A Vector-Product Lie Algebra of a Reductive Homogeneous Space and Its Applications

Abstract

A new vector-product Lie algebra is constructed for a reductive homogeneous space, which can lead to the presentation of two corresponding loop algebras. As a result, two integrable hierarchies of evolution equations are derived from a new form of zero-curvature equation. These hierarchies can be reduced to the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation. Notably, one of them exhibits a pseudo-Hamiltonian structure, which is derived from a new vector-product identity proposed in this paper.

1. Introduction

Concerning the generation of (1+1)-dimensional integrable equations, lots of approaches have been developed. For example, Tu [1] applied certain Lie subalgebras of the Lie algebra $A_1=\{A:trA=0,A={(a_{ij})}_{2\times 2}\}$ to generate integrable hierarchies of evolution equations within the framework of zero-curvature equations. For $\forall A,B\in A_1$, a commutator is defined as $[A,B]=AB-BA$. A basis of $A_1$ reads as follows:

$$h=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right),e=\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right),f=\left(\begin{array}{cc}0& 0\\ 1& 0\end{array}\right),$$

which satisfies that $[h,e]=2e,[h,f]=-2f,[e,f]=h.$ In particular, the Hamiltonian structures of the obtained hierarchies were generated by the well-known trace identity presented in [1]. The approach was called the Tu scheme by Ma [2]. Based on the method, some interesting integrable hierarchies and their Hamiltonian structures, hereditary symmetries, and so on were obtained [3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Athome and Dorfman [20] and Zhang et al. [21] proposed the concept of symmetric and homogeneous spaces. A homogeneous space of a Lie group, G, is a differential manifold M on which G has the property: $\forall p_1,p_2\in M,$ $\exists g\in G,$ so that $g.p_1=p_2.$ Assume the subgroup K of G is an isotropy group of the Lie group G, then $K=K_{p_0}=\{g\in G:g.p_0=p_0,p_0\in M\}$ is isotropy. Let g and k be the Lie algebras of G and K, respectively, and let m be the vector space complement of k and in g. Then, we have

$$g=k\oplus m,[k,k]\subset k.$$

If $G/K$ is a reduced homogeneous space, then g satisfies the following condition

$$g=k\oplus m,[k,k]\subset k,[k,m]\subset m.$$

(1)

We usually make use of the condition to derive some integrable couplings of the known integrable systems. For example, Fuchssteiner [18] and Ma [19] just right employed the Lie algebra to obtain the integrable couplings of some integrable hierarchies. However, if $G/K$ is a symmetric space, the corresponding Lie algebra g satisfies the condition as follows:

$$g=k\oplus m,[k,k]\subset k,[k,m]\subset m,[m,m]\subset k,$$

(2)

which is called a symmetric algebra. It is remarkable that the Lie subalgebra m satisfying (1) may be an Abel ideal of the Lie algebra g if, just as in the situation, we could only deduce linear integrable couplings under the framework of the Tu scheme. Otherwise, we could generate nonlinear integrable couplings. If the k in (2) is a single Lie algebra, then the Lie subalgebra m satisfying (2) must be an ideal of the Lie algebra g. Therefore, condition (2) could generate nonlinear integrable couplings under the appropriate choice of isospectral Lax pair using the Tu scheme. In the paper, a new vector-product Lie algebra of a reductive homogeneous space is constructed, for which two corresponding loop algebras are introduced. As a result, two integrable hierarchies of evolution equations are derived. The Hamiltonian structure of one of them is obtained through a new vector-product identity that we introduced in the paper. To do this, we first present a three-dimensional vector Lie algebra. Now, let us review some basic notations and computing formulas.

Assume ${\overrightarrow{e}}_1,{\overrightarrow{e}}_2$ and ${\overrightarrow{e}}_3$ are unit coordinate vectors in the right-handed vectical coordinate system $\left\{0;{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3\right\}$, then for arbitrary vectors $\overrightarrow{a}=(a_1,a_2,a_3),\overrightarrow{b}=(b_1,b_2,b_3)\in \left\{0;{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3\right\},$ where $a_i,b_i$ are coordinates of the vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, respectively, the inner product is given by

$$\overrightarrow{a}\ast \overrightarrow{b}=a_1{b}_1+a_2{b}_2+a_3{b}_3.$$

(3)

The vector product of the vectors $\overrightarrow{a},\overrightarrow{b}$ is given by

$$\overrightarrow{a}\times \overrightarrow{b}=\left|\begin{array}{ccc}{\overrightarrow{e}}_1& {\overrightarrow{e}}_2& {\overrightarrow{e}}_3\\ a_1& a_2& a_3\\ b_1& b_2& b_3\end{array}\right|$$

(4)

For arbitrary three vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},$ the following identity holds

$$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})=(\overrightarrow{a}\ast \overrightarrow{c})\overrightarrow{b}-(\overrightarrow{a}\ast \overrightarrow{b})\overrightarrow{c}.$$

(5)

For arbitrary vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$, the Jacobi identity holds

$$\overrightarrow{a}\times (\overrightarrow{b}\times \overrightarrow{c})+\overrightarrow{b}\times (\overrightarrow{c}\times \overrightarrow{a})+\overrightarrow{c}\times (\overrightarrow{a}\times \overrightarrow{b})=\overrightarrow{0}.$$

(6)

2. 6-Dimensional Vector-Product Lie Algebra G and Its Applications

Denote a vector space by $\mathcal{S}$, i.e.,

$$\mathcal{S}=\mathrm{span}\{{\overrightarrow{e}}_1,{\overrightarrow{e}}_2,{\overrightarrow{e}}_3\}.$$

Then, for arbitrary vectors $\overrightarrow{a},\overrightarrow{b}\in \mathcal{S}$, we define a commutation operation using vector product

$${[\overrightarrow{a},\overrightarrow{b}]}_{\mathcal{V}}=\overrightarrow{a}\times \overrightarrow{b}.$$

(7)

It can be verified that $\mathcal{S}$ is a Lie algebra, i.e.,

Theorem 1.

For three-dimensional row-vector space $\mathcal{S}$, if $\overrightarrow{a},\overrightarrow{b}\in \mathcal{S}$ satisfy (7), then $\mathcal{S}$ is a Lie algebra.

Proof. 

According to the properties of vector product, we have

$${[\overrightarrow{a},\overrightarrow{b}]}_{\mathcal{V}}=-{[\overrightarrow{b},\overrightarrow{a}]}_{\mathcal{V}},$$

$${[\alpha \overrightarrow{a}+\beta \overrightarrow{b},\overrightarrow{c}]}_{\mathcal{V}}=\alpha {[\overrightarrow{a},\overrightarrow{c}]}_{\mathcal{V}}+\beta {[\overrightarrow{b},\overrightarrow{c}]}_{\mathcal{V}}.$$

Again, using (6), the Jacobi identity holds. Hence, $\mathcal{S}$ is a Lie algebra with the commutator (7). □

According to relation (2), we extend the vector-product Lie algebra $\mathcal{S}$ to a 6-dimensional case. To write conveniently, we omit the symbol “→” above every vector. For example, the vector ${\overrightarrow{E}}_i$ is simply written as $E_i(i=1,2,\cdots ,6)$. Denote by

$$G=\mathrm{span}\{E_1,E_2,E_3,E_4,E_5,E_6\},$$

which is vector space. In terms of the vector product (7), we define explicitly the following vector-product relations

$$E_1\times E_2=E_3,E_2\times E_3=E_1,E_3\times E_1=E_2,E_1\times E_4=E_5,E_1\times E_5=-E_4,$$

$$E_1\times E_6=0,E_2\times E_4=0,E_2\times E_5=E_6,E_2\times E_6=-E_5,E_3\times E_4=-E_6,$$

$$E_3\times E_5=0,E_3\times E_6=E_4,E_4\times E_5=E_1,E_4\times E_6=-E_3,E_5\times E_6=E_2.$$

It can be proved by the Jacobi identity (6) that the vector space G is a Lie algebra with the aforementioned computational relations. Additionally, assuming that

$$G=G_1\oplus G_2,G_1=\mathrm{span}\{E_1,E_2,E_3\},G_2=\mathrm{span}\{E_4,E_5,E_6\},$$

we observe that

$$G_1\cong \mathcal{S},G_1\times G_2\subset G_2,G_2\times G_2\subset G_1,$$

satisfying the relation (2). This implies that the Lie algebra represents a symmetric Lie algebra of a reductive homogeneous space, as illustrated above. It is remarkable that the relation involves Lie bracket computation. Here, G possesses the vector products.

Now, we construct two loop algebras corresponding to the Lie algebra G. The first vector-product loop algebra is given by

$$\tilde{G}=\mathrm{span}\{E_1(n),E_2(n),\dots ,E_6(n)\},$$

with $E_i(n)=E_i\otimes {\lambda }^n,n\in N,i=1,2,\dots ,6.$ The degree of every element in the loop algebra $\tilde{G}$ is defined as $\mathrm{deg}{E}_i(n)=n,n\in N.$ The resulting the vector-product computations are as follows

$$E_1(n)\times E_2(m)=E_3(m+n),E_2(m)\times E_3(n)=E_1(m+n),$$

$$E_3(m)\times E_1(n)=E_2(m+n),E_1(m)\times E_4(n)=E_5(m+n),$$

$$E_1(m)\times E_5(n)=-E_4(m+n),E_1(m)\times E_6(n)=0,E_2(m)\times E_4(n)=0,$$

$$E_2(m)\times E_5(n)=E_6(m+n),E_2(m)\times E_6(n)=-E_5(m+n),$$

$$E_3(m)\times E_4(n)=-E_6(m+n),E_3(m)\times E_5(n)=0,E_3(m)\times E_6(n)=E_4(m+n),$$

$$E_4(m)\times E_5(n)=E_1(m+n),E_4(m)\times E_6(n)=-E_3(m+n),$$

$$E_5(m)\times E_6(n)=E_2(m+n),m,n\in N.$$

The second vector-product loop algebra is represented as

$$\overline{G}=\mathrm{span}\{F_1(n),F_2(n),\dots ,F_6(n)\},$$

along with the degree distributions as follows

$$F_1(n)=E_1\otimes {\lambda }^{2n},F_2(n)=E_2\otimes {\lambda }^{2n+1},F_3(n)=E_3\otimes {\lambda }^{2n+1},F_4(n)=E_4\otimes {\lambda }^{2n},$$

$$F_5(n)=E_5\otimes {\lambda }^{2n},F_6(n)=E_6\otimes {\lambda }^{2n+1}.$$

The resulting vector-product computations are presented in the following

$$F_1(m)\times F_2(n)=F_3(m+n),F_2(m)\times F_3(n)=F_1(m+n+1),$$

$$F_3(m)\times F_1(n)=F_2(m+n),F_1(m)\times F_4(n)=F_5(m+n),$$

$$F_1(m)\times F_5(n)=-F_4(m+n),F_1(m)\times F_6(n)=0,F_2(m)\times F_4(n)=0,$$

$$F_2(m)\times F_5(n)=F_6(m+n),F_2(m)\times F_6(n)=-F_5(m+n+1),$$

$$F_3(m)\times F_4(n)=-F_6(m+n),F_3(m)\times F_5(n)=0,F_3(m)\times F_6(n)=F_4(m+n+1),$$

$$F_4(m)\times F_5(n)=F_1(m+n),F_4(m)\times F_6(n)=-F_3(m+n),$$

$$F_5(m)\times F_6(n)=F_2(m+n),m,n\in N.$$

Theorem 2.

For arbitrary vectors $\overrightarrow{U},\overrightarrow{V},\overrightarrow{\varphi }\in \tilde{G}or\overline{G}$ as above, suppose that

$${\overrightarrow{\varphi }}_x=\overrightarrow{U}\times \overrightarrow{\varphi },{\overrightarrow{\varphi }}_t=\overrightarrow{V}\times \overrightarrow{\varphi },$$

(8)

then the compatibility condition of (8) leads to a new form of zero-curvature equation

$${\overrightarrow{U}}_t-{\overrightarrow{V}}_x+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0},$$

(9)

which is called a vector-product zero-curvature equation.

Proof. 

From (5) and (8), one has

$${\overrightarrow{\varphi }}_{xt}={\overrightarrow{U}}_t\times \overrightarrow{\varphi }+\overrightarrow{U}\times \overrightarrow{{\varphi }_t}={\overrightarrow{U}}_t\times \overrightarrow{\varphi }+\overrightarrow{U}\times (\overrightarrow{V}\times \overrightarrow{\varphi })={\overrightarrow{U}}_t\times \overrightarrow{\varphi }+(\overrightarrow{U}\ast \overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{U}\ast \overrightarrow{V})\overrightarrow{\varphi },$$

$${\overrightarrow{\varphi }}_{tx}={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+\overrightarrow{V}\times \overrightarrow{{\varphi }_x}={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+\overrightarrow{V}\times (\overrightarrow{U}\times \overrightarrow{\varphi })={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+(\overrightarrow{V}\ast \overrightarrow{\varphi })\overrightarrow{U}-(\overrightarrow{V}\ast \overrightarrow{U})\overrightarrow{\varphi }.$$

Since

$${\overrightarrow{\varphi }}_{xt}={\overrightarrow{\varphi }}_{tx},$$

it follows that

$${\overrightarrow{\varphi }}_{xt}={\overrightarrow{U}}_t\times \overrightarrow{\varphi }-{\overrightarrow{V}}_x\times \overrightarrow{\varphi }-(\overrightarrow{U}\ast \overrightarrow{V}-\overrightarrow{V}\ast \overrightarrow{U})\overrightarrow{\varphi }+(\overrightarrow{U}\times \overrightarrow{V})\times \overrightarrow{\varphi }=\overrightarrow{0}.$$

Since $\overrightarrow{U},\overrightarrow{V}$ are both vectors, their dot product is commutative, that is $\overrightarrow{U}\ast \overrightarrow{V}=\overrightarrow{V}\ast \overrightarrow{U}$. Again, $\overrightarrow{\varphi }$ is an arbitrary vector, thus the (9) holds. □

In the theory of integrable systems, the zero-curvature equation stands as a pivotal equation that characterizes the integrability of the system. The vector-product zero-curvature equation introduced in Theorem 2 opens up possibilities for constructing novel integrable systems or integrable hierarchies. By selecting diverse vectors $\overrightarrow{U}$ and $\overrightarrow{V}$, along with an appropriate field $\overrightarrow{\varphi }$, one can generate a series of new integrable models, which may possess unique physical properties and practical applications.

In what follows, we shall make use of the two vector-product loop algebras to generate two kinds of expanding integrable hierarchies by applying the vector-product zero-curvature Equation (9).

Case 1: The application of the first loop algebra $\tilde{G}$. Assume that

$$\overrightarrow{U}=-E_1(1)+E_2(0)-w{E}_3(0)+{\displaystyle \frac{v}{2}}{E}_1(0)+u{E}_4(0)+s{E}_5(0),$$

$$\overrightarrow{V}=\sum _{m\ge 0}(a_m{E}_1(0)+b_m{E}_2(0)+c_m{E}_3(0)+d_m{E}_4(0)+e_m{E}_5(0)+f_m{E}_6(0)){\lambda }^{-m}.$$

With the help of equation $\overrightarrow{U}\times \overrightarrow{V}={\overrightarrow{V}}_x$, a constrained relation among the coefficients and the potential functions is given by

$$\left\{\begin{array}{c}{a}_{mx}=c_m+w{b}_m+u{e}_m-s{d}_m,\hfill \\ f_{mx}=e_m+w{d}_m+u{c}_m-s{b}_m,\hfill \\ c_{m+1}=b_{mx}+w{a}_m+{\displaystyle \frac{v}{2}}{c}_m-s{f}_m\hfill \\ b_{m+1}=-c_{mx}-a_m+{\displaystyle \frac{v}{2}}{b}_m-u{f}_m,\hfill \\ e_{m+1}=d_{mx}+w{f}_m+{\displaystyle \frac{v}{2}}{e}_m-s{a}_m,\hfill \\ d_{m+1}=-e_{mx}-f_m+{\displaystyle \frac{v}{2}}{d}_m-u{a}_m.\hfill \end{array}\right.$$

(10)

Taking $a_0={\alpha }_0(t),b_0=c_0=d_0=e_0=0,$ then (10) has some special values, such as

$$c_1={\alpha }_{0}w,b_1=-{\alpha }_0,e_1=-{\alpha }_{0}s,d_1=-{\alpha }_{0}u,a_1={\alpha }_1(t),f_1={\beta }_1(t),$$

$$c_2={\alpha }_{1}w+{\displaystyle \frac{{\alpha }_0}{2}}vw-{\beta }_{1}s,b_2=-{\alpha }_0{w}_x-{\alpha }_1-{\displaystyle \frac{{\alpha }_0}{2}}v-{\beta }_{1}u,e_2=-{\alpha }_0{u}_x+{\beta }_{1}w-{\displaystyle \frac{{\alpha }_0}{2}}vs-{\alpha }_{1}s,$$

$$d_2={\alpha }_0{s}_x-{\beta }_1-{\displaystyle \frac{{\alpha }_0}{2}}uv-{\alpha }_{1}u,a_2=-{\displaystyle \frac{{\alpha }_0}{2}}{w}^2-{\displaystyle \frac{{\alpha }_0}{2}}{s}^2-{\displaystyle \frac{{\alpha }_0}{2}}{u}^2+{\alpha }_2(t),f_2=-{\alpha }_0(u-sw)+{\beta }_2(t),$$

$$\dots$$

Denote that

$${\overrightarrow{V}}^{(n)}=\sum _{m=0}^n(a_m{E}_1(-m)+b_m{E}_2(-m)+c_m{E}_3(-m)+d_m{E}_4(-m)+e_m{E}_5(-m)+f_m{E}_6(-m)){\lambda }^{2n}-{\displaystyle \frac{c_{n+1}}{w}}{E}_1(0),$$

then the special vector-product zero-curvature equation

$${\overrightarrow{U}}_t-{\overrightarrow{V}}_x^{(n)}+\overrightarrow{U}\times {\overrightarrow{V}}^{(n)}=\overrightarrow{0}$$

(11)

admits the following isospectral integrable hierarchy

$$\left\{\begin{array}{c}{w}_{t_n}=b_{n+1}+{\displaystyle \frac{c_{n+1}}{w}}\hfill \\ v_{t_n}=-2{\left({\displaystyle \frac{c_{n+1}}{w}}\right)}_x\hfill \\ u_{t_n}=e_{n+1}+s{\displaystyle \frac{s{c}_{n+1}}{w}}\hfill \\ s_{t_n}=-d_{n+1}-{\displaystyle \frac{u{c}_{n+1}}{w}}\hfill \end{array}\right.$$

(12)

We consider a reduction of the hierarchy when $n=2:$

$$\begin{array}{ccc}& & w_{t_2}=-{\alpha }_1{w}_x-{\displaystyle \frac{{\alpha }_0}{2}}{(vw)}_x+{\beta }_1{s}_x+{\alpha }_0(w^2+s^2+u^2)-{\displaystyle \frac{{\alpha }_0}{2}}v{w}_x-{\displaystyle \frac{{\beta }_1}{2}}uv-{\alpha }_{0}suw-{\beta }_{2}u\hfill \\ & & -{\alpha }_0{\displaystyle \frac{w_{xx}}{w}}-{\displaystyle \frac{{\alpha }_0{v}_x}{2w}}-{\beta }_1{\displaystyle \frac{u_x}{w}}+{\alpha }_0{\displaystyle \frac{su}{w}}-{\alpha }_{0}s-{\beta }_2{\displaystyle \frac{s}{w}}+{\displaystyle \frac{{\alpha }_0}{2}}{v}^2-{\displaystyle \frac{{\beta }_{1}vs}{2w}},\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & v_{t_2}=-2(-{\alpha }_0{\displaystyle \frac{w_{xx}}{w}}-{\displaystyle \frac{{\alpha }_0{v}_x}{2w}}-{\beta }_1{\displaystyle \frac{u_x}{w}}-{\displaystyle \frac{{\alpha }_0}{2}}(w^2+s^2+u^2)+{\alpha }_2+{\displaystyle \frac{{\alpha }_1}{2}}v+{\displaystyle \frac{{\alpha }_0}{4}}{v}^2\hfill \\ & & -{\displaystyle \frac{{\beta }_{1}vs}{2w}}+{\alpha }_0{\displaystyle \frac{su}{w}}-{\alpha }_0{s}^2-{\beta }_2{\displaystyle \frac{s}{w}}{)}_x,\hfill \end{array}$$

(14)

$$\begin{array}{ccc}& & u_{t_2}={\alpha }_0{s}_{xx}-{\displaystyle \frac{{\alpha }_0}{2}}{(uv)}_x-{\alpha }_1{u}_x-{\alpha }_{0}wu+{\displaystyle \frac{3{\alpha }_0}{2}}s{w}^2+{\beta }_{2}w-{\displaystyle \frac{{\alpha }_0}{2}}v{u}_x+{\displaystyle \frac{{\beta }_1}{2}}vw-{\displaystyle \frac{{\alpha }_1}{2}}vs\hfill \\ & & -{\displaystyle \frac{{\alpha }_0}{2}}{s}^3+{\displaystyle \frac{{\alpha }_0}{2}}s{u}^2+{\displaystyle \frac{s}{w}}(-{\alpha }_0{w}_{xx}+{\alpha }_{0}su-{\beta }_{2}s-{\displaystyle \frac{{\alpha }_0}{2}}{v}_x-{\beta }_1{u}_x)-{\displaystyle \frac{{\alpha }_0}{2}}(w^2+s^2+u^2)\hfill \\ & & +{\displaystyle \frac{{\alpha }_1}{2}}sv-{\displaystyle \frac{{\beta }_{1}v{s}^2}{2w}},\hfill \end{array}$$

(15)

$$\begin{array}{ccc}& & s_{t_2}=-{\alpha }_0{u}_{xx}+{\beta }_1{w}_x-{\displaystyle \frac{{\alpha }_0}{2}}{(vs)}_x-{\alpha }_1{s}_x-{\displaystyle \frac{{\alpha }_0}{2}}v{s}_x+{\displaystyle \frac{{\beta }_{1}v}{2}}+{\displaystyle \frac{{\alpha }_{0}v}{2}}-{\alpha }_{0}u+{\alpha }_{0}sw+{\alpha }_{0}u{s}^2\hfill \\ & & -{\displaystyle \frac{u}{w}}(-{\alpha }_0{w}_{xx}-{\displaystyle \frac{{\alpha }_0}{2}}{v}_x-{\beta }_1{u}_x+{\displaystyle \frac{{\beta }_1}{2}}vs-{\alpha }_{0}su-{\beta }_{2}s)-{\displaystyle \frac{{\alpha }_0}{4}}{v}^{2}u+{\beta }_2.\hfill \end{array}$$

(16)

A reduction of Equations (13)–(16) is presented when $u=s=0,{\alpha }_1={\alpha }_2={\beta }_1={\beta }_2=0:$

$$\left\{\begin{array}{c}{w}_{t_2}={\alpha }_0\left(-{\displaystyle \frac{1}{2}}v{w}_x-{\displaystyle \frac{w_{xx}}{w}}+w^2-{\displaystyle \frac{1}{2}}{(vw)}_x+{\displaystyle \frac{1}{2}}{v}^2\right),\hfill \\ v_{t_2}=-2{\alpha }_0{\left(-{\displaystyle \frac{w_{xx}}{w}}-{\displaystyle \frac{v_x}{2w}}-{\displaystyle \frac{1}{2}}{w}^2+{\displaystyle \frac{1}{4}}{v}^2\right)}_x.\hfill \end{array}\right.$$

(17)

If $w=1,$ then we obtain

$$v_t={\displaystyle \frac{{\alpha }_0}{2}}{v}_{xx},$$

which is the standard heat equation when ${\alpha }_0=2.$

If $v=0,$ a reduced equation is given by

$$w_t=-3{\alpha }_0{\displaystyle \frac{w_{xx}}{w}},$$

which is a special case of the general diffusion equation presented as follows in [24]:

$$u_t=\alpha u^p{u}_{xx}+\beta u^q{u}_x^2.$$

Case 2: The application of the second loop algebra $\overline{G}.$ Assume that

$$\overrightarrow{U}=F_1(0){\lambda }^2+q{F}_2(0)\lambda +r{F}_3(0)\lambda +u{F}_4(0)+v{F}_5(0),$$

$$\overrightarrow{V}=\sum _{m\ge 0}(a_m{F}_1(0)+b_m{F}_2(0)\lambda +c_m{F}_3(0)\lambda +d_m{F}_4(0)+e_m{F}_5(0)+f_m{F}_6(0)\lambda ){\lambda }^{-2m},$$

then constrained relations among the coefficients of the vector $\overrightarrow{V}$ and the potential functions in the vector $\overrightarrow{U}$ are given by

$$\left\{\begin{array}{c}{a}_{mx}=q{c}_m-r{b}_m+u{e}_m-v{d}_m,\hfill \\ c_{m+1}=-b_{mx}+r{a}_m+v{f}_m,\hfill \\ b_{m+1}=c_{mx}+q{a}_m+u{f}_m,\hfill \\ e_{m+1}=-d_{mx}+r{f}_{m+1}+v{a}_m,\hfill \\ d_{m+1}=e_{mx}+q{f}_{m+1}+u{a}_m,\hfill \\ f_{m+1,x}=-q{d}_{mx}-r{e}_{mx}-u{b}_{mx}-v{c}_{mx}.\hfill \end{array}\right.$$

(18)

Set $b_0=c_0=d_0=e_0=f_0=0,a_0={\alpha }_0(t),$ then from (18) we obtain

$$c_1={\alpha }_{0}r,b_1={\alpha }_{0}q,f_1=0,e_1={\alpha }_{0}v,d_1={\alpha }_{0}u,a_1=:{\alpha }_1(t),c_2=-{\alpha }_0{q}_x+{\alpha }_{1}r,$$

$$b_2={\alpha }_0{r}_x+{\alpha }_{1}q,f_2=-{\alpha }_{0}rv-{\alpha }_{0}qu,e_2=-{\alpha }_0{u}_x-{\alpha }_{0}r(rv+qu)+{\alpha }_{1}v,$$

$$d_2={\alpha }_0{v}_x-{\alpha }_{0}q(rv+qu)+{\alpha }_{1}u,a_2=-{\displaystyle \frac{{\alpha }_0}{2}}(q^2+r^2+u^2+v^2)+{\alpha }_2(t),\dots .$$

Denote

$${\overrightarrow{V}}^{(n)}=\sum _{m=0}^n(a_m{F}_1(0)+b_m{F}_2(0)\lambda +c_m{F}_3(0)\lambda +d_m{F}_4(0)+e_m{F}_5(0)+f_m{F}_6(0)\lambda ){\lambda }^{2n-2m},$$

then, a direct calculation reads that

$$-{\overrightarrow{V}}_x^{(n)}+\overrightarrow{U}\times {\overrightarrow{V}}^{(n)}=-b_{n+1}{F}_3(0)+c_{n+1}{F}_2(0)+(e_{n+1}-r{f}_{n+1})F_4(0)+(q{f}_{n+1}-d_{n+1})F_5(0).$$

Thus, the vector-product zero-curvature equation

$${\overrightarrow{U}}_t-{\overrightarrow{V}}_x^{(n)}+\overrightarrow{U}\times {\overrightarrow{V}}^{(n)}=\overrightarrow{0}$$

(19)

admits the following isospectral integrable hierarchy

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=\left(\begin{array}{c}-c_{n+1}\\ b_{n+1}\\ r{f}_{n+1}-e_{n+1}=d_{nx}-v{a}_n\\ d_{n+1}-q{f}_{n+1}=e_{nx}+u{a}_n\end{array}\right).$$

(20)

Consider reductions of the hierarchy (20) now. When $u=v=0$ and $n=2,$ (20) reduces to

$$\left\{\begin{array}{c}{q}_t={\alpha }_0{r}_{xx}+{\alpha }_{1}q+{\displaystyle \frac{{\alpha }_0}{2}}r\left(q^2+r^2\right)+{\alpha }_{2}r,\hfill \\ r_t=-{\alpha }_0{q}_{xx}+{\alpha }_1{r}_x-{\displaystyle \frac{{\alpha }_0}{2}}q\left(q^2+r^2\right)+{\alpha }_{2}q.\hfill \end{array}\right.$$

(21)

When $r=\pm iq,$ the (21) reduces to a general linear Schrödinger equation

$$q_t=\pm i{\alpha }_0{q}_{xx}+{\alpha }_1{q}_x\mp i{\alpha }_{2}q.$$

Taking ${\alpha }_0=i,{\alpha }_2=0$ and $r=\overline{q}$, the (21) becomes a nonlinear Schrödinger-type equation

$$q_t=i{\overline{q}}_{xx}+{\alpha }_1{q}_x+{\displaystyle \frac{i}{2}}({\overline{q}}^3+{\left|q\right|}^{2}q).$$

(22)

3. Nonisospectral Integrable Hierarchies

Zhang et al. [25] developed a scheme (an improved Tu scheme) for seeking nonisospectral integrable hierarchies of evolution equations based on the work by Li [26], which has been used to generate the nonisospectral integrable KdV hierarchy and some symmetries. Zhang et al. [27] applied the scheme to obtain the nonisospectral integrable nonlinear Schrödinger hierarchy and its expanding integrable model, as well as some reductions. In Refs. [28,29], Ma made use of the zero-curvature equation to present an approach for generating nonisospectral integrable hierarchies under the spectral evolution ${\lambda }_t={\lambda }^n$. In Refs. [30,31], Qiao adopted the Lax equation under the spectral evolution ${\lambda }_t={\lambda }^{m+1)\eta }M$ to introduce a way for nonisospectral integrable hierarchies. However, we found that the works by Ma and Qiao are special cases of spectral evolution ${\lambda }_t={\displaystyle \sum _{j=0}^n}{k}_j(t){\lambda }^{n-j}$ which appears in [9,32,33,34]. In this subsection, we shall only investigate the nonisospectral integrable hierarchy that corresponds to the isospectral hierarchy (20). We first extend the Tu scheme to the case where the nonisospectral integrable hierarchies are produced by the vector product equation as follows

$${\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t-{\overrightarrow{V}}_x+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0}.$$

(23)

In fact, set

$${\overrightarrow{\varphi }}_x=\overrightarrow{U}\times \overrightarrow{\varphi },{\overrightarrow{\varphi }}_t=\overrightarrow{V}\times \overrightarrow{\varphi },\overrightarrow{U}=\overrightarrow{U}(u,\lambda ),$$

then

$$\begin{array}{ccc}& & {\overrightarrow{\varphi }}_{xt}=({\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t)\times \overrightarrow{\varphi }+\overrightarrow{U}\times (\overrightarrow{V}\times \overrightarrow{\varphi }),\hfill \\ & & {\overrightarrow{\varphi }}_{tx}={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+\overrightarrow{V}\times {\overrightarrow{\varphi }}_x={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+\overrightarrow{V}\times (\overrightarrow{U}\times \overrightarrow{\varphi }),\hfill \\ & & {\overrightarrow{\varphi }}_{xt}={\overrightarrow{\varphi }}_{tx}({\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t)\times \overrightarrow{\varphi }+(\overrightarrow{U}\ast \overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{U}\ast \overrightarrow{V})\overrightarrow{\varphi }\hfill \\ & & ={\overrightarrow{V}}_x\times \overrightarrow{\varphi }+(\overrightarrow{V}\ast \overrightarrow{\varphi })\overrightarrow{U}-(\overrightarrow{V}\ast \overrightarrow{U})\overrightarrow{\varphi },({\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t)\times \overrightarrow{\varphi }-{\overrightarrow{V}}_x\times \overrightarrow{\varphi }+(\overrightarrow{V}\ast \overrightarrow{U}-\overrightarrow{U}\ast \overrightarrow{V})\overrightarrow{\varphi }\hfill \\ & & +(\overrightarrow{U}\ast \overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{V}\ast \overrightarrow{\varphi })\overrightarrow{U}=\overrightarrow{0}.\hfill \end{array}$$

(24)

It is easy to find that

$$(\overrightarrow{U}\ast \overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{V}\ast \overrightarrow{\varphi })\overrightarrow{U}=-\overrightarrow{\varphi }\times (\overrightarrow{U}\times \overrightarrow{V}),$$

that is,

$$(\overrightarrow{U}\times \overrightarrow{V})\times \overrightarrow{\varphi }=(\overrightarrow{U}\ast \overrightarrow{\varphi })\overrightarrow{V}-(\overrightarrow{V}\ast \overrightarrow{\varphi })\overrightarrow{U}.$$

Hence, we have

$$({\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t)\times \overrightarrow{\varphi }-{\overrightarrow{V}}_x\times \overrightarrow{\varphi }+(\overrightarrow{U}\times \overrightarrow{V})\times \overrightarrow{\varphi }=\overrightarrow{0}.$$

Since $\overrightarrow{\varphi }$ is arbitrary, (23) holds.

Next, we derive a nonisospectral integrable hierarchy by applying (23). Denote ${\lambda }_t={\displaystyle \sum _{m\ge 0}}{k}_m(t){\lambda }^{-2m+1},$ then the equation

$${\overrightarrow{U}}_{\lambda }{\lambda }_t-{\overrightarrow{V}}_x+\overrightarrow{U}\times \overrightarrow{V}=\overrightarrow{0}$$

leads to the following recursive relations among the coefficients of the vector $\overrightarrow{V}:$

$$\left\{\begin{array}{c}{a}_{mx}=q{c}_m-r{b}_m+u{e}_m-v{d}_m+2k_{j+1}(t),\hfill \\ c_{m+1}=-b_{mx}+r{a}_m+v{f}_m+q{k}_i(t),\hfill \\ b_{m+1}=c_{mx}+q{a}_m+u{f}_m,\hfill \\ e_{m+1}=-d_{mx}+r{f}_{m+1}+v{a}_m,\hfill \\ d_{m+1}=e_{mx}+q{f}_{m+1}+u{a}_m,\hfill \\ f_{m+1,x}=-q{d}_{mx}-r{e}_{mx}-u{b}_{mx}-v{c}_{mx}+(qu+vr)k_i(t).\hfill \end{array}\right.$$

(25)

Let $b_0=c_0=d_0=e_0=0,a_0={\sigma }_0(t),f_0={\delta }_0(t),$ then (25) admit that

$$c_1={\sigma }_{0}r+{\delta }_{0}v+q{k}_0(t),b_1={\sigma }_{0}q+{\delta }_{0}u-r{k}_0(t),f_1=k_0{\partial }^{-1}(qu+vr)+{\delta }_0,$$

$$d_1=k_{0}q{\partial }^{-1}(qu+vr)+{\delta }_{0}q+{\sigma }_{0}u,e_1=k_{0}r{\partial }^{-1}(qu+vr)+{\delta }_{0}r+{\sigma }_{0}v,a_1=k_{0}M+2k_2(t)x,$$

$$f_2=N-{\displaystyle \frac{{\delta }_0}{2}}(q^2+r^2+u^2+v^2)-{\sigma }_0(qu+vr),$$

$$c_2=-{\sigma }_0{q}_x-{\delta }_0{u}_x-k_0{r}_x+k_{0}rM+2k_{2}xr+k_{0}v{\partial }^{-1}(qu+vr)+{\delta }_{0}v+k_{1}q,$$

$$b_2={\sigma }_0{r}_x+{\delta }_0{v}_x+k_0{q}_x+k_{0}qM+2k_{2}xq+k_{0}u{\partial }^{-1}(qu+vr)-k_{1}r+{\delta }_{0}u,$$

$$e_2=-k_0{q}_x{\partial }^{-1}(qu+vr)-k_{0}q(qu+vr)-{\delta }_0{q}_x-{\sigma }_0{u}_x+rN-{\displaystyle \frac{{\delta }_0}{2}}r(q^2+r^2+u^2+v^2)-{\sigma }_{0}r(qu+vr)+k_{0}vM+2k_{2}xv,$$

$$d_2=k_0{r}_x{\partial }^{-1}(qu+vr)+k_{0}r(qu+vr)+{\delta }_0{r}_x+{\sigma }_x{v}_x+qN-{\displaystyle \frac{{\delta }_0}{2}}q(q^2+r^2+u^2+v^2)-{\sigma }_{0}q(qu+vr)+k_{0}uM+2k_{2}xu,$$

$$a_2=-{\displaystyle \frac{{\sigma }_0}{2}}(q^2+r^2+u^2+v^2)-{\delta }_0(qu+vr)-k_{0}qr+2k_{3}x+Q,\dots ,$$

where

$$M_x=q^2+r^2+(ur-qv){\partial }^{-1}(qu+vr),$$

$$N_x=-k_0[{\displaystyle \frac{1}{2}}{(q^2+r^2)}_x{\partial }^{-1}(qu+vr)+(q^2+r^2)(qu+vr)-u{r}_x+v{q}_x]+k_1(qu+vr),$$

$Q_x=k_{0}qv{\partial }^{-1}(qu+vr)+{\delta }_{0}qv+k_1{q}^2-k_{0}ur{\partial }^{-1}(qu+vr)-{\delta }_{0}ur+k_1{r}^2-k_{0}u{q}_x{\partial }^{-1}(qu+vr)-k_{0}uq(qu+vr)+(ur-qv)N+{\displaystyle \frac{{\delta }_0}{2}}(qv-ur)(q^2+r^2+u^2+v^2)+{\sigma }_0(qv-ur)(qu+vr)-k_{0}vr(qu+vr).$

Denoting that

$${\overrightarrow{V}}^{(n)}=\sum _{m=0}^n(a_m{E}_1(-m)+b_m{E}_2(-m)+c_m{E}_3(-m)+d_m{E}_4(-m)+e_m{E}_5(-m)+f_m{E}_6(-m)){\lambda }^{2n},$$

then the special vector-product zero-curvature equation

$${\overrightarrow{U}}_u{u}_t+{\overrightarrow{U}}_{\lambda }{\lambda }_t-{\overrightarrow{V}}_x^{(n)}+\overrightarrow{U}\times {\overrightarrow{V}}^{(n)}=\overrightarrow{0}.$$

(26)

admits the nonisospectral integrable hierarchy

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=\left(\begin{array}{c}-c_{n+1}\\ b_{n+1}\\ d_{nx}-v{a}_n\\ e_{nx}+u{a}_n\end{array}\right).$$

(27)

When $n=2,$ a reduction of (27) is presented as

$$\begin{array}{ccc}& & q_t={\sigma }_0{r}_{xx}+{\delta }_0{v}_{xx}+k_0{q}_{xx}+k_0{(qM)}_x+2k_2{(xq)}_x+k_0{[u{\partial }^{-1}(qu+vr)]}_x-k_1{r}_x\hfill \\ & & +{\displaystyle \frac{{\sigma }_0}{2}}r(q^2+r^2+u^2+v^2)+{\delta }_{0}r(qu+vr)+k_{0}q{r}^2-2k_{3}xr-rQ-vN\hfill \\ & & +{\displaystyle \frac{{\delta }_0}{2}}v(q^2+r^2+u^2+v^2)+{\sigma }_{0}v(qu+vr)-q{k}_2,\hfill \end{array}$$

(28)

$$\begin{array}{ccc}& & r_t=-{\sigma }_0{q}_{xx}-{\delta }_0{u}_{xx}-k_0{r}_{xx}+k_0{(rM)}_x+2k_2{(xr)}_x+k_0{[v{\partial }^{-1}(qu+vr)]}_x+{\delta }_0{v}_x+k_1{q}_x\hfill \\ & & -{\displaystyle \frac{{\sigma }_0}{2}}q(q^2+r^2+u^2+v^2)-{\delta }_{0}q(qu+vr)-k_0{q}^{2}r+2k_{3}xq+qQ-r{k}_2,\hfill \end{array}$$

(29)

$$\begin{array}{ccc}& & u_t={\delta }_0{r}_{xx}+{\sigma }_0{v}_{xx}+{(qN)}_x+k_0{[r_x{\partial }^{-1}(qu+vr)]}_x+k_0{(qru+v{r}^2)}_x\hfill \\ & & -{\displaystyle \frac{{\delta }_0}{2}}{[q(q^2+r^2+u^2+v^2)]}_x-{\sigma }_0{(q^{2}u+qrv)}_x+k_0{(uM)}_x+2k_2{(xu)}_x\hfill \\ & & +{\displaystyle \frac{{\sigma }_0}{2}}v(q^2+r^2+u^2+v^2)+{\delta }_{0}v(qu+vr)+k_{0}qrv-2k_{3}xv-vQ,\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& & v_t=-{\delta }_0{q}_{xx}-{\sigma }_0{u}_{xx}-k_0{[q_x{\partial }^{-1}(qu+vr)]}_x-k_0{(q^{2}u+qrv)}_x+{(rN)}_x\hfill \\ & & -{\displaystyle \frac{{\delta }_0}{2}}{[r(q^2+r^2+u^2+v^2)]}_x-{\sigma }_0{(qru+v{r}^2)}_x+k_0{(vM)}_x+2k_2{(xv)}_x\hfill \\ & & -{\displaystyle \frac{{\sigma }_0}{2}}u(q^2+r^2+u^2+v^2)+{\delta }_{0}u(qu+vr)-k_{0}qrv+2k_{3}xu+uQ.\hfill \end{array}$$

(31)

Set $u=v=0,$ the (28)–(31) again reduces to

$$\left\{\begin{array}{cc}\hfill q_t& ={\sigma }_0{r}_{xx}+k_0{q}_{xx}+k_0{\left[q{\partial }^{-1}\left(q^2+r^2\right)\right]}_x+2k_2{(xq)}_x-k_1{r}_x\hfill \\ \hfill & +{\displaystyle \frac{{\sigma }_0}{2}}r\left(q^2+r^2\right)+k_{0}q{r}^2-2k_{3}xr-k_{2}q-k_{1}r{\partial }^{-1}\left(q^2+r^2\right),\hfill \\ \hfill r_t& =-{\sigma }_0{q}_{xx}-k_0{r}_{xx}+k_0{\left[r{\partial }^{-1}\left(q^2+r^2\right)\right]}_x+2k_2{(xr)}_x+k_1{q}_x\hfill \\ \hfill & -{\displaystyle \frac{{\sigma }_0}{2}}\left(q^2+r^2\right)-k_0{q}^{2}r+2k_{3}xq-k_{2}r+k_{1}q{\partial }^{-1}\left(q^2+r^2\right).\hfill \end{array}\right.$$

(32)

When $k_0=k_1=0,{\sigma }_0={\overline{k}}_3=i,$ the (32) become

$$q_t=i{\overline{q}}_{xx}+2k_2{(xq)}_x+{\displaystyle \frac{i}{2}}\overline{q}(q^2+{\overline{q}}^2)-2ix\overline{q}-k_{2}q,$$

(33)

which is a new nonisospectral integrable nonlinear Schrödinger-type equation.

4. Pseudo-Hamiltonian Structure of the Isospectral Integrable Hierarchy

The trace identity [1] proposed by Tu Guizhang is a powerful tool for generating the Hamiltonian structure of integrable hierarchies produced by the zero-curvature equation. A generalized identity called the quadratic-form identity was presented by Guo and Zhang [17]. In this section, we follow the idea for deriving the quadratic-form identity to deduce a new vector-product identity for generating the Hamiltonian structure of the integrable hierarchies obtained by the vector-product zero-curvature Equation (11).

Assume $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\in \mathcal{S},$ and a linear functional ${\{\overrightarrow{a},\overrightarrow{b}\}}_{\mathcal{V}}=:\overrightarrow{a}F{\overrightarrow{b}}^T,$ where F is a square matrix with constant entries. Two requirements are imposed on the functional ${\{\overrightarrow{a},\overrightarrow{b}\}}_{\mathcal{V}},$ which are listed as follows

$${\{\overrightarrow{a},\overrightarrow{b}\}}_{\mathcal{V}}={\{\overrightarrow{b},\overrightarrow{a}\}}_{\mathcal{V}},{\{\overrightarrow{a},{[\overrightarrow{b},\overrightarrow{c}]}_{\mathcal{V}}\}}_{\mathcal{V}}={\{{[\overrightarrow{a},\overrightarrow{b}]}_{\mathcal{V}},\overrightarrow{c}\}}_{\mathcal{V}}.$$

(34)

(34) requires that F is symmetric, i.e., $F^T=F$. Since the vector product of $\overrightarrow{a}$ and $\overrightarrow{b}$ can be written as

$${[\overrightarrow{a},\overrightarrow{b}]}_{\mathcal{V}}=\overrightarrow{a}\times \overrightarrow{b}=(a_2{b}_3-a_3{b}_2,a_3{b}_1-b_3{a}_1,a_1{b}_2-b_1{a}_2)$$

$$=(a_1,a_2,a_3)\left(\begin{array}{ccc}0& -b_3& b_2\\ b_3& 0& -b_1\\ -b_2& b_1& 0\end{array}\right)=:\overrightarrow{a}R(b),$$

(35)

where $\overrightarrow{a}=(a_1,a_2,a_3),\overrightarrow{b}=(b_1,b_2,b_3).$ Therefore, (35) requires that

$$R(b)F=-{(R(b)F)}^T$$

(36)

A linear functional is introduced as follows

$$Q={\left\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda }\right\}}_{\mathcal{V}}+{\left\{\overrightarrow{\mathsf{\Lambda }},{\overrightarrow{V}}_x-{[\overrightarrow{U},\overrightarrow{V}]}_{\mathcal{V}}\right\}}_{\mathcal{V}},$$

(37)

where ${\overrightarrow{U}}_{\lambda }$ denotes the partial derivative with respect to spectral parameter $\lambda$, and $\overrightarrow{V},\overrightarrow{\mathsf{\Lambda }}\in \mathcal{W}$ are two vectors to be determined. We also present the variational derivative

$${\displaystyle \frac{\delta }{\delta u}}=\sum _{k=0}^{\infty }{(-D)}^k{\displaystyle \frac{\partial }{\partial (D^{k}u)}},D={\displaystyle \frac{d}{dx}},$$

and apply it to (34), two variational equations are obtained

$${\displaystyle \frac{\delta }{\delta \overrightarrow{V}}}Q={\overrightarrow{U}}_{\lambda }-{\overrightarrow{\mathsf{\Lambda }}}_x+{[\overrightarrow{U},\overrightarrow{\mathsf{\Lambda }}]}_{\mathcal{V}},{\displaystyle \frac{\delta }{\delta \overrightarrow{\mathsf{\Lambda }}}}Q={\overrightarrow{V}}_x-{[\overrightarrow{U},\overrightarrow{V}]}_{\mathcal{V}}.$$

(38)

Given the following constrained conditions

$${\displaystyle \frac{\delta }{\delta \overrightarrow{V}}}Q=0\Rightarrow {\overrightarrow{U}}_{\lambda }-{\overrightarrow{\mathsf{\Lambda }}}_x+{[\overrightarrow{U},\overrightarrow{\mathsf{\Lambda }}]}_{\mathcal{V}}=\overrightarrow{0}.$$

(39)

$${\displaystyle \frac{\delta }{\delta \overrightarrow{\mathsf{\Lambda }}}}Q=0\Rightarrow {\overrightarrow{V}}_x-{[\overrightarrow{U},\overrightarrow{V}]}_{\mathcal{V}}=\overrightarrow{0}.$$

(40)

Using (38) and (34), we have

$${\displaystyle \frac{\delta }{\delta u}}\left\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda }\right\}={\displaystyle \frac{\delta Q}{\delta u}}={\left\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda u}\right\}}_{\mathcal{V}}+{\left\{{[\overrightarrow{\mathsf{\Lambda }},\overrightarrow{V}]}_{\mathcal{V}},{\overrightarrow{U}}_u\right\}}_{\mathcal{V}}.$$

(41)

Using (39), (40) and the Jacobi identity, one can infer that

$${[\overrightarrow{\mathsf{\Lambda }},\overrightarrow{V}]}_{\mathcal{V}}={[{\overrightarrow{U}}_{\lambda }+{[\overrightarrow{U},\overrightarrow{\mathsf{\Lambda }}]}_{\mathcal{V}}]}_{\mathcal{V}}+{[\overrightarrow{\mathsf{\Lambda }},{[\overrightarrow{U},\overrightarrow{V}]}_{\mathcal{V}}]}_{\mathcal{V}}={[{\overrightarrow{U}}_{\lambda },\overrightarrow{V}]}_{\lambda }+{[\overrightarrow{U},{[\overrightarrow{\mathsf{\Lambda }},\overrightarrow{V}]}_{\mathcal{V}}]}_{\mathcal{V}}.$$

(42)

Again from (40), we have

$${\overrightarrow{V}}_{\lambda ,x}={\overrightarrow{V}}_{x,\lambda }={[{\overrightarrow{U}}_{\lambda },\overrightarrow{V}]}_{\mathcal{V}}+{[\overrightarrow{U},{\overrightarrow{V}}_{\lambda }]}_{\mathcal{V}}.$$

Hence, ${[\overrightarrow{\mathsf{\Lambda }},\overrightarrow{V}]}_{\mathcal{V}}-{\overrightarrow{V}}_{\lambda }=:\overrightarrow{P}$ is a solution to equation

$${\overrightarrow{V}}_x={[\overrightarrow{U},\overrightarrow{V}]}_{\mathcal{V}}.$$

(43)

As we all know, if ${\overrightarrow{V}}_1,{\overrightarrow{V}}_2$ are solutions of (43), then there exists a constant $\gamma$ so that

$${\overrightarrow{V}}_1=\gamma {\overrightarrow{V}}_2.$$

(44)

Therefore, using (44) and $rank(\overrightarrow{P})=rank({\overrightarrow{V}}_{\lambda })=rank({\displaystyle \frac{1}{\lambda }}\overrightarrow{V}),$ we have

$$\overrightarrow{P}={[\overrightarrow{\mathsf{\Lambda }},\overrightarrow{V}]}_{\mathcal{V}}-{\overrightarrow{V}}_{\lambda }={\displaystyle \frac{\gamma }{\lambda }}\overrightarrow{V}.$$

(45)

Since ${\displaystyle \frac{1}{\lambda }}\overrightarrow{V}$ is also a solution of (44), we find that (41) can be written as follows

$${\displaystyle \frac{\delta }{\delta u}}{\left\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda }\right\}}_{\mathcal{V}}={\displaystyle \frac{\partial }{\partial \lambda }}{\left\{\overrightarrow{V},{\overrightarrow{U}}_u\right\}}_{\mathcal{V}}+({\lambda }^{-\gamma }{\displaystyle \frac{\partial }{\partial \lambda }}{\lambda }^{\gamma }){\left\{\overrightarrow{V},{\overrightarrow{U}}_u\right\}}_{\mathcal{V}}={\lambda }^{-\gamma }{\displaystyle \frac{\partial }{\partial \lambda }}{\lambda }^{\gamma }{\left\{\overrightarrow{V},{\overrightarrow{U}}_u\right\}}_{\mathcal{V}}.$$

(46)

Summarizing the above discussion, we obtain the following consequence.

Theorem 3

(a vector-product trace identity). Let $\mathcal{S}$ be a Lie algebra with row vectors, $\overrightarrow{U}=\overrightarrow{U}(u,\lambda )\in \mathcal{S}$ be homogeneous in rank and ${\left\{·,·\right\}}_{\mathcal{V}}$ denote a non-degenerate symmetric form invariant under the Lie algebra $\mathcal{S}$. Then we have the following vector-product trace identity

$${\displaystyle \frac{\delta }{\delta u}}{\left\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda }\right\}}_{\mathcal{V}}={\lambda }^{-\gamma }{\displaystyle \frac{\partial }{\partial \lambda }}{\lambda }^{\gamma }{\left\{\overrightarrow{V},{\overrightarrow{U}}_u\right\}}_{\mathcal{V}}.$$

(47)

Proof. 

For $\overrightarrow{a},\overrightarrow{b}\in G,$ $\overrightarrow{a}=(a_1,a_2,a_3,a_4,a_5,a_6),b=(b_1,b_2,b_3,b_4,b_5,b_6),$ the (35) is extended as the following form

$${[\overrightarrow{a},\overrightarrow{b}]}_{\mathcal{V}}=\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{a}R(b){\overrightarrow{E}}^T,$$

(48)

where

$$\overrightarrow{E}=(E_1,E_2,\dots ,E_6),R(b)=\left(\begin{array}{cccccc}0& b_3& -b_2& b_5& -b_4& 0\\ -b_3& 0& b_1& 0& b_6& -b_5\\ b_2& -b_1& 0& -b_6& 0& b_4\\ -b_5& 0& b_6& 0& b_1& -b_3\\ b_4& -b_6& 0& -b_1& 0& b_2\\ 0& b_5& -b_4& b_3& -b_2& 0\end{array}\right),$$

which satisfies (36). Solving (36) for F using software Maple (latest v. 2024), we find that

$$F=\left(\begin{array}{cccccc}1& 0& 0& 0& 0& 1\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 0& 1& 0& 1& 0& 0\\ 0& 0& 1& 0& 1& 0\\ 1& 0& 0& 0& 0& 1\end{array}\right).$$

□

For the integrable hierarchy (20), it is easy to calculate that

$${\{\overrightarrow{V},{\overrightarrow{U}}_q\}}_{\mathcal{V}}=\lambda (\lambda b+d),{\{\overrightarrow{V},{\overrightarrow{U}}_r\}}_{\mathcal{V}}=\lambda (\lambda c+e),{\{\overrightarrow{V},{\overrightarrow{U}}_u\}}_{\mathcal{V}}=\lambda (\lambda b+d),$$

$${\{\overrightarrow{V},{\overrightarrow{U}}_v\}}_{\mathcal{V}}=\lambda (\lambda c+e),{\{\overrightarrow{V},{\overrightarrow{U}}_{\lambda }\}}_{\mathcal{V}}=2\lambda (\lambda (a+\lambda f)+q(\lambda b+d)+r(\lambda c+e).$$

Substituting the above results into (47) and comparing the coefficients of ${\lambda }^{-2n}$, we have

$${\displaystyle \frac{\delta }{\delta \overrightarrow{u}}}(2f_{n+1}+q{d}_n+r{e}_n)=(-2n+1+\gamma ){(d_n,e_n,b_n,c_n)}^T.$$

(49)

The isospectral integrable hierarchy (20) can be written as

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=\left(\begin{array}{c}A\\ B\\ -v{\partial }^{-1}q{c}_n+v{\partial }^{-1}r{b}_n+(\partial +v{\partial }^{-1}v)d_n-v{\partial }^{-1}u{e}_n\\ (\partial +u{\partial }^{-1}u)e_n+u{\partial }^{-1}q{c}_n-u{\partial }^{-1}r{b}_n-u{\partial }^{-1}v{d}_n\end{array}\right),$$

(50)

where

$$A=(\partial +r{\partial }^{-1}r+v{\partial }^{-1}v)b_n+(-r{\partial }^{-1}q-v{\partial }^{-1}u)c_n+(-r{\partial }^{-1}u-v{\partial }^{-1}q)e_n+(r{\partial }^{-1}v+v{\partial }^{-1}r)d_n,$$

$$B=-(q{\partial }^{-1}r+u{\partial }^{-1}v)b_n+(\partial +q{\partial }^{-1}q+u{\partial }^{-1}u)c_n-(q{\partial }^{-1}v+u{\partial }^{-1}r)d_n+(q{\partial }^{-1}u+u{\partial }^{-1}q)e_n.$$

Therefore, (50) can be written in a pseudo-Hamiltonian form using (49)

$${\left(\begin{array}{c}q\\ r\\ u\\ v\end{array}\right)}_t=J\left(\begin{array}{c}{d}_n\\ e_n\\ b_n\\ c_n\end{array}\right)=J{\displaystyle \frac{\delta H_n}{\delta \overrightarrow{u}}},$$

(51)

where

$$H_n={\displaystyle \frac{2f_{n+1}+q{d}_n+r{e}_n}{-2n+1+\gamma }},$$

$$J=\left(\begin{array}{cccc}r{\partial }^{-1}v+v{\partial }^{-1}r& -r{\partial }^{-1}u-v{\partial }^{-1}q& \partial +r{\partial }^{-1}r+v{\partial }^{-1}v& -r{\partial }^{-1}q-v{\partial }^{-1}u\\ -q{\partial }^{-1}v-u{\partial }^{-1}r& q{\partial }^{-1}u+u{\partial }^{-1}q& -q{\partial }^{-1}r-u{\partial }^{-1}v& \partial +q{\partial }^{-1}q+u{\partial }^{-1}u\\ \partial +v{\partial }^{-1}v& -v{\partial }^{-1}u& v{\partial }^{-1}r& -v{\partial }^{-1}q\\ -u{\partial }^{-1}v& \partial +u{\partial }^{-1}u& -u{\partial }^{-1}r& u{\partial }^{-1}q\end{array}\right).$$

5. Conclusions

In this paper, we have presented a novel vector-product Lie algebra constructed specifically for a reductive homogeneous space, which has provided a rich mathematical framework for exploring integrable systems. The construction of this Lie algebra has not only led to the definition of two distinct loop algebras but also facilitated the derivation of two integrable hierarchies of evolution equations through a novel form of the zero-curvature equation. These hierarchies encompass a diverse range of physically relevant equations, including the heat equation, a special diffusion equation, a general linear Schrödinger equation, and a nonlinear Schrödinger-type equation, highlighting the broad applicability of our approach.