{r}_{2}},\ldots ,{\theta }^{-N+1}{{\rm{I}}}_{{r}_{N}})$, and ${\theta }^{2}=\omega ,\,{{\rm{\Theta }}}^{2}={\rm{\Omega }}$. 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 $\widehat{V}(x,y)$ satisfy

(4.10)$\begin{eqnarray}\left\{\begin{array}{l}\overline{V}=V,\,\,{\rm{\Omega }}V{{\rm{\Omega }}}^{-1}=\pm V,\,\,{V}^{{\rm{T}}}{KV}+\varepsilon {\widehat{V}}^{{\rm{T}}}\widehat{K}V+\varepsilon ({\widehat{V}}^{{\rm{T}}}K\widehat{V}+{V}^{{\rm{T}}}\widehat{K}\widehat{V})=K,\\ \overline{\widehat{V}}=\widehat{V},\,\,{\rm{\Omega }}\widehat{V}{{\rm{\Omega }}}^{-1}=\pm \widehat{V},\,\,{\widehat{V}}^{{\rm{T}}}{KV}+{V}^{{\rm{T}}}\widehat{K}V+{V}^{{\rm{T}}}K\widehat{V}+\varepsilon {\widehat{V}}^{{\rm{T}}}\widehat{K}\widehat{V}=\widehat{K},\end{array}\right.\end{eqnarray}$

then $P,\widehat{P}$, $Q,\widehat{Q}$ satisfy

(4.11)$\begin{eqnarray}\left\{\begin{array}{l}\overline{P}=P,\,\,{\rm{\Omega }}P{{\rm{\Omega }}}^{-1}=P,\,\,{KPK}+\varepsilon \widehat{K}\widehat{P}K+\varepsilon (K\widehat{P}\widehat{K}+\widehat{K}{PK})=-{P}^{{\rm{T}}},\\ \overline{\widehat{P}}=\widehat{P},\,\,{\rm{\Omega }}\widehat{P}{{\rm{\Omega }}}^{-1}=\widehat{P},\,\,\widehat{K}{PK}+K\widehat{P}K+\varepsilon (\widehat{K}\widehat{P}\widehat{K}+{KP}\widehat{K})=-\varepsilon {\widehat{P}}^{{\rm{T}}},\end{array}\right.\end{eqnarray}$

(4.12)$\begin{eqnarray}\left\{\begin{array}{l}\overline{Q}=Q,\,\,{\rm{\Omega }}Q{{\rm{\Omega }}}^{-1}=Q,\,\,{KPK}+\varepsilon \widehat{K}\widehat{Q}K+\varepsilon (K\widehat{Q}\widehat{K}+\widehat{K}{QK})={Q}^{{\rm{T}}},\\ \overline{\widehat{Q}}=\widehat{Q},\,\,{\rm{\Omega }}\widehat{Q}{{\rm{\Omega }}}^{-1}=\widehat{Q},\,\,\widehat{K}{QK}+K\widehat{Q}K+\varepsilon (\widehat{K}\widehat{Q}\widehat{K}+{KQ}\widehat{K})=\varepsilon {\widehat{Q}}^{{\rm{T}}}.\end{array}\right.\end{eqnarray}$

Lemma 2.

(4.13)$\begin{eqnarray}\left\{\begin{array}{l}{K}^{{\prime} }={\theta }^{m-2}{\rm{\Theta }}K{\rm{\Theta }},\,\,{J}^{{\prime} }={\theta }^{-1}{{\rm{\Theta }}}^{-1}J{\rm{\Theta }},\,\,{Q}^{{\prime} }=\theta {{\rm{\Theta }}}^{-1}Q{\rm{\Theta }},\\ {\widehat{K}}^{{\prime} }={\theta }^{m-2}{\rm{\Theta }}\widehat{K}{\rm{\Theta }},\,\,{\widehat{Q}}^{{\prime} }=\theta {{\rm{\Theta }}}^{-1}\widehat{Q}{\rm{\Theta }},\end{array}\right.\end{eqnarray}$

are real matrices, and ${K}^{{\prime} }$, ${\widehat{K}}^{{\prime} }$ are symmetric matrices.

Proof.By calculating the symmetry of the previous assumption, we can obtain lemma 1. According to (4.7), ${\omega }^{-(j-1)}{K}_{{jk}}{\omega }^{-(k-1)}={\omega }^{2-m}{K}_{{jk}}$, ${\omega }^{-(j-1)}{\widehat{K}}_{{jk}}{\omega }^{-(k-1)}={\omega }^{2-m}{\widehat{K}}_{{jk}}$, so ${K}_{{jk}}\ne 0$, ${\widehat{K}}_{{jk}}\ne 0$; we have $j+k\equiv m\,{mod}\,N$ because of ${\theta }^{N}=-1$. When ${K}_{{jk}}\ne 0$, ${\widehat{K}}_{{jk}}\ne 0$, ${\omega }^{-(j-1)}{\widehat{K}}_{{jk}}{\omega }^{-(k-1)}=\pm {\omega }^{2-(j+k)}{\widehat{K}}_{{jk}}=\pm {\omega }^{2-m}{\widehat{K}}_{{jk}}$, so ${\widehat{K}}_{{jk}}^{{\prime} }=\pm {\widehat{K}}_{{jk}}$; therefore, ${\widehat{K}}^{{\prime} }$ is a real symmetry matrix, and the expression of ${K}^{{\prime} }$ tells us that ${K}^{{\prime} }$ is also a real symmetry matrix. From (4.8), ${\omega }^{-(j-1)}{\widehat{J}}_{{jk}}{\omega }^{k-1}=\omega {\widehat{J}}_{{jk}}$, we obtain $K\equiv j+1\,{mod}\,N$ as ${\widehat{J}}_{{jk}}\ne 0$. When ${\widehat{J}}_{{jk}}\ne 0$, ${\omega }^{-(j-1)}{\widehat{J}}_{{jk}}{\omega }^{k-1}=\pm \theta {\widehat{J}}_{{jk}}$ then ${\widehat{J}}^{{\prime} }={\theta }^{-1}{{\rm{\Theta }}}^{-1}\widehat{J}{\rm{\Theta }}$ is a real matrix. ${Q}^{{\prime} },{\widehat{Q}}^{{\prime} },{J}^{{\prime} }$ can be proved similarly.

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

	(1)${\rm{\Omega }}{\rm{\Phi }},\,{\rm{\Omega }}\widehat{{\rm{\Phi }}}$ are the solutions of (4.3) when $\lambda ={\lambda }_{0}$;
	(2)$\overline{{\rm{\Phi }}},\,\overline{\widehat{{\rm{\Phi }}}}$ are the solutions of (4.3) when $\lambda ={\overline{\lambda }}_{0}$;
	(3)$\left\{\begin{array}{l}{\rm{\Psi }}=K{\rm{\Phi }}+\varepsilon \widehat{K}\widehat{{\rm{\Phi }}},\\ \widehat{{\rm{\Psi }}}=\widehat{K}{\rm{\Phi }}+K\widehat{{\rm{\Phi }}},\end{array}\right.$ are the solutions of the conjugated Lax pairs when $\lambda =-{\lambda }_{0}$:

(4.14)$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Psi }}}_{x}=(-{\lambda }_{0}{J}^{{\rm{T}}}+{P}^{{\rm{T}}}){\rm{\Psi }}+(-{\lambda }_{0}{J}^{{\rm{T}}}+{\widehat{P}}^{{\rm{T}}})\widehat{{\rm{\Psi }}},\\ {\widehat{{\rm{\Psi }}}}_{x}=(-{\lambda }_{0}{J}^{{\rm{T}}}+\varepsilon {\widehat{P}}^{{\rm{T}}}){\rm{\Psi }}+(-{\lambda }_{0}{J}^{{\rm{T}}}+{P}^{{\rm{T}}})\widehat{{\rm{\Psi }}},\\ {{\rm{\Psi }}}_{y}=-\displaystyle \frac{1}{{\lambda }_{0}}({Q}^{{\rm{T}}}{\rm{\Psi }}+{\widehat{Q}}^{{\rm{T}}}\widehat{{\rm{\Psi }}}),\\ {\widehat{{\rm{\Psi }}}}_{y}=-\displaystyle \frac{1}{{\lambda }_{0}}({Q}^{{\rm{T}}}\widehat{{\rm{\Psi }}}+\varepsilon {\widehat{Q}}^{{\rm{T}}}{\rm{\Psi }});\end{array}\right.\end{eqnarray}$

(1)$\left\{\begin{array}{l}{{\rm{\Phi }}}^{{\prime} }={{\rm{\Theta }}}^{-1}{\rm{\Phi }},\\ {\widehat{{\rm{\Phi }}}}^{{\prime} }={{\rm{\Theta }}}^{-1}\widehat{{\rm{\Phi }}},\end{array}\right.$ are the solutions of

(4.15)$\begin{eqnarray}\left\{\begin{array}{l}{{\rm{\Phi }}}_{x}^{{\prime} }=(\lambda {J}^{{\prime} }+P){{\rm{\Phi }}}^{{\prime} }+\varepsilon (\lambda {J}^{{\prime} }+\widehat{P}){\widehat{{\rm{\Phi }}}}^{{\prime} },\\ {\widehat{{\rm{\Phi }}}}_{x}^{{\prime} }=(\lambda {J}^{{\prime} }+\widehat{P}){{\rm{\Phi }}}^{{\prime} }+(\lambda {J}^{{\prime} }+P){\widehat{{\rm{\Phi }}}}^{{\prime} },\\ {{\rm{\Phi }}}_{y}^{{\prime} }=\displaystyle \frac{1}{\lambda }({Q}^{{\prime} }{{\rm{\Phi }}}^{{\prime} }+\varepsilon {\widehat{Q}}^{{\prime} }{\widehat{{\rm{\Phi }}}}^{{\prime} }),\\ {\widehat{{\rm{\Phi }}}}_{y}=\displaystyle \frac{1}{\lambda }({\widehat{Q}}^{{\prime} }{{\rm{\Phi }}}^{{\prime} }+{Q}^{{\prime} }{\widehat{{\rm{\Phi }}}}^{{\prime} }),\end{array}\right.{as}\,\lambda ={\theta }^{-1}{\lambda }_{0},\end{eqnarray}$

where ${J}^{{\prime} }$, ${Q}^{{\prime} }$ and ${\widehat{Q}}^{{\prime} }$ are given by lemma 2; when ${\lambda }_{0}$ is real, ${{\rm{\Phi }}}^{{\prime} }$, ${\widehat{{\rm{\Phi }}}}^{{\prime} }$ are real solutions.

Proof.From

(4.16)$\begin{eqnarray}\left\{\begin{array}{l}{\rm{\Omega }}{{\rm{\Phi }}}_{x}={\rm{\Omega }}({\lambda }_{0}J+P){\rm{\Phi }}+{\rm{\Omega }}({\lambda }_{0}J+\varepsilon \widehat{P})\widehat{{\rm{\Phi }}}=(\omega {\lambda }_{0}J+P){\rm{\Omega }}{\rm{\Phi }}+(\omega {\lambda }_{0}J+\varepsilon \widehat{P}){\rm{\Omega }}\widehat{{\rm{\Phi }}},\\ {\rm{\Omega }}{\widehat{{\rm{\Phi }}}}_{x}={\rm{\Omega }}({\lambda }_{0}J+\widehat{P}){\rm{\Phi }}+{\rm{\Omega }}({\lambda }_{0}J+P)\widehat{{\rm{\Phi }}}=(\omega {\lambda }_{0}J+\widehat{P}){\rm{\Omega }}{\rm{\Phi }}+(\omega {\lambda }_{0}J+P){\rm{\Omega }}\widehat{{\rm{\Phi }}},\end{array}\right.\end{eqnarray}$

and

(4.17)$\begin{eqnarray}\left\{\begin{array}{l}{\left({\rm{\Omega }}{\rm{\Phi }}\right)}_{x}=\displaystyle \frac{1}{{\lambda }_{0}}\left(Q{\rm{\Phi }}+\varepsilon \widehat{Q}\widehat{{\rm{\Phi }}}\right)=\displaystyle \frac{1}{\omega {\lambda }_{0}}(Q{\rm{\Omega }}{\rm{\Phi }}+\varepsilon \widehat{Q}{\rm{\Omega }}\widehat{{\rm{\Phi }}}),\\ {\left({\rm{\Omega }}\widehat{{\rm{\Phi }}}\right)}_{x}=\displaystyle \frac{1}{{\lambda }_{0}}\left(Q{\rm{\Phi }}+\varepsilon \widehat{Q}\widehat{{\rm{\Phi }}}\right)=\displaystyle \frac{1}{\omega {\lambda }_{0}}\left(\widehat{Q}{\rm{\Omega }}{\rm{\Phi }}+Q{\rm{\Omega }}\widehat{{\rm{\Phi }}}\right),\end{array}\right.\end{eqnarray}$

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

(4.18)$\begin{eqnarray}\left\{\begin{array}{l}K{{\rm{\Phi }}}_{x}+\varepsilon \widehat{K}{\widehat{{\rm{\Phi }}}}_{x}=K({\lambda }_{0}J+P){\rm{\Phi }}+\varepsilon \widehat{K}({\lambda }_{0}J+\widehat{P}){\rm{\Phi }}+\widehat{K}({\lambda }_{0}J+P)\widehat{{\rm{\Phi }}}+\varepsilon K({\lambda }_{0}J+\widehat{P})\widehat{{\rm{\Phi }}}\\ \widehat{K}{{\rm{\Phi }}}_{x}+K{\widehat{{\rm{\Phi }}}}_{x}=\widehat{K}({\lambda }_{0}J+P){\rm{\Phi }}+K({\lambda }_{0}J+\widehat{P}){\rm{\Phi }}+K({\lambda }_{0}J+P)\widehat{{\rm{\Phi }}}+\varepsilon \widehat{K}({\lambda }_{0}J+\widehat{P})\widehat{{\rm{\Phi }}}\\ =({\lambda }_{0}{J}^{T}-{P}^{T})K{\rm{\Phi }}+({\lambda }_{0}{J}^{T}-{P}^{T})\widehat{K}{\rm{\Phi }}+({\lambda }_{0}{J}^{T}-{\widehat{P}}^{T})K\widehat{{\rm{\Phi }}}+\varepsilon ({\lambda }_{0}{J}^{T}-{P}^{T})\widehat{K}\widehat{{\rm{\Phi }}}\\ =({\lambda }_{0}{J}^{T}-\varepsilon {\widehat{P}}^{T})K{\rm{\Phi }}+({\lambda }_{0}{J}^{T}-{P}^{T})\widehat{K}{\rm{\Phi }}+\varepsilon ({\lambda }_{0}{J}^{T}-\varepsilon {\widehat{P}}^{T})\widehat{K}\widehat{{\rm{\Phi }}}+({\lambda }_{0}{J}^{T}-{P}^{T})K\widehat{{\rm{\Phi }}},\end{array}\right.\end{eqnarray}$

and

(4.19)$\begin{eqnarray}\left\{\begin{array}{l}K{{\rm{\Phi }}}_{y}+\varepsilon \widehat{K}{\widehat{{\rm{\Phi }}}}_{y}=\displaystyle \frac{1}{{\lambda }_{0}}({KQ}{\rm{\Phi }}+\varepsilon \widehat{K}\widehat{Q}{\rm{\Phi }}+\widehat{K}Q\widehat{{\rm{\Phi }}}+\varepsilon K\widehat{Q}\widehat{{\rm{\Phi }}}),\\ \widehat{K}{{\rm{\Phi }}}_{y}+K{\widehat{{\rm{\Phi }}}}_{y}=\displaystyle \frac{1}{{\lambda }_{0}}(\widehat{K}Q{\rm{\Phi }}+K\widehat{Q}{\rm{\Phi }}+{KQ}\widehat{{\rm{\Phi }}}+\varepsilon \widehat{K}\widehat{Q}\widehat{{\rm{\Phi }}}),\end{array}\right.\end{eqnarray}$

we have proved conclusion (3). According to the expressions of ${J}^{{\prime} },{\widehat{J}}^{{\prime} },{Q}^{{\prime} }$ and ${\widehat{Q}}^{{\prime} }$, (4) is obvious.

Assume $G(x,y,\lambda ),\widehat{G}(x,y,\lambda )$ are M × M matrices given by:where ${G}_{i},{\widehat{G}}_{i}$ $(i=1,2,\ldots ,L)$ are independent of λ and ${G}_{0}=I$, ${\widehat{G}}_{0}=0$. Thus, $V(x,y)$, $\widehat{V}(x,y)$ have a similar symmetry satisfying (4.10):where $\widetilde{V}(x,y),\widehat{\widetilde{V}}(x,y)$ are partition diagonal matrices.

Theorem 1.Let Φ, $\widehat{{\rm{\Phi }}}$ be the solutions of (4.3), so $\widetilde{{\rm{\Phi }}}=G{\rm{\Phi }}+\varepsilon \widehat{G}\widehat{{\rm{\Phi }}}$, $\widetilde{\widehat{{\rm{\Phi }}}}=\widehat{G}{\rm{\Phi }}+G\widehat{{\rm{\Phi }}}$ satisfy

(4.22)$\begin{eqnarray}\left\{\begin{array}{l}{\widetilde{{\rm{\Phi }}}}_{x}=(\lambda J+\widetilde{P})\widetilde{{\rm{\Phi }}}+\varepsilon (\lambda J+\widehat{\widetilde{P}})\widehat{\widetilde{{\rm{\Phi }}}},\\ {\widehat{\widetilde{{\rm{\Phi }}}}}_{x}=(\lambda J+\widehat{\widetilde{P}})\widetilde{{\rm{\Phi }}}+(\lambda J+\widetilde{P})\widehat{\widetilde{{\rm{\Phi }}}},\\ {\widetilde{{\rm{\Phi }}}}_{y}=\displaystyle \frac{1}{\lambda }(\widetilde{Q}\widetilde{{\rm{\Phi }}}+\varepsilon \widehat{\widetilde{Q}}\widehat{\widetilde{{\rm{\Phi }}}}),\\ {\widehat{\widetilde{{\rm{\Phi }}}}}_{y}=\displaystyle \frac{1}{\lambda }(\widehat{\widetilde{Q}}\widetilde{{\rm{\Phi }}}+\widetilde{Q}\widehat{\widetilde{{\rm{\Phi }}}}),\end{array}\right.\end{eqnarray}$

where

(4.23)$\begin{eqnarray}\left\{\begin{array}{l}\widetilde{P}=\displaystyle \frac{{\widetilde{V}}_{x}\widetilde{V}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}-\displaystyle \frac{\varepsilon {\widehat{\widetilde{V}}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ \widehat{\widetilde{P}}=\displaystyle \frac{{\widehat{\widetilde{V}}}_{x}\widetilde{V}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}-\displaystyle \frac{{\widetilde{V}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ \end{array}\right.\left\{\begin{array}{l}\widetilde{Q}=\displaystyle \frac{\widetilde{V}{\widetilde{J}}^{{\rm{T}}}\widetilde{V}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}-\displaystyle \frac{\varepsilon \widehat{\widetilde{V}}{J}^{{\rm{T}}}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ \widehat{\widetilde{Q}}=\displaystyle \frac{\widehat{\widetilde{V}}{J}^{{\rm{T}}}\widetilde{V}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}-\displaystyle \frac{\widetilde{V}{J}^{{\rm{T}}}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}.\end{array}\right.\end{eqnarray}$

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

(4.24)$\begin{eqnarray}\left\{\begin{array}{l}G(\lambda )=\prod _{l=1}^{L}(\lambda +{\overline{\lambda }}_{l})[E-\sum _{j,k=1}^{L}\displaystyle \frac{1}{(\lambda +{\overline{\lambda }}_{k})({T}^{2}-\varepsilon {\widehat{T}}^{2})}({H}_{j}{{TH}}_{k}^{* }K-\varepsilon {\widehat{H}}_{j}\widehat{T}{H}_{k}^{* }K+{\varepsilon }^{2}{\widehat{H}}_{j}{{TH}}_{k}^{* }K\\ \,\,\,\,\,\,\,\,\,-{\varepsilon }^{2}{H}_{j}\widehat{T}{H}_{k}^{* }K+{H}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+\varepsilon {\widehat{H}}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {H}_{j}\widehat{T}{H}_{k}^{* }\widehat{K})],\\ \widehat{G}(\lambda )=\prod _{l=1}^{L}(\lambda +{\overline{\lambda }}_{l})\sum _{j,k=1}^{L}\displaystyle \frac{-1}{(\lambda +{\overline{\lambda }}_{k})({T}^{2}-\varepsilon {\widehat{T}}^{2})}({\widehat{H}}_{j}{{TH}}_{k}^{* }K-{H}_{j}\widehat{T}{H}_{k}^{* }K+\varepsilon {H}_{j}T{\widehat{H}}_{k}^{* }K-\\ \,\,\,\,\,\,\,\,\,\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }K+{\widehat{H}}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}),\end{array}\right.\end{eqnarray}$

where

(4.25)$\begin{eqnarray}\left\{\begin{array}{l}{T}_{{jk}}=\displaystyle \frac{1}{{\overline{\lambda }}_{j}+{\lambda }_{k}}({H}_{j}^{* }{{KH}}_{k}+{\widehat{H}}_{j}^{* }\widehat{K}{H}_{k}+{\widehat{H}}_{j}^{* }K{\widehat{H}}_{k}+\varepsilon {H}_{j}^{* }\widehat{K}{\widehat{H}}_{k}),\\ {\widehat{T}}_{{jk}}=\displaystyle \frac{1}{{\overline{\lambda }}_{j}+{\lambda }_{k}}({H}_{j}^{* }\widehat{K}{H}_{k}+\varepsilon {\widehat{H}}_{j}^{* }{{KH}}_{k}+{H}_{j}^{* }K{\widehat{H}}_{k}+{\varepsilon }^{2}{\widehat{H}}_{j}^{* }\widehat{K}{\widehat{H}}_{k}),\end{array}\right.j,k=1,\ldots ,L.\end{eqnarray}$

Since $G(\lambda ),\,\widehat{G}(\lambda )$ are the L-degree polynomials of λ, this can be also written as:

(4.26)$\begin{eqnarray}\left\{\begin{array}{l}G(\lambda )={\lambda }^{L}E+{\lambda }^{L-1}{G}_{1}+{\lambda }^{L-2}{G}_{2}+\cdots +\lambda {G}_{L-1}+{G}_{L},\\ \widehat{G}(\lambda )={\lambda }^{L-1}{\widehat{G}}_{1}+{\lambda }^{L-2}{\widehat{G}}_{2}+\cdots +\lambda {\widehat{G}}_{L-1}+{\widehat{G}}_{L}.\end{array}\right.\end{eqnarray}$

From (4.24), we have

(4.27)$\begin{eqnarray}\left\{\begin{array}{l}{G}_{1}=\sum _{l=1}^{L}{\overline{\lambda }}_{l}-\sum _{j,k=1}^{L}\displaystyle \frac{1}{{T}^{2}-\varepsilon {\widehat{T}}^{2}}({H}_{j}{{TH}}_{k}^{* }K-\varepsilon {\widehat{H}}_{j}\widehat{T}{H}_{k}^{* }K+{\varepsilon }^{2}{\widehat{H}}_{j}{{TH}}_{k}^{* }K-{\varepsilon }^{2}{H}_{j}\widehat{T}{H}_{k}^{* }K+\\ \,\,\,\,\,\,\,\,{H}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+\varepsilon {\widehat{H}}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {H}_{j}\widehat{T}{H}_{k}^{* }\widehat{K}),\\ {\widehat{G}}_{1}=\sum _{j,k=1}^{L}\displaystyle \frac{-1}{{T}^{2}-\varepsilon {\widehat{T}}^{2}}({\widehat{H}}_{j}{{TH}}_{k}^{* }K-{H}_{j}\widehat{T}{H}_{k}^{* }K+\varepsilon {H}_{j}T{\widehat{H}}_{k}^{* }K-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }K+\\ \,\,\,\,\,\,\,\,\,{\widehat{H}}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}),\\ {G}_{L}=(\prod _{l=1}^{L}{\overline{\lambda }}_{l})[E-\sum _{j,k=1}^{L}\displaystyle \frac{1}{{\overline{\lambda }}_{k}({T}^{2}-\varepsilon {\widehat{T}}^{2})}({H}_{j}{{TH}}_{k}^{* }K-\varepsilon {\widehat{H}}_{j}\widehat{T}{H}_{k}^{* }K+{\varepsilon }^{2}{\widehat{H}}_{j}{{TH}}_{k}^{* }K-{\varepsilon }^{2}{H}_{j}\widehat{T}{H}_{k}^{* }K+\\ \,\,\,\,\,\,\,\,\,{H}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+\varepsilon {\widehat{H}}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {H}_{j}\widehat{T}{H}_{k}^{* }\widehat{K})],\\ {\widehat{G}}_{L}=(\prod _{l=1}^{L}{\overline{\lambda }}_{l})\sum _{j,k=1}^{L}\displaystyle \frac{-1}{{\overline{\lambda }}_{k}({T}^{2}-\varepsilon {\widehat{T}}^{2})}({\widehat{H}}_{j}{{TH}}_{k}^{* }K-{H}_{j}\widehat{T}{H}_{k}^{* }K+\varepsilon {H}_{j}T{\widehat{H}}_{k}^{* }K-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }K+\\ \,\,\,\,\,\,\,\,\,{\widehat{H}}_{j}T{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}+{H}_{j}{{TH}}_{k}^{* }\widehat{K}-\varepsilon {\widehat{H}}_{j}\widehat{T}{\widehat{H}}_{k}^{* }\widehat{K}).\end{array}\right.\end{eqnarray}$

Assuming $G(\lambda )$, $\widehat{G}(\lambda )$ are given by (4.24), then theorem 1 is established. According to [24], we have

(4.28)$\begin{eqnarray}\left\{\begin{array}{l}\widetilde{P}=P-[J,{G}_{1}],\\ \widehat{\widetilde{P}}=\widehat{P}-[J,{\widehat{G}}_{1}],\end{array}\right.\left\{\begin{array}{l}\widetilde{Q}=\displaystyle \frac{{G}_{L}{{QG}}_{L}-\varepsilon ({\widehat{G}}_{L}\widehat{Q}{G}_{L}+{G}_{L}\widehat{Q}{\widehat{G}}_{L}+{\widehat{G}}_{L}Q{\widehat{G}}_{L})}{{G}_{L}^{2}-\varepsilon {\widehat{G}}_{L}^{2}},\\ \widehat{\widetilde{Q}}=\displaystyle \frac{{\widehat{G}}_{L}{{QG}}_{L}+{G}_{L}\widehat{Q}{G}_{L}-{G}_{L}Q{\widehat{G}}_{L}-\varepsilon {\widehat{G}}_{L}\widehat{Q}{\widehat{G}}_{L}}{{G}_{L}^{2}-\varepsilon {\widehat{G}}_{L}^{2}},\end{array}\right.\end{eqnarray}$

and if $\widetilde{V},\widehat{\widetilde{V}}$ satisfy

(4.29)$\begin{eqnarray}\left\{\begin{array}{l}\widetilde{V}={G}_{L}V+\varepsilon {\widehat{G}}_{L}\widehat{V},\\ \widehat{\widetilde{V}}={\widehat{G}}_{L}V+{G}_{L}\widehat{V},\end{array}\right.\end{eqnarray}$

then

(4.30)$\begin{eqnarray}\left\{\begin{array}{l}\widetilde{P}=\displaystyle \frac{{\widetilde{V}}_{x}\widetilde{V}-\varepsilon {\widehat{\widetilde{V}}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ \widehat{\widetilde{P}}=\displaystyle \frac{{\widehat{\widetilde{V}}}_{x}\widetilde{V}-{\widetilde{V}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\end{array}\right.\left\{\begin{array}{l}\widetilde{Q}=\displaystyle \frac{\widetilde{V}{J}^{{\rm{T}}}\widetilde{V}-\varepsilon \widehat{\widetilde{V}}{J}^{{\rm{T}}}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ \widehat{\widetilde{Q}}=\displaystyle \frac{\widehat{\widetilde{V}}{J}^{{\rm{T}}}\widetilde{V}-\widetilde{V}{J}^{{\rm{T}}}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\end{array}\right.\end{eqnarray}$

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

(4.31)$\begin{eqnarray}\left\{\begin{array}{l}{T}_{{jk},x}={H}_{j}^{* }{{KJH}}_{k}+{\widehat{H}}_{j}^{* }\widehat{K}{{JH}}_{k}+{\widehat{H}}_{j}^{* }{KJ}{\widehat{H}}_{k}+\varepsilon {H}_{j}^{* }\widehat{K}J{\widehat{H}}_{k},\\ {\widehat{T}}_{{jk},x}={H}_{j}^{* }\widehat{K}{{JH}}_{k}+{H}_{j}^{* }{KJ}{\widehat{H}}_{k}+\varepsilon {\widehat{H}}_{j}^{* }{{KJH}}_{k}+{\varepsilon }^{2}{\widehat{H}}_{j}^{* }\widehat{K}J{\widehat{H}}_{k},\end{array}\right.\end{eqnarray}$

(4.32)$\begin{eqnarray}\left\{\begin{array}{l}{T}_{{jk},y}=\displaystyle \frac{1}{{\overline{\lambda }}_{j}{\lambda }_{k}}[{H}_{j}^{* }{{KQH}}_{k}+{\widehat{H}}_{j}^{* }\widehat{K}{{QH}}_{k}+{\widehat{H}}_{j}^{* }K\widehat{Q}{H}_{k}+{\widehat{H}}_{j}^{* }{KQ}{\widehat{H}}_{k}+\\ \,\,\,\,\,\,\,\,\,\varepsilon ({H}_{j}^{* }\widehat{K}\widehat{Q}{H}_{k}+{H}_{j}^{* }K\widehat{Q}{\widehat{H}}_{k}+{\widehat{H}}_{j}^{* }\widehat{K}\widehat{Q}{\widehat{H}}_{k}+{H}_{j}^{* }\widehat{K}Q{\widehat{H}}_{k})],\\ {\widehat{T}}_{{jk},y}=\displaystyle \frac{1}{{\overline{\lambda }}_{j}{\lambda }_{k}}[{H}_{j}^{* }\widehat{K}{{QH}}_{k}+{H}_{j}^{* }K\widehat{Q}{H}_{k}+{H}_{j}^{* }{KQ}{\widehat{H}}_{k}+\varepsilon ({\widehat{H}}_{j}^{* }{{KQH}}_{k}+{H}_{j}^{* }\widehat{K}\widehat{Q}{\widehat{H}}_{k})\\ +{\varepsilon }^{2}({\widehat{H}}_{j}^{* }\widehat{K}\widehat{Q}{H}_{k}+{H}_{j}^{* }\widehat{K}Q{\widehat{H}}_{k}+{\widehat{H}}_{j}^{* }K\widehat{Q}{\widehat{H}}_{k})].\end{array}\right.\end{eqnarray}$

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

(4.33)$\begin{eqnarray}\left\{\begin{array}{l}(\lambda J+\widetilde{P})G+\varepsilon \widehat{\widetilde{P}}\widehat{G}-G(\lambda J+P)-\varepsilon \widehat{G}\widehat{P}-{G}_{x}=0,\\ (\lambda J+\widetilde{P})\widehat{G}+\widehat{\widetilde{P}}G-\widehat{G}(\lambda J+P)-G\widehat{P}-{\widehat{G}}_{x}=0,\\ {\lambda }^{-1}(\widetilde{Q}G+\varepsilon \widehat{\widetilde{Q}}\widehat{G})-{\lambda }^{-1}({GQ}+\varepsilon \widehat{G}\widehat{Q})-{G}_{y}=0,\\ {\lambda }^{-1}(\widehat{\widetilde{Q}}G+\widetilde{Q}\widehat{G})-{\lambda }^{-1}(\widehat{G}Q+G\widehat{Q})-{\widehat{G}}_{y}=0,\end{array}\right.\end{eqnarray}$

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

(4.34)$\begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{\widetilde{V}J\widetilde{V}-\varepsilon \widehat{\widetilde{V}}J\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}=\displaystyle \frac{{G}_{L}{{QG}}_{L}+\varepsilon {\widehat{G}}_{L}\widehat{Q}{G}_{L}-\varepsilon ({G}_{L}\widehat{Q}{\widehat{G}}_{L}+{\widehat{G}}_{L}Q{\widehat{G}}_{L})}{{G}_{L}^{2}-\varepsilon {\widehat{G}}_{L}^{2}},\\ \displaystyle \frac{\widehat{\widetilde{V}}J\widetilde{V}-\widetilde{V}J\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}=\displaystyle \frac{{\widehat{G}}_{L}{{QG}}_{L}+{G}_{L}\widehat{Q}{G}_{L}-{G}_{L}Q{\widehat{G}}_{L}-\varepsilon {\widehat{G}}_{L}\widehat{Q}{\widehat{G}}_{L}}{{G}_{L}^{2}-\varepsilon {\widehat{G}}_{L}^{2}},\end{array}\right.\end{eqnarray}$

are obvious; what we need to prove is that

(4.35)$\begin{eqnarray}\left\{\begin{array}{l}\displaystyle \frac{{\widetilde{V}}_{x}\widetilde{V}-\varepsilon {\widehat{\widetilde{V}}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}=P-[J,{G}_{1}],\\ \displaystyle \frac{{\widehat{\widetilde{V}}}_{x}\widetilde{V}-{\widetilde{V}}_{x}\widehat{\widetilde{V}}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}}=\widehat{P}-[J,{\widehat{G}}_{1}].\end{array}\right.\end{eqnarray}$

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

(4.36)$\begin{eqnarray}\left\{\begin{array}{l}{G}_{x}+G(\lambda J+P)+\varepsilon \widehat{G}\widehat{P}=(\lambda J+P-[J,{G}_{1}])G+\varepsilon (\widehat{P}-[J,{\widehat{G}}_{1}])\widehat{G},\\ {\widehat{G}}_{x}+\widehat{G}(\lambda J+P)+G\widehat{P}=(\lambda J+P-[J,{G}_{1}])\widehat{G}+(\widehat{P}-[J,{\widehat{G}}_{1}])G,\end{array}\right.\end{eqnarray}$

when $\lambda =0$. On the other hand, from (4.30), we get

(4.37)$\begin{eqnarray}\left\{\begin{array}{l}{G}_{L,x}+{G}_{L}P+\varepsilon {\widehat{G}}_{L}\widehat{P}=\displaystyle \frac{{\widetilde{V}}_{x}\widetilde{V}{G}_{L}-\varepsilon ({\widehat{\widetilde{V}}}_{x}\widehat{\widetilde{V}}{G}_{L}+{\widetilde{V}}_{x}\widehat{\widetilde{V}}{\widehat{G}}_{L}-{\widehat{\widetilde{V}}}_{x}\widetilde{V}{\widehat{G}}_{L})}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\\ {\widehat{G}}_{L,x}+{\widehat{G}}_{L}P+{G}_{L}\widehat{P}=\displaystyle \frac{{\widehat{\widetilde{V}}}_{x}\widetilde{V}{G}_{L}-{\widetilde{V}}_{x}\widehat{\widetilde{V}}{G}_{L}+{\widetilde{V}}_{x}\widetilde{V}{\widehat{G}}_{L}-\varepsilon {\widehat{\widetilde{V}}}_{x}\widehat{\widetilde{V}}{\widehat{G}}_{L}}{{\widetilde{V}}^{2}-\varepsilon {\widehat{\widetilde{V}}}^{2}},\end{array}\right.\end{eqnarray}$

and the proof is complete.

Remarks: