1. Introduction
The 2D Toda lattice hierarchy [1, 2] is one of the most fundamental topics in the theory of integrable systems. This hierarchy is a completely integrable system, which has many important applications in mathematics and physics including the representation theory of Lie algebras, orthogonal polynomials, random matrix models [3–6], and the theory of Gromov-Witten invariants [7]. Many known integrable nonlinear partial differential and difference equations are reductions or special cases of 2D Toda lattice. For example, the 1D Toda hierarchy [8–10], interestingly, the topological sigma model (geometrically, the Gromov-Witten invariants) of the Riemann sphere CP1, which is related to the 1D Toda hierarchy, the Ablowitz-Ladik hierarchy [11], which can control the Gromov-Witten theory [12], the fractional Volterra (FV) hierarchy [13] and the multi-component extension version of this hierarchy [14], and the limit fractional Volterra (LFV) hierarchy [15], which can control the linear Hodge integral. Particularly, the LFV hierarchy is a new system defined by Liu in 2021 [15], which can be as a certain reduction of the 2D Toda hierarchy. Moreover, the reduced equations can be further reduced to finite-dimensional systems like Newton equations for a system of particles, Garnier system and other models [16].
There are many good properties of the LFV hierarchy, such as it's a tau-symmetric Hamiltonian integrable hierarchy, and it contains two sub-hierarchies: the intermediate long wave hierarchy (ILW) [17] and the limit of the FV hierarchy [18]. It is proved by Buryak in [17, 19] that the Hodge hierarchy for linear Hodge integrals is equivalent to the ILW hierarchy, and the relationship between the LFV hierarchy and the ILW hierarchy is established by the Miura-type transformation, the equations of the flows t1,n in the LFV hierarchy are transformed to the symmetries of the ILW hierarchy. In other words, the LFV hierarchy can be an extension of the ILW hierarchy. The FV hierarchy is an integrable hierarchy, which can be regarded as a fractional generalization of the Volterra lattice hierarchy [20], and which is also regarded as a reduction of the bigraded Toda hierarchy [21, 22]. Liu [18] gave the definition of this integrable hierarchy in terms of Lax pair and Hamiltonian formalisms, and he presented its tau functions and multi-soliton solutions. Moreover, the generating function of cubic Hodge integrals satisfies the local Calabi-Yau condition, which is conjectured to be a tau function of the FV hierarchy.
Based on the excellent properties of the LFV hierarchy, we want to explore its other properties. It was pointed out that the Darboux transformation was an efficient method to generate soliton solutions of integrable equations. The multi-solitons can be obtained by this Darboux transformation from a trivial seed solution. If we give the Lax pair of this hierarchy, on this basis, we can construct the Darboux transformation and obtain the soliton solutions using the Darboux transformation of the LFV hierarchy. Further, we want to extend this hierarchy to the noncommutative version [23–25].
Figure 1. When parameters are p = 2, λ1 = 1 and ϵ = 0.5, u[1] is the soliton solution. |
Figure 2. When parameters are p = 2, λ1 = 1, λ2 = 2 and ϵ = 0.5, u[2] is the two-soliton solution. |
This paper is arranged as follows: In section 2 , summarizing the related lemmas and definition of the LFV hierarchy, we construct the Darboux transformation [22, 26] of the LFV hierarchy, and obtain the corresponding relationship between the old and new solutions. Furthermore, the one and two soliton solutions are obtained from the LFV hierarchy. In section 3 , we extend the LFV hierarchy to the NCLFV hierarchy. And we construct the Darboux transformation of the noncommutative version [27, 28]. Meanwhile, we obtain the quasi determinant [29] solution of the NCLFV hierarchy. Finally, we draw a brief conclusion of this paper.
2. The LFV hierarchy
2.1. The definition of the LFV hierarchy
We recall the two Lax operators of 2D Toda hierarchy2.1 ) and (2.2 ) can be written as the dressing operators2.1 ) and (2.2 ) is defined by 2.1 ) satisfies2.1 ) can be obtained, the recurrence relations are 2.2 ) satisfies2.5 ), and bk satisfies the following recursion relations 2.5 ), L is determined by (2.1 ) and satisfies the conditions in lemma 1 , $\bar{L}$ is determined by (2.2 ) and satisfies the conditions in lemma 2 . For $A={\sum }_{i\in Z}{a}_{i}(x){{\rm{\Lambda }}}^{i}$, ${A}_{+}={\sum }_{i\geqslant 0}{a}_{i}(x){{\rm{\Lambda }}}^{i}$, ${A}_{-}={\sum }_{i\lt 0}{a}_{i}(x){{\rm{\Lambda }}}^{i}$. Meanwhile, the operators L and $\bar{L}$ satisfy the Lax equations 2.9 ). 2.14a ) into equation (2.15 ), we have2.14b ) are the equivalent form of ${t}_{2,n}$ flow equations in the LFV hierarchy. □
$\begin{eqnarray}L={\rm{\Lambda }}+{a}_{0}+{a}_{1}{{\rm{\Lambda }}}^{-1}+\cdots ,\end{eqnarray}$
$\begin{eqnarray}\bar{L}={{\rm{e}}}^{u}{{\rm{\Lambda }}}^{-1}+{b}_{0}+{b}_{1}{\rm{\Lambda }}+\cdots ,\end{eqnarray}$
where Λ represents shift operator ${\rm{\Lambda }}:= {{\rm{e}}}^{\epsilon {\partial }_{x}}$, which acts on the function f(x) with one variable x, Λnf(x) = f(x + nε), and ai = ai(u), bi = bi(u) for i ≥ 0. The Lax operators ( $\begin{eqnarray}L=P{\rm{\Lambda }}{P}^{-1},\,P=1+\displaystyle \sum _{k\geqslant 1}{p}_{k}{{\rm{\Lambda }}}^{-k},\end{eqnarray}$
$\begin{eqnarray}\bar{L}=Q{{\rm{\Lambda }}}^{-1}{Q}^{-1},\,Q=\displaystyle \sum _{k\geqslant 0}{q}_{k}{{\rm{\Lambda }}}^{k}.\end{eqnarray}$
The logarithm of the operators ($\begin{eqnarray*}\begin{array}{l}\mathrm{log}L:= P(\varepsilon {\partial }_{x}){P}^{-1}=\varepsilon {\partial }_{x}-\varepsilon {P}_{x}{P}^{-1},\\ \mathrm{log}\bar{L}:= Q(-\varepsilon {\partial }_{x}){Q}^{-1}=-\varepsilon {\partial }_{x}+\varepsilon {Q}_{x}{Q}^{-1},\end{array}\end{eqnarray*}$
where Px = ∑k≥1(∂xpk)Λ−k and Qx = ∑k≥0(∂xqk)Λk.([15]) There exist differential polynomials ak such that
$\begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{a}_{k}=\displaystyle \frac{{p}^{k+1}}{(k+1)!}{{\rm{e}}}^{(k+1)u},\end{eqnarray*}$
and the difference operator L in ($\begin{eqnarray*}\displaystyle \frac{1}{p}\mathrm{log}L=K,\end{eqnarray*}$
where K is a differential-difference operator, which is defined by $\begin{eqnarray}K=\displaystyle \frac{1}{p}\epsilon {\partial }_{x}+{{\rm{e}}}^{u}{{\rm{\Lambda }}}^{-1}.\end{eqnarray}$
Meanwhile, the expression of ai in ( $\begin{eqnarray}\displaystyle \frac{1}{p}\epsilon {\partial }_{x}{a}_{i+1}={a}_{k}{{\rm{\Lambda }}}^{-k}{{\rm{e}}}^{u}-{{\rm{e}}}^{u}{{\rm{\Lambda }}}^{-1}{a}_{i},\,i\geqslant 0,\end{eqnarray}$
with $\begin{eqnarray}{a}_{0}=p\displaystyle \frac{{\rm{\Lambda }}-1}{\epsilon {\partial }_{x}}{{\rm{e}}}^{u}.\end{eqnarray}$
([15]) There exist unique differential polynomials bk such that
$\begin{eqnarray*}\mathop{\mathrm{lim}}\limits_{\varepsilon \to 0}{b}_{k}=\displaystyle \frac{{{\rm{e}}}^{-{ku}}}{{p}^{k+1}}{\beta }_{k},\end{eqnarray*}$
and the difference operator $\bar{L}$ in ($\begin{eqnarray*}\bar{L}-\displaystyle \frac{1}{p}\mathrm{log}\bar{L}=K,\end{eqnarray*}$
where K is defined by ( $\begin{eqnarray}\begin{array}{l}\displaystyle \frac{1}{p}\epsilon {\partial }_{x}{b}_{k}={b}_{k+1}{{\rm{\Lambda }}}^{k+1}{{\rm{e}}}^{u}-{{\rm{e}}}^{u}{{\rm{\Lambda }}}^{-1}{b}_{k+1},\\ \,k\geqslant -1,\end{array}\end{eqnarray}$
with ${b}_{-1}={{\rm{e}}}^{u}$.The LFV hierarchy consists of the flows
$\begin{eqnarray}\begin{array}{l}\varepsilon \displaystyle \frac{\partial K}{\partial {t}_{1,n}}=[{\left({L}^{n}\right)}_{+},K],\\ \varepsilon \displaystyle \frac{\partial K}{\partial {t}_{2,n}}=-[{\left({\bar{L}}^{n}\right)}_{-},K],\,n\geqslant 1,\end{array}\end{eqnarray}$
where K is defined by (When n = 1, the equations of ${t}_{\mathrm{1,1}}$ and ${t}_{2.1}$ flows of the LFV hierarchy respectively are
$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial {t}_{\mathrm{1,1}}}=p\displaystyle \frac{(1-{{\rm{\Lambda }}}^{-1})({\rm{\Lambda }}-1)}{{\varepsilon }^{2}{\partial }_{x}}{{\rm{e}}}^{u},\end{eqnarray}$
$\begin{eqnarray}\displaystyle \frac{\partial u}{\partial {t}_{\mathrm{2,1}}}=\displaystyle \frac{1}{p}{u}_{x}.\end{eqnarray}$
$\begin{eqnarray}\varepsilon \displaystyle \frac{\partial L}{\partial {t}_{1,n}}=[{\left({L}^{n}\right)}_{+},L],\,\varepsilon \displaystyle \frac{\partial L}{\partial {t}_{2,n}}=[-{\left({\bar{L}}^{n}\right)}_{-},L];\end{eqnarray}$
$\begin{eqnarray}\varepsilon \displaystyle \frac{\partial \bar{L}}{\partial {t}_{1,n}}=[{\left({L}^{n}\right)}_{+},\bar{L}],\,\varepsilon \displaystyle \frac{\partial \bar{L}}{\partial {t}_{2,n}}=[-{\left({\bar{L}}^{n}\right)}_{-},\bar{L}].\end{eqnarray}$
Actually, the LFV hierarchy (
$\begin{eqnarray}\left\{\begin{array}{l}K\phi =\lambda \phi ,\\ \varepsilon \displaystyle \frac{\partial \phi }{\partial {t}_{1,n}}={\left({L}^{n}\right)}_{+}\phi ,\end{array}\right.\end{eqnarray}$
$\begin{eqnarray}\left\{\begin{array}{l}K\phi =\lambda \phi ,\\ \varepsilon \displaystyle \frac{\partial \phi }{\partial {t}_{2,n}}=-{\left({\bar{L}}^{n}\right)}_{-}\phi ,\end{array}\right.\end{eqnarray}$
where λ is a parameter, and $\phi =\phi (x,t)$ is the wave function of the LFV hierarchy (For the first equation of (
$\begin{eqnarray}{K}_{{t}_{1,n}}\phi +K{\phi }_{{t}_{1,n}}=\lambda {\phi }_{{t}_{1,n}}.\end{eqnarray}$
Substituting the second equation of ( $\begin{eqnarray}\begin{array}{l}{K}_{{t}_{1,n}}\phi +\displaystyle \frac{1}{\varepsilon }K{\left({L}^{n}\right)}_{+}\phi =\lambda {\phi }_{{t}_{1,n}}{\left({L}^{n}\right)}_{+}\phi \\ \,=\,\displaystyle \frac{1}{\varepsilon }{\left({L}^{n}\right)}_{+}K\phi ,\end{array}\end{eqnarray}$
thus $\begin{eqnarray}\begin{array}{l}\varepsilon {K}_{{t}_{1,n}}={\left({L}^{n}\right)}_{+}K-K{\left({L}^{n}\right)}_{+}\\ \,=\,[{\left({L}^{n}\right)}_{+},K].\end{array}\end{eqnarray}$
Using the similar calculation method, we can prove that the linear equations (2.2. Darboux transformation of the LFV hierarchy
In this section, we construct the n-fold Darboux transformation of the LFV hierarchy, and obtain the relationship between new and old solutions.
We consider the Darboux transformation of the LFV hierarchy on Lax operator K in (2.5 ), let2.9 ) invariant, we have 2.19 ), we have
$\begin{eqnarray}{K}^{[1]}=\displaystyle \frac{1}{p}\varepsilon {\partial }_{x}+{{\rm{e}}}^{{u}^{[1]}}{{\rm{\Lambda }}}^{-1}={{TKT}}^{-1},\end{eqnarray}$
where T is the Darboux transformation operator. In order to keep the Lax equations ( $\begin{eqnarray}\begin{array}{l}\varepsilon \displaystyle \frac{\partial {K}^{[1]}}{\partial {t}_{1,n}}=[{\left({\left({L}^{[1]}\right)}^{n}\right)}_{+},K],\\ \varepsilon \displaystyle \frac{\partial {K}^{[1]}}{\partial {t}_{2,n}}=-[{\left({\left({\bar{L}}^{[1]}\right)}^{n}\right)}_{-},K],\,n\geqslant 1,\end{array}\end{eqnarray}$
where L[1] = TLT−1 and ${\bar{L}}^{[1]}=T\bar{L}{T}^{-1}$.The Darboux transformation operator T should satisfy the following equations
$\begin{eqnarray}\varepsilon {T}_{{t}_{q,n}}=-T{\left({L}_{q}^{n}\right)}_{+}+{\left({{TL}}_{q}^{n}{T}^{-1}\right)}_{+}T,\,q=1,2,\,n\geqslant 1,\end{eqnarray}$
where ${L}_{1}=L$, ${L}_{2}=\bar{L}$.From (
$\begin{eqnarray*}\begin{array}{l}\varepsilon \displaystyle \frac{\partial {K}^{[1]}}{\partial {t}_{q,n}}=\varepsilon {T}_{{t}_{q,n}}{{KT}}^{-1}+\varepsilon {{TK}}_{{t}_{q,n}}{T}^{-1}\\ \,-\,\varepsilon {{TKT}}^{-1}{T}_{{t}_{q,n}}{T}^{-1}\\ \,=\,\varepsilon {T}_{{t}_{q,n}}{T}^{-1}{{TKT}}^{-1}+\varepsilon {{TK}}_{{t}_{q,n}}{T}^{-1}\\ \,-\,\varepsilon {{TKT}}^{-1}{T}_{{t}_{q,n}}{T}^{-1}\\ \,=\,\varepsilon [{T}_{{t}_{q,n}}{T}^{-1},{K}^{[1]}]+T[{\left({L}_{q}^{n}\right)}_{+},K]{T}^{-1}\\ \,=\,[\varepsilon {T}_{{t}_{q,n}}{T}^{-1},{K}^{[1]}]+[T{\left({L}_{q}^{n}\right)}_{+}{T}^{-1},{K}^{[1]}]\\ \,=\,[\varepsilon {T}_{{t}_{q,n}}{T}^{-1}+T{\left({L}_{q}^{n}\right)}_{+}{T}^{-1},{K}^{[1]}],\end{array}\end{eqnarray*}$
from ( $\begin{eqnarray}\varepsilon \displaystyle \frac{\partial {K}^{[1]}}{\partial {t}_{q,n}}=[{\left({{TL}}_{q}^{n}{T}^{-1}\right)}_{+},{K}^{[1]}],\end{eqnarray}$
so that $\begin{eqnarray}\varepsilon {T}_{{t}_{q,n}}{T}^{-1}+T{\left({L}_{q}^{n}\right)}_{+}{T}^{-1}={\left({{TL}}_{q}^{n}{T}^{-1}\right)}_{+},\end{eqnarray}$
thus $\begin{eqnarray}\varepsilon {T}_{{t}_{q,n}}=-T{\left({L}_{q}^{n}\right)}_{+}+{\left({{TL}}_{q}^{n}{T}^{-1}\right)}_{+}T.\end{eqnarray}$
□Let $A={\sum }_{n=0}^{\infty }{a}_{n}(x){{\rm{\Lambda }}}^{n}$ is a non-negative differential operator. For simplicity, we abbreviate f(x) to f. The following equalities hold
$\begin{eqnarray}\begin{array}{l}{\left({Af}\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g\right)}_{-}=A(f)\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g,\\ {\left(f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}{gA}\right)}_{-}=f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}{A}^{* }(g),\end{array}\end{eqnarray}$
where ${A}^{* }={\sum }_{n=0}^{\infty }{{\rm{\Lambda }}}^{-n}{a}_{n}$.We note that Af or $A\circ f$ means the operator A multiplies function f, but A(f) or $A\cdot f$ means that the operator A acts on function f(x), such as ${\rm{\Lambda }}f={\rm{\Lambda }}\circ f\,=f(x+\varepsilon ){\rm{\Lambda }}$, and ${\rm{\Lambda }}(f)=f(x+\varepsilon )$.
We proof (
$\begin{eqnarray}\begin{array}{l}{\left({Af}\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g\right)}_{-}=\displaystyle \sum _{n=0}^{\infty }{a}_{n}{\left(f(x+n\varepsilon ){{\rm{\Lambda }}}^{n}\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g\right)}_{-}\\ \,=\,\displaystyle \sum _{n=0}^{\infty }{a}_{n}f(x+n\varepsilon ){\left(\displaystyle \frac{{{\rm{\Lambda }}}^{n-1}}{1-{{\rm{\Lambda }}}^{-1}}\right)}_{-}g\\ \,=\,\displaystyle \sum _{n=0}^{\infty }{a}_{n}f(x+n\varepsilon )\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g\\ \,=\,A(f)\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left(f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}{gA}\right)}_{-}=\displaystyle \sum _{n=0}^{\infty }{\left(f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}{{ga}}_{n}{{\rm{\Lambda }}}^{n}\right)}_{-}\\ \,=\,\displaystyle \sum _{n=0}^{\infty }{\left(f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{n-1}}g(x-n\varepsilon ){a}_{n}(x-n\varepsilon )\right)}_{-}\\ \,=\,\displaystyle \sum _{n=0}^{\infty }{\left(f\displaystyle \frac{{{\rm{\Lambda }}}^{n-1}}{1-{{\rm{\Lambda }}}^{-1}}\right)}_{-}g(x-n\varepsilon ){a}_{n}(x-n\varepsilon )\\ \,=\,\displaystyle \sum _{n=0}^{\infty }f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}g(x-n\varepsilon ){a}_{n}(x-n\varepsilon )\\ \,=\,f\displaystyle \frac{{{\rm{\Lambda }}}^{-1}}{1-{{\rm{\Lambda }}}^{-1}}{A}^{* }(g).\end{array}\end{eqnarray}$
□There is an operator
$\begin{eqnarray}T(\phi )=1-\displaystyle \frac{\phi }{{{\rm{\Lambda }}}^{-1}\phi }{{\rm{\Lambda }}}^{-1}=\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1},\end{eqnarray}$
which can be as a Darboux transformaton operator of the LFV hierarchy, where φ is the wave function of the LFV hierarchy, which is defined by (From (
$\begin{eqnarray*}\begin{array}{l}\varepsilon {T}_{{t}_{q,n}}{T}^{-1}=\varepsilon {\left(\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1}\right)}_{{t}_{q,n}}\phi \\ \,\circ \,{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\\ \,=\,(({\left({L}_{q}^{n}\right)}_{+}\phi )\circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1})\phi \\ \,\circ \,{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\\ \,-\,\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1}({\left({L}_{q}^{n}\right)}_{+}\phi ){\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\\ \,=\,({\left({L}_{q}^{n}\right)}_{+}\phi ){\phi }^{-1}-\phi \circ (1-{{\rm{\Lambda }}}^{-1})\\ \,\circ \,{\phi }^{-1}({\left({L}_{q}^{n}\right)}_{+}\phi ){\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\\ \,=\,-{\left(\phi \circ [(1-{{\rm{\Lambda }}}^{-1})\cdot {\phi }^{-1}({\left({L}_{q}^{n}\right)}_{+}\phi )]{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\right)}_{-}\\ \,=\,-{\left(\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1}{\left({L}_{q}^{n}\right)}_{+}\circ \phi ){\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\right)}_{-}\\ \,=\,-\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1}{\left({L}_{q}^{n}\right)}_{+}\circ \phi ){\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\\ \,+\,{\left(\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1}{\left({L}_{q}^{n}\right)}_{+}\circ \phi ){\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{-1}\circ {\phi }^{-1}\right)}_{+}\\ \,=\,-T{\left({L}_{q}^{n}\right)}_{+}{T}^{-1}+{\left({{TL}}_{q}^{n}{T}^{-1}\right)}_{+}.\end{array}\end{eqnarray*}$
According to theorem $\begin{eqnarray*}T(\phi )=\phi \circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }^{-1},\end{eqnarray*}$
which can be as a Darboux transformation operator of the LFV hierarchy. □Further, we can obtain that the relation between the new solution u[1] and the old solution u is
$\begin{eqnarray}{u}^{[1]}={{\rm{\Lambda }}}^{-1}u+{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\mathrm{log}\phi .\end{eqnarray}$
If we define $\begin{eqnarray}{\phi }_{i}^{[j-1]}={\phi }^{[j-1]}:= {\phi }^{[j-1]}{| }_{\lambda ={\lambda }_{i}},\end{eqnarray}$
let j = 1, we have the one-fold Darboux transformation of of the LFV hierarchy $\begin{eqnarray}{T}_{1}({\phi }_{1})={\phi }_{1}\circ (1-{{\rm{\Lambda }}}^{-1})\circ {\phi }_{1}^{-1}=\displaystyle \frac{{C}_{1}}{{D}_{1}},\end{eqnarray}$
where $\begin{eqnarray}{C}_{1}=\left|\begin{array}{cc}1 & {{\rm{\Lambda }}}^{-1}\\ {\phi }_{1} & {\phi }_{1}(x-\varepsilon )\end{array}\right|,\end{eqnarray}$
and $\begin{eqnarray}{D}_{1}={\phi }_{1}(x-\varepsilon ).\end{eqnarray}$
Meanwhile, we can also obtain the expression of the wave function φ after Darboux transformation, which is $\begin{eqnarray}{\phi }^{[1]}=\left(1-\displaystyle \frac{{\phi }_{1}^{[0]}}{{{\rm{\Lambda }}}^{-1}{\phi }_{1}^{[0]}}{{\rm{\Lambda }}}^{-1}\right)\phi .\end{eqnarray}$
Using Darboux transformation alternately, we can obtain $\begin{eqnarray}{\phi }^{[j]}=\left(1-\displaystyle \frac{{\phi }_{j}^{[j-1]}}{{{\rm{\Lambda }}}^{-1}{\phi }_{j}^{[j-1]}}{{\rm{\Lambda }}}^{-1}\right){\phi }^{[j-1]},\end{eqnarray}$
$\begin{eqnarray}{u}^{[n]}={{\rm{\Lambda }}}^{-1}{u}^{[n-1]}+{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\mathrm{log}{\phi }_{j}^{[j-1]}.\end{eqnarray}$
After using Darboux transformations twice, the two-fold Darboux transformation will become $\begin{eqnarray}{T}_{2}({\phi }_{1},{\phi }_{2})=T({\phi }_{2}^{[1]})T({\phi }_{1})=\displaystyle \frac{{C}_{2}}{{D}_{2}},\end{eqnarray}$
with $\begin{eqnarray}{C}_{2}=\left|\begin{array}{ccc}1 & {{\rm{\Lambda }}}^{-1} & {{\rm{\Lambda }}}^{-2}\\ {\phi }_{1} & {\phi }_{1}(x-\varepsilon ) & {\phi }_{1}(x-2\varepsilon )\\ {\phi }_{2} & {\phi }_{2}(x-\varepsilon ) & {\phi }_{2}(x-2\varepsilon )\end{array}\right|,\end{eqnarray}$
and $\begin{eqnarray}{D}_{2}=\left|\begin{array}{cc}{\phi }_{1}(x-\varepsilon ) & {\phi }_{1}(x-2\varepsilon )\\ {\phi }_{2}(x-\varepsilon ) & {\phi }_{2}(x-2\varepsilon )\end{array}\right|.\end{eqnarray}$
Similarly, we can obtain the n-fold Darboux transformation, which is $\begin{eqnarray}\begin{array}{l}{T}_{n}({\phi }_{1},\cdots ,{\phi }_{n})=T({\phi }_{n}^{[n-1]})\\ \circ \cdots \circ T({\phi }_{2}^{[1]})T({\phi }_{1})=\displaystyle \frac{{C}_{n}}{{D}_{n}},\end{array}\end{eqnarray}$
with $\begin{eqnarray}{C}_{n}=\left|\begin{array}{ccccc}1 & {{\rm{\Lambda }}}^{-1} & {{\rm{\Lambda }}}^{-2} & \cdots & {{\rm{\Lambda }}}^{-n}\\ {\phi }_{1} & {\phi }_{1}(x-\varepsilon ) & {\phi }_{1}(x-2\varepsilon ) & \cdots & {\phi }_{1}(x-n\varepsilon )\\ {\phi }_{2} & {\phi }_{2}(x-\varepsilon ) & {\phi }_{2}(x-2\varepsilon ) & \cdots & {\phi }_{2}(x-n\varepsilon )\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\phi }_{n} & {\phi }_{n}(x-\varepsilon ) & {\phi }_{n}(x-2\varepsilon ) & \cdots & {\phi }_{n}(x-n\varepsilon )\end{array}\right|,\end{eqnarray}$
and $\begin{eqnarray}{D}_{n}=\left|\begin{array}{ccccc}{\phi }_{1} & {\phi }_{1}(x-\varepsilon ) & {\phi }_{1}(x-2\varepsilon ) & \cdots & {\phi }_{1}(x-n\varepsilon )\\ {\phi }_{2} & {\phi }_{2}(x-\varepsilon ) & {\phi }_{2}(x-2\varepsilon ) & \cdots & {\phi }_{2}(x-n\varepsilon )\\ \vdots & \vdots & \vdots & \ddots & \vdots \\ {\phi }_{n} & {\phi }_{n}(x-\varepsilon ) & {\phi }_{n}(x-2\varepsilon ) & \cdots & {\phi }_{n}(x-n\varepsilon )\end{array}\right|.\end{eqnarray}$
We can obtain the n-soliton solutions from the seed solution u $\begin{eqnarray}{u}^{[n]}={{\rm{\Lambda }}}^{-n}u+{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\mathrm{log}{Wr}({\phi }_{1},\cdots ,{\phi }_{n}),\end{eqnarray}$
where Wr(φ1, ⋯ ,φn) is the discrete Wronskian $\begin{eqnarray}{Wr}({\phi }_{1},\cdots ,{\phi }_{n})=\det {\left({{\rm{\Lambda }}}^{-j+1}{\phi }_{n+1-i}\right)}_{1\leqslant i,j\leqslant n}.\end{eqnarray}$
2.3. Soliton solutions of the LFV hierarchy
In this section, we give the soliton solutions of the LFV hierarchy. 2.44 )2.10 ) as follows (figure 1)
For the equation (
$\begin{eqnarray}\left\{\begin{array}{l}\left(\displaystyle \frac{1}{p}\epsilon {\partial }_{x}+{{\rm{\Lambda }}}^{-1}\right)(\phi )=\lambda \phi ;\\ \varepsilon \displaystyle \frac{\partial \phi }{\partial {t}_{\mathrm{1,1}}}={\rm{\Lambda }}(\phi ).\end{array}\right.\end{eqnarray}$
We can obtain a solution of equations ( $\begin{eqnarray}\phi (\lambda )={{\rm{e}}}^{\left[\tfrac{\lambda p}{\varepsilon }+\frac{1}{\varepsilon }W(-p{{\rm{e}}}^{-\lambda p})\right]x+\left[\frac{1}{\varepsilon }{{\rm{e}}}^{\lambda p+W(-p{{\rm{e}}}^{-\lambda p})}\right]{t}_{\mathrm{1,1}}},\end{eqnarray}$
where W(x) is the Lambert function. For $y=W(x)$, it satisfies $y{{\rm{e}}}^{y}=x$. We choose $\begin{eqnarray}{\phi }_{1}=\phi ({\lambda }_{1})={{\rm{e}}}^{\left[\tfrac{{\lambda }_{1}p}{\varepsilon }+\frac{1}{\varepsilon }W(-p{{\rm{e}}}^{-{\lambda }_{1}p})\right]x+\left[\frac{1}{\varepsilon }{{\rm{e}}}^{{\lambda }_{1}p+W(-p{{\rm{e}}}^{-{\lambda }_{1}p})}\right]{t}_{\mathrm{1,1}}},\end{eqnarray}$
so that, we obtain the one soliton solution of the equation ( $\begin{eqnarray}\begin{array}{c}{u}^{[1]}={\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\left(\left[\frac{{\lambda }_{1}p}{\varepsilon }+\frac{1}{\varepsilon }W(-p{{\rm{e}}}^{-{\lambda }_{1}p})\right]x\right.\\ \left.+\left[\frac{1}{\varepsilon }{{\rm{e}}}^{{\lambda }_{1}p+W(-p{{\rm{e}}}^{-{\lambda }_{1}p})}\right]{t}_{\mathrm{1,1}}\right).\end{array}\end{eqnarray}$
Furthermore, we can obtain the two soliton solution of t1,1 flows by using the Darboux transformation again from the one soliton solution u[1] (figure 2)2.10 )
$\begin{eqnarray}{\phi }^{[2]}=\left(1-\displaystyle \frac{{\phi }_{2}^{[1]}}{{{\rm{\Lambda }}}^{-1}{\phi }_{2}^{[1]}}{{\rm{\Lambda }}}^{-1}\right){\phi }^{[1]},\end{eqnarray}$
$\begin{eqnarray}{u}^{[2]}={{\rm{\Lambda }}}^{-1}{u}^{[1]}+{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\mathrm{log}{\phi }_{2}^{[1]},\end{eqnarray}$
where φ2 is as follows $\begin{eqnarray}{\phi }_{2}={{\rm{e}}}^{\left[\tfrac{{\lambda }_{2}p}{\varepsilon }+\frac{1}{\varepsilon }W(-p{{\rm{e}}}^{-{\lambda }_{2}p})\right]x+\left[\frac{1}{\varepsilon }{{\rm{e}}}^{{\lambda }_{2}p+W(-p{{\rm{e}}}^{-{\lambda }_{2}p})}\right]{t}_{\mathrm{1,1}}}.\end{eqnarray}$
We can obtain the two soliton solution of the equation ( $\begin{eqnarray}{u}^{[2]}={\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}\mathrm{log}{Wr}({\phi }_{1},{\phi }_{2}),\end{eqnarray}$
where ${Wr}({\phi }_{1},{\phi }_{2})=\left|\begin{array}{cc}{\phi }_{1} & {\phi }_{2}\\ {\phi }_{1}(x-\varepsilon ) & {\phi }_{2}(x-\varepsilon )\end{array}\right|$.3. The NCLFV hierarchy
3.1. The definition of the NCLFV hierarchy
Introducing two operators3.1 ) and (3.2 ) satisfy the following Lax equations
$\begin{eqnarray}{{\mathbb{L}}}_{1}={\rm{\Lambda }}+{u}_{0}+{u}_{1}\star {{\rm{\Lambda }}}^{-1}+\cdots ,\end{eqnarray}$
$\begin{eqnarray}{{\mathbb{L}}}_{2}={v}_{-1}\star {{\rm{\Lambda }}}^{-1}+{v}_{0}+{v}_{1}\star {\rm{\Lambda }}+\cdots ,\end{eqnarray}$
we define the two operators $\begin{eqnarray}{{\mathbb{L}}}_{i}^{n}=\mathop{\underbrace{{{\mathbb{L}}}_{i}\star {{\mathbb{L}}}_{i}\star \cdots \star {{\mathbb{L}}}_{i}}}\limits_{n\,{times}},\,i=1,2.\end{eqnarray}$
The Lax operators ( $\begin{eqnarray}\begin{array}{l}\varepsilon \displaystyle \frac{\partial {{\mathbb{L}}}_{1}}{\partial {t}_{2,n}}=-{\left[{\left({{\mathbb{L}}}_{2}^{n}\right)}_{-},{{\mathbb{L}}}_{1}\right]}_{\star },\\ \varepsilon \displaystyle \frac{\partial {{\mathbb{L}}}_{2}}{\partial {t}_{1,n}}={\left[{\left({{\mathbb{L}}}_{1}^{n}\right)}_{+},{{\mathbb{L}}}_{2}\right]}_{\star },\,n\geqslant 1,\end{array}\end{eqnarray}$
and $\begin{eqnarray}\varepsilon \displaystyle \frac{\partial {{\mathbb{L}}}_{r}}{\partial {t}_{r,n}}={\left[{\left({{\mathbb{L}}}_{r}^{n}\right)}_{+},{{\mathbb{L}}}_{r}\right]}_{\star },\,r=1,\,2,\,n\geqslant 1,\end{eqnarray}$
where [A, B]⋆ = A⋆B − B⋆A. The star-product [25] is defined by$\begin{eqnarray*}\begin{array}{l}f(x)\star g(x):=\exp \left(\displaystyle \frac{i}{2}{\theta }^{{ij}}{\partial }_{i}^{(x^{\prime} )}{\partial }_{j}^{(x^{\prime\prime} )}\right)f(x^{\prime} )g(x^{\prime\prime} ){| }_{x^{\prime} =x^{\prime\prime} =x}\\ \,=\,f(x)g(x)+\displaystyle \frac{i}{2}{\theta }^{{ij}}{\partial }_{i}f(x){\partial }_{j}g(x)+{ \mathcal O }({\theta }^{2}),\end{array}\end{eqnarray*}$
with ${\partial }_{i}^{(x^{\prime} )}=\partial /\partial {x}^{i^{\prime} }$ and θij is called the noncommutative parameter.Consider the Lax operator of the NCLFV hierarchy
$\begin{eqnarray}{\mathbb{K}}=\displaystyle \frac{1}{q}\varepsilon {\partial }_{x}+{{\rm{e}}}_{\star }^{h}\star {{\rm{\Lambda }}}^{-1},\end{eqnarray}$
where ${{\rm{e}}}_{\star }^{h}=1+{\sum }_{n=1}^{\infty }\tfrac{1}{n!}\underset{n\,{times}}{\overset{h\star h\star \cdots \star h}{\unicode{x0FE38}}}$, and q is a parameter.We define the Lax equation of the NCLFV hierarchy3.7 ) can be also rewritten as the linear differential equations
$\begin{eqnarray}\varepsilon \displaystyle \frac{\partial {\mathbb{K}}}{\partial {t}_{r,n}}=-{\left[{\left({{\mathbb{L}}}_{r}^{n}\right)}_{-},{\mathbb{K}}\right]}_{\star },\,r=1,\,2,\,n\geqslant 1,\end{eqnarray}$
the NCLFV hierarchy ( $\begin{eqnarray}\left\{\begin{array}{l}{\mathbb{K}}\psi =\kappa \psi ,\\ \varepsilon \displaystyle \frac{\partial \psi }{\partial {t}_{r,n}}=-{\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi ,\end{array}\right.\end{eqnarray}$
where ψ is the wave function of the NCLFV hierarchy, and r = 1, 2.3.2. Darboux transformation of the NCLFV hierarchy
We consider the Darboux transformation of the NCLFV hierarchy on Lax operator (3.6 )3.9 ) should satisfy the following theorem: 3.11 ) and (3.12 ), we have
$\begin{eqnarray}{{\mathbb{K}}}^{[1]}=\displaystyle \frac{1}{q}\varepsilon {\partial }_{x}+{{\rm{e}}}_{\star }^{{h}^{[1]}}\star {{\rm{\Lambda }}}^{-1}=W\star {\mathbb{K}}\star {W}^{-1},\end{eqnarray}$
where W is the Darboux transformation operator. The spectral problem $\begin{eqnarray}{\mathbb{K}}\psi =\displaystyle \frac{1}{q}\varepsilon {\partial }_{x}\psi +{{\rm{e}}}_{\star }^{h}\star {{\rm{\Lambda }}}^{-1}\psi =\kappa \psi ,\end{eqnarray}$
will become $\begin{eqnarray}{{\mathbb{K}}}^{[1]}{\psi }^{[1]}=\displaystyle \frac{1}{q}\varepsilon {\partial }_{x}{\psi }^{[1]}+{{\rm{e}}}_{\star }^{{h}^{[1]}}\star {{\rm{\Lambda }}}^{-1}{\psi }^{[1]}=\kappa {\psi }^{[1]}.\end{eqnarray}$
In order to keep the Lax equations invariant $\begin{eqnarray}\varepsilon \displaystyle \frac{\partial {\psi }^{[1]}}{\partial {t}_{r,n}}=-{\left({\left({{\mathbb{L}}}_{r}^{[1]}\right)}^{n}\right)}_{-}\cdot {\psi }^{[1]},\end{eqnarray}$
where ${{\mathbb{L}}}_{r}^{[1]}=W{{\mathbb{L}}}_{r}{W}^{-1}$, the Darboux transformation operator W in (The Darboux transformation operator W should satisfy the following equations
$\begin{eqnarray}\varepsilon {W}_{{t}_{r,n}}\star {W}^{-1}=W\star {\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\star {W}^{-1}-{\left(W\star {{\mathbb{L}}}_{r}^{n}\star {W}^{-1}\right)}_{-},\end{eqnarray}$
where ${{\mathbb{L}}}_{1}={\mathbb{L}}$, ${{\mathbb{L}}}_{2}=\bar{{\mathbb{L}}}$.From (
$\begin{eqnarray*}\begin{array}{l}\varepsilon \displaystyle \frac{\partial {{\mathbb{K}}}^{[1]}}{\partial {t}_{r,n}}=\varepsilon {W}_{{t}_{r,n}}\star {\mathbb{K}}\star {W}^{-1}+\varepsilon W\star {{\mathbb{K}}}_{{t}_{r,n}}\star {W}^{-1}\\ \,-\,\varepsilon W\star {\mathbb{K}}\star {W}^{-1}\star {W}_{{t}_{r,n}}\star {W}^{-1}\\ \,=\,\varepsilon {W}_{{t}_{r,n}}\star {W}^{-1}\star W\star {\mathbb{K}}\star {W}^{-1}\\ \,+\,\varepsilon W\star {{\mathbb{K}}}_{{t}_{r,n}}\star {W}^{-1}-\varepsilon {{\mathbb{K}}}^{[1]}\star {W}_{{t}_{r,n}}\star {W}^{-1}\\ \,=\,\varepsilon {\left[{W}_{{t}_{r,n}}\star {W}^{-1},{{\mathbb{K}}}^{[1]}\right]}_{\star }-\varepsilon W\star {\left[{\left({{\mathbb{L}}}_{r}^{n}\right)}_{-},{\mathbb{K}}\right]}_{\star }\star {W}^{-1}\\ \,=\,{\left[\varepsilon {W}_{{t}_{r,n}}\star {W}^{-1},{{\mathbb{K}}}^{[1]}\right]}_{\star }-{\left[W\star {\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\star {W}^{-1},{{\mathbb{K}}}^{[1]}\right]}_{\star }\\ \,=\,{\left[\varepsilon {W}_{{t}_{r,n}}\star {W}^{-1}-W\star {\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\star {W}^{-1},{{\mathbb{K}}}^{[1]}\right]}_{\star },\end{array}\end{eqnarray*}$
from ( $\begin{eqnarray}\varepsilon \displaystyle \frac{\partial {{\mathbb{K}}}^{[1]}}{\partial {t}_{r,n}}=-{\left[{\left(W\star {{\mathbb{L}}}_{r}^{n}\star {W}^{-1}\right)}_{-},{{\mathbb{K}}}^{[1]}\right]}_{\star },\end{eqnarray}$
thus $\begin{eqnarray}\begin{array}{l}\varepsilon {W}_{{t}_{r,n}}\star {W}^{-1}-W\star {\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\star {W}^{-1}\\ \,=\,-{\left(W\star {{\mathbb{L}}}_{r}^{n}\star {W}^{-1}\right)}_{-}.\end{array}\end{eqnarray}$
□We firstly give the necessary lemma for giving the Darboux transformation of the NCLFV hierarchy.
Let $B={\sum }_{n=1}^{\infty }{b}_{n}(x)\star {{\rm{\Lambda }}}^{-n}$ is a negative difference operator, then the following equalities hold
$\begin{eqnarray}\begin{array}{l}{\left(B\star f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\right)}_{+}\\ \,=\,B(f)\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left(f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\star B\right)}_{+}\\ =\,f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {B}^{* }(g),\end{array}\end{eqnarray}$
where ${B}^{* }={\sum }_{n=1}^{\infty }{{\rm{\Lambda }}}^{n}\star {b}_{n}(x)$, and ${B}^{* }(g)={\sum }_{n\,=\,1}^{\infty }g(x\,+n\varepsilon )\star {b}_{n}(x+n\varepsilon )$.Similar to the proof of (
$\begin{eqnarray}\begin{array}{l}{\left(B\star f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\right)}_{+}\\ =\,\displaystyle \sum _{n=0}^{\infty }{b}_{n}\star {\left(f(x-n\varepsilon )\star {{\rm{\Lambda }}}^{-n}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\right)}_{+}\\ =\,\displaystyle \sum _{n=0}^{\infty }{b}_{n}\star f(x-n\varepsilon )\star {\left({{\rm{\Lambda }}}^{-n}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\right)}_{+}\star g\\ =\,\displaystyle \sum _{n=0}^{\infty }{b}_{n}\star f(x-n\varepsilon )\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\\ =\,B(f)\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g,\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}{\left(f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\star B\right)}_{+}\\ =\,\displaystyle \sum _{n=0}^{\infty }{\left(f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g\star {b}_{n}\star {{\rm{\Lambda }}}^{-n}\right)}_{+}\\ =\,\displaystyle \sum _{n=0}^{\infty }{\left(f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {{\rm{\Lambda }}}^{-n}\star g(x+n\varepsilon )\star {b}_{n}(x+n\varepsilon )\right)}_{+}\\ =\,\displaystyle \sum _{n=0}^{\infty }{\left(f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {{\rm{\Lambda }}}^{-n}\right)}_{+}\star g(x+n\varepsilon )\star {b}_{n}(x+n\varepsilon )\\ =\,\displaystyle \sum _{n=0}^{\infty }f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star g(x+n\varepsilon )\star {b}_{n}(x+n\varepsilon )\\ =\,f\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {B}^{* }(g).\end{array}\end{eqnarray}$
□The operator
$\begin{eqnarray}\begin{array}{l}W(\psi )=\psi (x+\varepsilon )\star {\psi }^{-1}(x)-{\rm{\Lambda }}\\ \,\,=\,\psi (x+\varepsilon )\star (1-{\rm{\Lambda }})\star {\psi }^{-1}(x),\end{array}\end{eqnarray}$
can be as a Darboux transformation operator of the NCLFV hierarchy.By direct computation using the lemma
$\begin{eqnarray*}\begin{array}{l}\varepsilon {W}_{{t}_{r,n}}^{-1}\star W=\varepsilon {\left(\psi (x+\varepsilon )\star (1-{\rm{\Lambda }})\star {\psi }^{-1}\right)}_{{t}_{r,n}}\\ \,\star \,\psi \star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,=\,-(({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi (x+\varepsilon ))\star (1-{\rm{\Lambda }})\star {\psi }^{-1})\\ \,\star \,\psi \star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,+\,\psi (x+\varepsilon )\star (1-{\rm{\Lambda }})\star {\psi }^{-1}\star \left({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi \right)\\ \,\star \,{\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,=\,-({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi (x+\varepsilon ))\star {\psi }^{-1}(x+\varepsilon )\\ \,+\,\psi (x+\varepsilon )\star (1-{\rm{\Lambda }})\star {\psi }^{-1}\star ({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi )\\ \,\star \,{\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,=\,-\psi (x+\varepsilon )\star \left[({\rm{\Lambda }}-1)\cdot \left({\psi }^{-1}\star \left({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi \right)\right)\right]\\ \,\star \,{\rm{\Lambda }}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,=\,\left(\psi (x+\varepsilon )\star \left[({\rm{\Lambda }}-1)\cdot \left({\psi }^{-1}\star \left({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi \right)\right)\right]\right.\\ {\left.\,\star \,{\rm{\Lambda }}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\right)}_{+}\\ \,=\,\psi (x+\varepsilon )\star \left[({\rm{\Lambda }}-1)\star \left({\psi }^{-1}\star \left({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi \right)\right)\right]\\ \,\star \,{\rm{\Lambda }}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\\ \,-\,\left(\psi (x+\varepsilon )\star \left[({\rm{\Lambda }}-1)\star \left({\psi }^{-1}\star \left({\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\psi \right)\right)\right]\right.\\ {\left.\,\star \,{\rm{\Lambda }}\star {\left(1-{\rm{\Lambda }}\right)}^{-1}\star {\psi }^{-1}(x+\varepsilon )\right)}_{-}\\ \,=\,W\star {\left({{\mathbb{L}}}_{r}^{n}\right)}_{-}\star {W}^{-1}-{\left(W\star ({{\mathbb{L}}}_{r}^{n})\star {W}^{-1}\right)}_{-}.\end{array}\end{eqnarray*}$
Therefore, the operator (
We can obtain the one-fold Darboux transformation of the NCLFV hierarchy as follows
$\begin{eqnarray}\begin{array}{l}{W}_{1}({\psi }_{1})={\psi }_{1}(x+\varepsilon )\star {\psi }_{1}^{-1}(x)-{\rm{\Lambda }}\\ \,=\,-\left|\begin{array}{cc}{\psi }_{1} & 1\\ {\psi }_{1}(x+\varepsilon ) & \boxed{{\rm{\Lambda }}}\end{array}\right|.\end{array}\end{eqnarray}$
Here for a 2 × 2 matrix $\begin{eqnarray}\begin{array}{l}A=\left(\begin{array}{cc}{a}_{11} & {a}_{12}\\ {a}_{21} & {a}_{22}\end{array}\right),\end{array}\end{eqnarray}$
there are four quasi determinants as follows $\begin{eqnarray}\begin{array}{l}| A{| }_{11}=\left|\begin{array}{cc}\boxed{{a}_{11}} & {a}_{12}\\ {a}_{21} & {a}_{22}\end{array}\right|\,=\,{a}_{11}-{a}_{12}\star {a}_{22}^{-1}\star {a}_{21},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}| A{| }_{12}=\left|\begin{array}{cc}{a}_{11} & \boxed{{a}_{12}}\\ {a}_{21} & {a}_{22}\end{array}\right|\,=\,{a}_{12}-{a}_{11}\star {a}_{21}^{-1}\star {a}_{22},\end{array}\end{eqnarray}$
$\begin{eqnarray}\begin{array}{l}| A{| }_{21}=\left|\begin{array}{cc}{a}_{11} & {a}_{12}\\ \boxed{{a}_{21}} & {a}_{22}\end{array}\right|\,=\,{a}_{21}-{a}_{22}\star {a}_{12}^{-1}\star {a}_{11},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{l}| A{| }_{22}=\left|\begin{array}{cc}{a}_{11} & {a}_{12}\\ {a}_{21} & \boxed{{a}_{22}}\end{array}\right|\,=\,{a}_{22}-{a}_{21}\star {a}_{11}^{-1}\star {a}_{12}.\end{array}\end{eqnarray}$
For a 3 × 3 matrix $\begin{eqnarray}\begin{array}{l}A=\left(\begin{array}{ccc}{a}_{11} & {a}_{12} & {a}_{13}\\ {a}_{21} & {a}_{22} & {a}_{23}\\ {a}_{31} & {a}_{32} & {a}_{33}\end{array}\right),\end{array}\end{eqnarray}$
there are nine quasi determinants as follows, such as $\begin{eqnarray}\begin{array}{l}| A{| }_{11}=\left|\begin{array}{ccc}\boxed{{a}_{11}} & {a}_{12} & {a}_{13}\\ {a}_{21} & {a}_{22} & {a}_{23}\\ {a}_{31} & {a}_{32} & {a}_{33}\end{array}\right|\\ \,=\,{a}_{11}-({a}_{12},{a}_{13})\star {\left|\begin{array}{cc}{a}_{22} & {a}_{23}\\ {a}_{32} & {a}_{33}\end{array}\right|}^{-1}\star \left(\begin{array}{c}{a}_{21}\\ {a}_{31}\end{array}\right)\\ \,=\,{a}_{11}-{a}_{12}\star \left|\begin{array}{cc}\boxed{{a}_{22}} & {a}_{23}\\ {a}_{32} & {a}_{33}\end{array}\right|\star {a}_{21}-{a}_{12}\star \left|\begin{array}{cc}{a}_{22} & {a}_{23}\\ \boxed{{a}_{32}} & {a}_{33}\end{array}\right|\star {a}_{31}\\ \,-\,{a}_{13}\star \left|\begin{array}{cc}{a}_{22} & \boxed{{a}_{23}}\\ {a}_{32} & {a}_{33}\end{array}\right|\star {a}_{21}-{a}_{13}\star \left|\begin{array}{cc}{a}_{22} & {a}_{23}\\ {a}_{32} & \boxed{{a}_{33}}\end{array}\right|\star {a}_{31}.\end{array}\end{eqnarray}$
After using once the one-fold Darboux transformation (3.21 ), the wave function will become 3.32 ) leads to new solution from seed solution h Using mathematical induction, we can obtain the n-fold Darboux transformation operator of the NCLFV hierarchy. 3.34 ) leads to new solution ${h}^{[n]}$ from the seed solution h
$\begin{eqnarray}{\psi }^{[1]}=[{\psi }_{1}(x+\varepsilon )\star {\psi }_{1}^{-1}(x)-{\rm{\Lambda }}]\psi .\end{eqnarray}$
Using an iteration on Darboux transformation, we have $\begin{eqnarray}{\psi }^{[j]}=[{\psi }_{j}^{[j-1]}(x+\varepsilon )\star {\left({\psi }_{j}^{[j-1]}\right)}^{-1}(x)-{\rm{\Lambda }}]{\psi }^{[j-1]},\end{eqnarray}$
$\begin{eqnarray}{h}^{[j]}={h}^{[j-1]}+{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}{\mathrm{log}}_{\star }\left({\left({\psi }_{j}^{[j-1]}\right)}^{-1}\right),\end{eqnarray}$
where ${\psi }_{i}^{[j]}:= {\psi }^{[j]}{| }_{\kappa ={\kappa }_{i}}$ is the wave function corresponding to different spectral with the jth solution h[j], and ${\mathrm{log}}_{\star }(f)\,={\sum }_{n\,=\,1}^{\infty }\tfrac{2}{2n-1}\underset{2n-1\,{times}}{\overset{[(f-1)\star {\left(f+1\right)}^{-1}]\star \cdots \star [(f-1)\star {\left(f+1\right)}^{-1}}{\unicode{x0FE38}}}]$.The two-fold Darboux transformation of the NCLFV hierarchy is
$\begin{eqnarray}\begin{array}{l}{W}_{2}({\psi }_{1},{\psi }_{2})=-\left|\begin{array}{ccc}{\psi }_{1} & {\psi }_{2} & 1\\ {\psi }_{1}(x+\varepsilon ) & {\psi }_{2}(x+\varepsilon ) & {\rm{\Lambda }}\\ {\psi }_{1}(x+2\varepsilon ) & {\psi }_{2}(x+2\varepsilon ) & \boxed{{{\rm{\Lambda }}}^{2}}\end{array}\right|.\end{array}\end{eqnarray}$
The Darboux transformation ( $\begin{eqnarray}\begin{array}{l}{h}^{[2]}=h-{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}{\mathrm{log}}_{\star }\left({\psi }_{1}\star \left|\begin{array}{cc}{\psi }_{1} & {\psi }_{2}\\ {\psi }_{1}(x+\varepsilon ) & \boxed{{\psi }_{2}(x+\varepsilon )}\end{array}\right|\right).\end{array}\end{eqnarray}$
The n-fold Darboux transformation operator of the NCLFV hierarchy is
$\begin{eqnarray}{W}_{n}=-\left|\begin{array}{ccccc}{\phi }_{1} & {\phi }_{2} & \cdots & {\phi }_{n} & 1\\ {\phi }_{1}(x+\varepsilon ) & {\phi }_{2}(x+\varepsilon ) & \cdots & {\phi }_{n}(x+\varepsilon ) & {\rm{\Lambda }}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\phi }_{1}(x+(n-1)\varepsilon ) & {\phi }_{2}(x+(n-1)\varepsilon ) & \cdots & {\phi }_{n}(x+(n-1)\varepsilon ) & \boxed{{{\rm{\Lambda }}}^{n-1}}\end{array}\right|.\end{eqnarray}$
The Darboux transformation ( $\begin{eqnarray}{h}^{[n]}=h-{\left(1-{{\rm{\Lambda }}}^{-1}\right)}^{2}{\mathrm{log}}_{\star }({Z}_{n}({\psi }_{1},{\psi }_{2},\cdots ,{\psi }_{n})),\end{eqnarray}$
where $\begin{eqnarray}{Z}_{n}={{\rm{\Pi }}}_{i=1}^{n}{\left(-1\right)}^{n}{Z}_{{n}_{i}}={Z}_{{n}_{1}}\star {Z}_{{n}_{2}}\star \cdots \star {Z}_{{n}_{n}},\end{eqnarray}$
with$\begin{eqnarray*}\begin{array}{l}{Z}_{{n}_{i}}({\psi }_{1},\cdots ,{\psi }_{i})=\left|\begin{array}{cccc}{\phi }_{1} & {\phi }_{2} & \cdots & {\phi }_{i}\\ {\phi }_{1}(x+\varepsilon ) & {\phi }_{2}(x+\varepsilon ) & \cdots & {\phi }_{i}(x+\varepsilon )\\ \vdots & \vdots & \ddots & \vdots \\ {\phi }_{1}(x+(i-1)\varepsilon ) & {\phi }_{2}(x+(i-1)\varepsilon ) & \cdots & \boxed{{\phi }_{i}(x+(i-1)\varepsilon )}\end{array}\right|.\end{array}\end{eqnarray*}$
The star-product will reduce to the ordinary product in the commutative limit ${\theta }^{{ij}}\to 0$. For an n × n matrix A, the quasi determinant $| A{| }_{i,j}$ will become
$\begin{eqnarray}| A{| }_{i,j}\to {\left(-1\right)}^{i+j}\displaystyle \frac{\det (A)}{\det ({\tilde{A}}^{i,j})},\,1\leqslant i,j\leqslant n,\end{eqnarray}$
with ${\tilde{A}}^{i,j}$ indicates that the i row and the j column are removed from the matrix A.3.3. Quasi determinant solution of the NCLFV hierarchy
In this section, we construct the exact solution of the NCLFV hierarchy in terms of the quasi determinant.
Let ${ \mathcal A }$ is a noncommutative associative algebra, there is a shift operator Λ on ${ \mathcal A }$ that satisfies the Leibnitz rule. Suppose f1, ⋯ ,fn is the element in ${ \mathcal A }$, we define the Wronski matrix W(f1, ⋯ ,fn) as follows The wave function of NCLFV hierarchy 3.48 ) is a shift operator of order greater than or equal to n, and the order of right hand is less than n. We have ${{\rm{\Psi }}}_{n}\star {\psi }_{k}=0$ obviously, thus3.5 , we have3.50 ) into the NCLFV hierarchy (3.7 ), we have3.44 ) and (3.45 ) are the solutions of the NCLFV hierarchy. □
$\begin{eqnarray}({f}_{1},\cdots ,{f}_{n})=\left(\begin{array}{ccc}{f}_{1} & \cdots & {f}_{n}\\ {\rm{\Lambda }}{f}_{1} & \cdots & {\rm{\Lambda }}{f}_{n}\\ \vdots & \ddots & \vdots \\ {{\rm{\Lambda }}}^{n-1}{f}_{1} & \cdots & {{\rm{\Lambda }}}^{n-1}{f}_{n}\end{array}\right),\end{eqnarray}$
where fi = fi(x). If the Wronski matrix W(f1, ⋯ ,fn) is an invertible matrix, then f1, ⋯ ,fn are said to be nondegenerate. We have the following lemma for the Wronski matrix.If ${f}_{1},\cdots ,{f}_{n}\in { \mathcal A }$ are sets of nondegenerate functions, then any of its subset ${f}_{{i}_{1}},\cdots ,{f}_{{i}_{m}},(m\leqslant n)$ are nondegenerate.
Wronski matrix $W({f}_{1},\cdots ,{f}_{n})$ is invertible if and only if $| W({f}_{1},\cdots ,{f}_{n}){| }_{n,n}\ne 0$.
Let ${f}_{1},\cdots ,{f}_{n}$ be the nondegenerate elements in set ${ \mathcal A }$, then there exists a unique shift operator ψ with n order satisfying ${\rm{\Psi }}{f}_{i}=0,(i=1,\cdots ,n)$, whose coefficients are the element in ${ \mathcal A }$ with first item is 1, and the shift operator ψ acts on any $f\in { \mathcal A }$ can be expressed as
$\begin{eqnarray}{\rm{\Psi }}f=| W({f}_{1},\cdots ,{f}_{n},f){| }_{n+1,n+1}.\end{eqnarray}$
$\begin{eqnarray}\psi (x,t,z)={\exp }_{\star }\left(\displaystyle \frac{1}{\varepsilon }{\mathrm{log}}_{\star }{wx}-\displaystyle \frac{1}{\varepsilon }{w}^{-1}{t}_{r,n}\right).\end{eqnarray}$
Suppose w1 and w2 are two real roots of the equation $\begin{eqnarray}{w}_{i}+{w}_{i}^{-1}={\kappa }_{i},\end{eqnarray}$
with κi(i = 1, ⋯ ,n) is the real number. We choose $\begin{eqnarray}{\psi }_{i}={a}_{i}\psi (x,t,{w}_{1}({\kappa }_{i}))+{b}_{i}\psi (x,t,{w}_{2}({\kappa }_{i})),\,(i=1,\cdots ,n),\end{eqnarray}$
where ai and bi are arbitrary real numbers.Let
$\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{n}=| W({\psi }_{1},\cdots ,{\psi }_{n},{\rm{\Lambda }}){| }_{n+1,n+1}\\ \,=\,\left|\begin{array}{ccccc}{\psi }_{1} & {\psi }_{2} & \cdots & {\psi }_{n} & 1\\ {\psi }_{1}(x+\varepsilon ) & {\psi }_{2}(x+\varepsilon ) & \cdots & {\psi }_{n}(x+\varepsilon ) & {\rm{\Lambda }}\\ \vdots & \vdots & \ddots & \vdots & \vdots \\ {\psi }_{1}(x+(n-1)\varepsilon ) & {\psi }_{2}(x+(n-1)\varepsilon ) & \cdots & {\psi }_{n}(x+(n-1)\varepsilon ) & {{\rm{\Lambda }}}^{n-1}\\ {\psi }_{1}(x+n\varepsilon ) & {\psi }_{2}(x+n\varepsilon ) & \cdots & {\psi }_{n}(x+n\varepsilon ) & \boxed{{{\rm{\Lambda }}}^{n}}\end{array}\right|,\end{array}\end{eqnarray}$
the quasi determinant solutions of the NCLFV hierarchy are $\begin{eqnarray}{\mathbb{K}}={{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}-\displaystyle \frac{1}{q}(\varepsilon {\partial }_{x}+\varepsilon {{\rm{\Psi }}}_{n,x}{{\rm{\Psi }}}_{n}^{-1}),\end{eqnarray}$
$\begin{eqnarray}{\mathbb{L}}={{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1},\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{l}{{\rm{\Psi }}}_{n}(\psi )=| W({\psi }_{1},\cdots ,{\psi }_{n},\psi ){| }_{n+1,n+1}\\ \,=\,\left|\begin{array}{cccc}{\psi }_{1} & \cdots & {\psi }_{n} & \psi \\ {\psi }_{1}(x+\varepsilon ) & \cdots & {\psi }_{n}(x+\varepsilon ) & \psi (x+\varepsilon )\\ \vdots & \ddots & \vdots & \vdots \\ {\psi }_{1}(x+(n-1)\varepsilon ) & \cdots & {\psi }_{n}(x+(n-1)\varepsilon ) & \psi (x+(n-1)\varepsilon )\\ {\psi }_{1}(x+n\varepsilon ) & \cdots & {\psi }_{n}(x+n\varepsilon ) & \boxed{\psi (x+n\varepsilon )}\end{array}\right|.\end{array}\end{eqnarray}$
Apparently, for any positive integer n, we have
$\begin{eqnarray}{{\mathbb{L}}}^{n}={{\rm{\Psi }}}_{n}\star {{\rm{\Lambda }}}^{n}\star {{\rm{\Psi }}}_{n}^{-1},\end{eqnarray}$
then $\begin{eqnarray}{\left({{\mathbb{L}}}^{n}\right)}_{+}\star {{\rm{\Psi }}}_{n}-{{\rm{\Psi }}}_{n}\star {{\rm{\Lambda }}}^{n}=-{\left({\mathbb{L}}\right)}_{-}^{n}\star {{\rm{\Psi }}}_{n}.\end{eqnarray}$
Notice that on the left hand side of equation ( $\begin{eqnarray}\begin{array}{l}0={\partial }_{{t}_{r,n}}({{\rm{\Psi }}}_{n}\star {\psi }_{k})\\ \,=\,{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\psi }_{k}+{{\rm{\Psi }}}_{n}\star {\partial }_{{t}_{r,n}}{\psi }_{k}\\ \,=\,{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\psi }_{k}+{{\rm{\Psi }}}_{n}\star \displaystyle \frac{1}{\varepsilon }{k}_{i}^{n}{\psi }_{k}\\ \,=\,{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\psi }_{k}+{{\rm{\Psi }}}_{n}\star \displaystyle \frac{1}{\varepsilon }({{\rm{\Lambda }}}^{n}{\psi }_{k})\\ \,=\,{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\psi }_{k}+\displaystyle \frac{1}{\varepsilon }({{\rm{\Psi }}}_{n}\star {{\rm{\Lambda }}}^{n}){\psi }_{k}\\ \,=\,{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\psi }_{k}+\displaystyle \frac{1}{\varepsilon }({{\mathbb{L}}}_{+}^{n}+{{\mathbb{L}}}_{-}^{n})\star {{\rm{\Psi }}}_{n}\star {\psi }_{k}\\ \,=\,({\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}+\displaystyle \frac{1}{\varepsilon }{{\mathbb{L}}}_{-}^{n}\star {{\rm{\Psi }}}_{n})\star {\psi }_{k}.\end{array}\end{eqnarray}$
Because ${{\mathbb{L}}}_{+}^{n}$ is a shift operator and ${{\rm{\Psi }}}_{n}\star {\psi }_{k}=0$, therefore ${{\mathbb{L}}}_{+}^{n}\star {{\rm{\Psi }}}_{n}\star {\psi }_{k}=0$. From theorem $\begin{eqnarray}{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}+\displaystyle \frac{1}{\varepsilon }{{\mathbb{L}}}_{-}^{n}\star {{\rm{\Psi }}}_{n}=0,\end{eqnarray}$
substituting ( $\begin{eqnarray}\begin{array}{l}\varepsilon {\partial }_{{t}_{r,n}}{\mathbb{K}}=\varepsilon {\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}-\varepsilon {{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}{\partial }_{{t}_{r,n}}{{\rm{\Psi }}}_{n}\star {{\rm{\Psi }}}_{n}^{-1}\\ \,-\,\displaystyle \frac{1}{p}{\partial }_{{t}_{r,n}}({\varepsilon }^{2}{\partial }_{x}+{\varepsilon }^{2}{{\rm{\Psi }}}_{n,x}{{\rm{\Psi }}}_{n}^{-1})\\ \,=\,-{{\mathbb{L}}}_{-}^{n}\star {{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}+{{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}{{\mathbb{L}}}_{-}^{n}\star {{\rm{\Psi }}}_{n}{{\rm{\Psi }}}_{n}^{-1}\\ \,=\,-{{\mathbb{L}}}_{-}^{n}\star {{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}+{{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}{{\mathbb{L}}}_{-}^{n}\\ \,=\,-{\left[{{\mathbb{L}}}_{-}^{n},{{\rm{\Psi }}}_{n}\star {\rm{\Lambda }}\star {{\rm{\Psi }}}_{n}^{-1}-\displaystyle \frac{1}{q}(\varepsilon {\partial }_{x}+\varepsilon {{\rm{\Psi }}}_{n,x}{{\rm{\Psi }}}_{n}^{-1})\right]}_{\star }\\ \,=\,-{\left[{{\mathbb{L}}}_{-}^{n},{\mathbb{K}}\right]}_{\star },\end{array}\end{eqnarray}$
thus, the operators (4. Conclusion
The LFV hierarchy can be written in the form of a Lax pair, which shows that the LFV hierarchy is integrable in the sense of a Lax pair. Based on the Darboux transformation, we give the soliton solutions of this hierarchy, this solutions may be related to the Gromov-Witten invariant. Furthermore, we extend the LFV hierarchy to the noncommutative version. The Darboux transformation and solutions are consistent with the results of the commutative version when taking the commutative limit θij → 0, thus the definition of the NCLFV hierarchy is well defined.
Can this hierarchy be extended to other versions, such as the supersymmetric version? We need further study.