Non-Autonomous Soliton Hierarchies

Abstract

A formalism for the systematic construction of integrable non-autonomous deformations of soliton hierarchies is presented. The theory is formulated as an initial value problem (IVP) for an associated Frobenius integrability condition on a Lie algebra. It is shown that this IVP has a formal solution, and within the framework of two particular subalgebras of the hereditary Lie algebra, the explicit forms of this formal solution are derived. Finally, this formalism is applied to the Korteveg-de Vries, dispersive water waves and Ablowitz–Kaup–Newell–Segur soliton hierarchies. The zero-curvature representations and Hamiltonian structures of the considered non-autonomous soliton hierarchies are also provided.

1. Introduction

In this article we present a systematic method of deforming commuting hierarchies of autonomous evolutionary flows, i.e., systems of evolutionary PDEs of the form

$$u_{t_n}=K_n\left[u\right],n=1,2,\dots ,$$

(1)

(where $u=u\left(x\right)={\left(u_1\left(x\right),\dots ,u_N\left(x\right)\right)}^T$ and where each $K_n\left[u\right]$ is some vector field depending on u and a finite number of its x-derivatives, but not explicitly on times (i.e., evolution parameters $t_i$), and such that

$$\left[K_m,K_n\right]=0,m,n=1,2,\dots ,$$

(2)

to the non-autonomous hierarchies of evolutionary flows

$$u_{t_n}={\mathbb{K}}_n\left(\left[u\right],x,t_1,\dots ,t_n\right)n=1,2,\dots ,$$

(3)

that satisfy the Frobenius integrability condition

$${\displaystyle \frac{\partial {\mathbb{K}}_n}{\partial t_m}}-{\displaystyle \frac{\partial {\mathbb{K}}_m}{\partial t_n}}+\left[{\mathbb{K}}_m,{\mathbb{K}}_n\right]=0,m,n=1,2,\dots .$$

(4)

In (1) and (3) $K_n$ and ${\mathbb{K}}_n$ are vector fields, depending on u and a finite number of its x-derivatives on some infinite-dimensional functional manifold, and $K_n$ does not depend explicitly on times $t_i$. Note that in (3) we assume triangular dependence of vector fields ${\mathbb{K}}_n$ on times $t_i$ for $1⩽i⩽n$, which consequently simplifies the Frobenius condition (4) to

$${\displaystyle \frac{\partial {\mathbb{K}}_n}{\partial t_m}}+\left[{\mathbb{K}}_m,{\mathbb{K}}_n\right]=0,m<n.$$

(5)

The condition (2) (which is nothing else besides the Frobenius integrability condition for the autonomous system (1)) means that the system (1) has a common, multi-time solution through each initial condition $u(x,0,0,\dots )=u_0\left(x\right)$. Likewise, the Frobenius condition (4) means that the system (3) has a common, multi-time solution through each initial condition $u(x,0,0,\dots )=u_0\left(x\right)$. If these compatibility conditions are not met it makes no sense to consider the systems in (1) or the systems in (3) as hierarchies; they are simply not compatible.

In order to highlight the main algebraic ingredients of our construction we first (Section 2) formulate our theory in a more general framework, as an initial-value problem (IVP) for $\mathcal{A}$-valued functions ${\mathbb{K}}_n$ satisfying (5), where $\mathcal{A}$ is a non-abelian Lie algebra. In Theorem 1 and Corollary 1 we show that this IVP has a formal solution. Next, we present solutions of the IVP for particular subalgebras of the hereditary Lie algebra [1,2,3] (Section 3) and then we apply these results to soliton hierarchies (Section 4). In Section 4 we also find the zero-curvature representations for the non-autonomous hierarchies (3) from zero-curvature representations of the corresponding autonomous hierarchies (1). We illustrate our method on three examples: KdV hierarchy (Section 5), dispersive water wave (DWW) hierarchy in the framework of [4,5] (Section 5) and Ablowitz–Kaup–Newell–Segur (AKNS) hierarchy [6] (Section 7). Finally, Section 8 contains a summary and some discussion.

We believe that the results presented in this article are important since the majority of research in the theory of integrable PDEs focuses on autonomous systems of type (1). To our best knowledge, the non-autonomous deformations (3) of soliton hierarchies have not been previously studied. The usual approach to non-autonomous soliton equations is to modify a single chosen soliton equation by assuming some time-dependence of one or more of its coefficients, see for example [7,8] or [9].

This article was inspired by previous results presented in [10,11,12], which we have considered polynomial for time deformations of autonomous Liouville integrable finite dimensional systems, i.e., systems of the form

$${\displaystyle \frac{dx}{d{t}_i}}=\pi d{h}_i,h_i=h_i\left(x\right),i=1,\dots ,n,$$

$${\left\{h_i,h_j\right\}}_{\pi }=0,i,j=1,\dots ,n,$$

on some $2n$-dimensional manifold equipped with a Poisson bivector $\pi$, to non-autonomous Frobenius integrable systems

$${\displaystyle \frac{dx}{d{t}_i}}=\pi d{H}_i,H_i=H_i(x,t_1,\dots ,t_n),i=1,\dots ,n,$$

$${\left\{H_i,H_j\right\}}_{\pi }+{\displaystyle \frac{\partial H_i}{\partial t_j}}-{\displaystyle \frac{\partial H_j}{\partial t_i}}=0,i,j=1,\dots ,n,$$

on the same manifold; here, x denotes points on this manifold. Another inspiration for this work was the article [13] in which the author constructed non-autonomous KdV hierarchies from Painlevé systems that were obtained as deformations of Stäckel separable systems. However, these hierarchies are only finite as the construction is dependent on the dimension n of the underlying Stäckel systems; increasing n led to a completely different (finite) hierarchy. This drawback is not present in the theory we develop in this article.

2. Frobenius Integrability Condition in Lie Algebras

In this section we present a formal solution to the Frobenius integrability condition (5) formulated as an initial-value problem (IVP) for finite or infinite sets $\left\{{\mathbb{K}}_0,{\mathbb{K}}_1,\dots \right\}$ of elements ${\mathbb{K}}_n$ that belong to a non-abelian Lie algebra $\mathcal{A}$ and such that each element ${\mathbb{K}}_n$ depends on (at most) $n+1$ real evolutionary parameters (times) $t_i$, so that

$${\mathbb{K}}_n={\mathbb{K}}_n(t_0,\dots ,t_n).$$

(6)

Thus, all elements ${\mathbb{K}}_n$ will be some $\mathcal{A}$-valued functions of a finite number of real parameters $t_i$. The word formal means in this context that we do not consider any convergence issues that may arise in the formulas presented in this section, such issues would depend on the properties of particular Lie algebras. However, in the next sections we will apply our theory to soliton hierarchies and then all the expressions appearing in this section will become convergent (in fact, finite); the $\mathcal{A}$-valued functions ${\mathbb{K}}_n$ will then become Frobenius integrable deformations of soliton hierarchies.

Definition 1.

We say that the set (finite or not) $\left\{{\mathbb{K}}_0,{\mathbb{K}}_1,\dots \right\}$ of $\mathcal{A}$-valued functions ${\mathbb{K}}_i={\mathbb{K}}_i(t_0,\dots ,t_i)$ satisfies the Frobenius integrability condition (in a triangular form) if

$${\displaystyle \frac{\partial {\mathbb{K}}_j}{\partial t_i}}+{ad}_{{\mathbb{K}}_i}{\mathbb{K}}_j=0,0⩽i<j.$$

(7)

Here and in what follows ${ad}_{{\mathbb{K}}_i}$ is the adjoint action in the Lie algebra $\mathcal{A}$ so that ${ad}_{{\mathbb{K}}_i}{\mathbb{K}}_j\equiv \left[{\mathbb{K}}_i,{\mathbb{K}}_j\right]$. Note that the triangular dependence of ${\mathbb{K}}_n$ on $t_i$ given in (6) means that (7) is the actual complete Frobenius condition

$${\displaystyle \frac{\partial {\mathbb{K}}_j}{\partial t_i}}-{\displaystyle \frac{\partial {\mathbb{K}}_i}{\partial t_j}}+\left[{\mathbb{K}}_i,{\mathbb{K}}_j\right]=0,0⩽i<j,$$

simply because (6) implies that ${\displaystyle \frac{\partial {\mathbb{K}}_i}{\partial t_j}}=0$ for $i<j$.

We begin our exposition by presenting a well-known fact.

Proposition 1.

Consider the following linear initial value problem

$${\displaystyle \frac{d\mathit{v}}{dt}}+\mathbb{A}\mathit{v}=0,\mathit{v}\left(0\right)={\mathit{v}}_0,$$

(8)

where $\mathbb{A}=\mathbb{A}\left(t\right)$ is some time-dependent linear operator acting in $\mathcal{A}$, $\mathit{v}=\mathit{v}\left(t\right)$ is an $\mathcal{A}$-valued function and where ${\mathit{v}}_0\in \mathcal{A}$. Then the formal solution of this problem is

$$\begin{array}{cc}\hfill \mathit{v}\left(t\right)& ={\mathit{v}}_0-{\partial }_t^{-1}\mathbb{A}{\mathit{v}}_0+{\partial }_t^{-1}\mathbb{A}{\partial }_t^{-1}\mathbb{A}{\mathit{v}}_0-{\partial }_t^{-1}\mathbb{A}{\partial }_t^{-1}\mathbb{A}{\partial }_t^{-1}\mathbb{A}{\mathit{v}}_0+\dots \hfill \\ \hfill & \equiv {(1+{\partial }_t^{-1}\mathbb{A})}^{-1}{\mathit{v}}_0,\hfill \end{array}$$

where

$${\partial }_t^{-1}={\int }_0^{t}d{t}^{\prime }$$

is the formal linear operator of definite integration with respect to the time variable t so that ${\partial }_t^{-1}\mathbb{A}$ is a t-dependent operator on $\mathcal{A}$.

Proof. 

Indeed, given that for any linear operator $\mathbb{B}$ in $\mathcal{A}$ we have, formally,

$${(1+\mathbb{B})}^{-1}\equiv 1-\mathbb{B}+{\mathbb{B}}^2-{\mathbb{B}}^3+\dots =1-\mathbb{B}{(1+\mathbb{B})}^{-1},$$

(9)

then, taking $\mathbb{B}={\partial }_t^{-1}\mathbb{A}$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \frac{d\mathit{v}\left(t\right)}{dt}}& ={\partial }_t{(1+{\partial }_t^{-1}\mathbb{A})}^{-1}{\mathit{v}}_0={\partial }_t({\mathit{v}}_0-{\partial }_t^{-1}\mathbb{A}{(1+{\partial }_t^{-1}\mathbb{A})}^{-1}{\mathit{v}}_0)\hfill \\ \hfill & =-\mathbb{A}{(1+{\partial }_t^{-1}\mathbb{A})}^{-1}{\mathit{v}}_0=-\mathbb{A}\mathit{v}\left(t\right).\hfill \end{array}$$

□

If case $\mathbb{A}$ does not depend on t, we obtain the well-known solution of the IVP (8) in the form of the exponential of $\mathbb{A}$:

$${\displaystyle \frac{d\mathbb{A}}{dt}}=0\Rightarrow \mathit{v}\left(t\right)={(1+{\partial }_t^{-1}\mathbb{A})}^{-1}{\mathit{v}}_0=exp(-t\mathbb{A}){\mathit{v}}_0.$$

(10)

We will now generalize this result to the case of a system of linear time-dependent equations. As notified above, we will work with $\mathcal{A}$-valued functions of some parameters $t_i$, which we will often call times.

Theorem 1.

Suppose that n $\mathcal{A}$-valued functions ${\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1}$ satisfy the following conditions

$$\begin{array}{cc}\hfill {\mathbb{K}}_i={\mathbb{K}}_i(t_0,\dots ,t_i),& i=0,1,\dots ,n-1,\hfill \\ \hfill {\displaystyle \frac{\partial {\mathbb{K}}_j}{\partial t_i}}+{ad}_{{\mathbb{K}}_i}{\mathbb{K}}_j& =0,0⩽i<j<n.\hfill \end{array}$$

(11)

Then the initial value problem

$$\begin{array}{cc}\hfill {\displaystyle \frac{\partial \mathbb{K}}{\partial t_i}}+{ad}_{{\mathbb{K}}_i}\mathbb{K}& =0,i=0,\dots ,n-1,\hfill \\ \hfill \mathbb{K}(0,\dots ,0)& =\overline{\mathbb{K}}\in \mathcal{A}.\hfill \end{array}$$

(12)

for the $\mathcal{A}$-valued function $\mathbb{K}=\mathbb{K}(t_0,\dots ,t_{n-1})$ has the formal solution

$$\mathbb{K}(t_0,\dots ,t_{n-1})={(1+{\partial }_{t_{n-1}}^{-1}{ad}_{{\mathbb{K}}_{n-1}})}^{-1}\cdots {(1+{\partial }_{t_1}^{-1}{ad}_{{\mathbb{K}}_1})}^{-1}{(1+{\partial }_{t_0}^{-1}{ad}_{{\mathbb{K}}_0})}^{-1}\overline{\mathbb{K}},$$

(13)

where

$${\partial }_{t_i}^{-1}f\left(t_i\right)\equiv {\int }_0^{t_i}f\left(t\right)dt.$$

Assuming that $\overline{\mathbb{K}}$ in (12) depends on an additional evolution parameter $t_n$ we arrive at the following corollary.

Corollary 1.

Suppose that n $\mathcal{A}$-valued functions ${\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1}$ satisfy the conditions (11). Suppose also that an $\mathcal{A}$-valued function ${\mathbb{K}}_n=$${\mathbb{K}}_n(t_0,\dots ,t_n)$ satisfies the following initial value problem

$$\begin{array}{cc}{\displaystyle \frac{\partial {\mathbb{K}}_n}{\partial t_i}}+{ad}_{{\mathbb{K}}_i}{\mathbb{K}}_n=0,\hfill & 0⩽i<n,\hfill \\ {\mathbb{K}}_n(\underset{n}{\underbrace{0,\dots ,0}},t_n)={\overline{\mathbb{K}}}_n\left(t_n\right),\hfill & \end{array}$$

(14)

where ${\overline{\mathbb{K}}}_n\left(t_n\right)$ is an $\mathcal{A}$-valued function of $t_n$. Then, the IVP (14) has the unique (formal) solution

$${\mathbb{K}}_n(t_0,\dots ,t_n)={(1+{\partial }_{t_{n-1}}^{-1}{ad}_{{\mathbb{K}}_{n-1}})}^{-1}\cdots {(1+{\partial }_{t_1}^{-1}{ad}_{{\mathbb{K}}_1})}^{-1}{(1+{\partial }_{t_0}^{-1}{ad}_{{\mathbb{K}}_0})}^{-1}{\overline{\mathbb{K}}}_n\left(t_n\right).$$

(15)

For the proof of Theorem 1, see Appendix A. Let us now comment this theorem and the corollary that follows. The conditions (11) means that the $\mathcal{A}$-valued functions ${\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1}$ satisfy the Frobenius integrability condition (7). Further in the article the $\mathcal{A}$-valued functions ${\mathbb{K}}_i$ are vector fields on some finite or infinite-dimensional manifold $\mathcal{M}$, so that ${\mathbb{K}}_i={\mathbb{K}}_i(t_0,\dots ,t_i,u)$ with $u\in \mathcal{M}$, and the Frobenius condition (7) in turn implies that the corresponding non-autonomous dynamical systems

$${\displaystyle \frac{du}{d{t}_i}}={\mathbb{K}}_i(t_0,\dots ,t_i,u),i=0,\dots ,n-1,$$

poses (at least locally) a common, multi-time solution $u=u(t_0,\dots ,t_{n-1};u_0)$ through each point $u_0$ of the manifold $\mathcal{M}$. Theorem 1 then yields us a tool to add one more vector field $\mathbb{K}$ to the set $\left\{{\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1}\right\}$ so that the set $\left\{{\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1},{\mathbb{K}}_n\equiv \mathbb{K}\right\}$ still satisfies the Frobenius integrability condition (7). If now the initial condition $\overline{\mathbb{K}}$ for (12) depends additionally on a new evolution parameter $t_n$, then the solution (15) in Corollary 1 provides us with the set $\left\{{\mathbb{K}}_0,\dots ,{\mathbb{K}}_{n-1},{\mathbb{K}}_n\equiv \mathbb{K}\right\}$ of non-autonomous vector fields, depending now on one more evolution parameter $t_n$, and such that they satisfy the Frobenius condition (7) for $0⩽i<j⩽n$.

Thus, Corollary 1 is a useful tool to recursively construct sets of non-autonomous vector fields, triangularly depending on $t_i$ and satisfying the Frobenius condition. This corollary will be used in the following sections to produce non-autonomous Frobenius integrable deformations of various soliton hierarchies.

Let us conclude this section with a specification of Theorem 1 to the situation when ${\mathbb{K}}_i={\mathbb{K}}_i(t_0,\dots ,t_{i-1})$ only (in such situation we say that neither of ${\mathbb{K}}_i$ depends on its own evolution parameter $t_i$). In this particular situation, in accordance with (10),

$${(1+{\partial }_{t_i}^{-1}{ad}_{{\mathbb{K}}_i})}^{-1}=exp(-t_i{ad}_{{\mathbb{K}}_i}),$$

and thus we have the following remark.

Remark 1.

If ${\mathbb{K}}_i={\mathbb{K}}_i(t_0,\dots ,t_{i-1})$ for $i=0,\dots ,n-1$ and ${\overline{\mathbb{K}}}_n\left(t_n\right)={\overline{\mathbb{K}}}_n$, i.e., if neither of ${\overline{\mathbb{K}}}_i$ depends on its own evolution parameter $t_i$, then the solution of the IVP (14) takes the form

$${\mathbb{K}}_n(t_0,\dots ,t_{n-1})=exp(-t_{n-1}{ad}_{{\mathbb{K}}_{n-1}})\cdots exp(-t_1{ad}_{{\mathbb{K}}_1})exp(-t_0{ad}_{{\mathbb{K}}_0}){\overline{\mathbb{K}}}_n.$$

(16)

Naturally, in this case

$${\mathbb{K}}_n(0,\dots ,0)={\overline{\mathbb{K}}}_n.$$

3. Frobenius Integrability in Hereditary Algebras

The Formulas (15) and (16) cannot be of any practical use until we have some method to compute the expressions on their right hand sides. To achieve this we will assume that our non-abelian Lie algebra $\mathcal{A}$ is a semi-product of a commutative algebra and the Witt algebra (a centerless Virasoro symmetry algebra), i.e., $\mathcal{A}$ is the so-called hereditary algebra [1,2,3]. Specifically, we assume that the hereditary algebra $\mathcal{A}$ is generated by the elements $K_n\in$$\mathcal{A}$, $n=1,2,\dots$, and ${\sigma }_m\in$$\mathcal{A}$, $m=-1,0,1,\dots$, such that

$$[K_n,K_m]=0,[{\sigma }_n,K_m]=(\alpha m+{\rho }^{\prime }-1)K_{n+m},[{\sigma }_n,{\sigma }_m]=\alpha (m-n){\sigma }_{n+m},$$

where ${\rho }^{\prime },\alpha \in \mathbb{R}$ and $\alpha \ne 0$. This choice is motivated by the fact that hereditary algebras of soliton hierarchies, that we will consider in this article, have this structure. By a simple rescaling of all ${\sigma }_n$ we can always set $\alpha =1$. Thus, in this article we will consider the hereditary algebra $\mathcal{A}$ with the generators $K_n$ and ${\sigma }_m$ satisfying the commutation relations

$$\begin{array}{cc}\hfill [K_n,K_m]& =0,m,n=1,2,\dots ,\hfill \\ \hfill [{\sigma }_n,K_m]& ={\kappa }_m{K}_{n+m},n=-1,0,1,\dots ,m=1,2,\dots ,\hfill \\ \hfill [{\sigma }_n,{\sigma }_m]& =(m-n){\sigma }_{n+m},m,n=-1,0,1,\dots ,\hfill \end{array}$$

(17)

(so that $\rho -1\equiv {\displaystyle \frac{1}{\alpha }}\left({\rho }^{\prime }-1\right)$) and where we denote $K_0\equiv 0$. Here and further on we use the notation

$${\kappa }_m\equiv \rho +m-1.$$

(18)

We can now choose our initial conditions ${\overline{\mathbb{K}}}_n\left(t_n\right)$ in (14) as some very particular deformations (time-dependent linear combinations) of the above generators of $\mathcal{A}$. It turns out that Formulas (15) and (16) attain a compact, finite form in two particular cases:

• ${\overline{\mathbb{K}}}_n\left(t_n\right)$ is a ${\mathcal{A}}_{-1}$-valued function for all n;
• ${\overline{\mathbb{K}}}_n\left(t_n\right)$ is a ${\mathcal{A}}_0$-valued function for all n.

where

$${\mathcal{A}}_{-1}:=span\left\{{\sigma }_{-1},K_1,K_2,\dots \right\}\mathrm{and}{\mathcal{A}}_0:=span\left\{{\sigma }_0,K_1,K_2,\dots \right\}$$

are two particular subalgebras of the hereditary algebra $\mathcal{A}$. They are exceptional in the sense that for any n the sets ${\mathcal{A}}_{\epsilon }^{\left(n\right)}\equiv span\left\{{\sigma }_i,K_1,K_2,\dots ,K_n\right\}$, where $\epsilon =-1$ or $\epsilon =0$ are finite dimensional subalgebras of $\mathcal{A}$. On the level of the algebras ${\mathcal{A}}_{\epsilon }$ we can interpret the solutions ${\mathbb{K}}_n$ of IVP (14) as leading to construction of new deformed bases of ${\mathcal{A}}_{\epsilon }$ satisfying the Frobenius integrability condition (7).

3.1. Frobenius Integrability in the Hereditary Subalgebra ${\mathcal{A}}_{-1}$

We begin with the first case, i.e., when ${\overline{\mathbb{K}}}_n\left(t_n\right)$ is a ${\mathcal{A}}_{-1}$-valued function.

Theorem 2.

Consider the IVP (14) with the initial conditions (14) in the form

$${\mathbb{K}}_n(0,\dots ,0,t_n)={\overline{\mathbb{K}}}_n\left(t_n\right)\equiv {\sigma }_{-1}+\sum _{i=1}^n{\mathbf{a}}_{n,i}\left(t_n\right)K_i,n=0,1,...,$$

(19)

where ${\mathbf{a}}_{n,i}\left(t_n\right)$ are arbitrary differentiable functions (thus ${\overline{\mathbb{K}}}_n\left(t_n\right)$ are ${\mathcal{A}}_{-1}$-valued functions). Then, the solution (15) of the IVP (14) is unique and attains the form

$${\mathbb{K}}_n={\sigma }_{-1}+\sum _{i=1}^n{\mathbf{u}}_{n,i}(t_0,\dots ,t_n)K_i,$$

(20)

where

$${\mathbf{u}}_{n,1}=-{\mathbf{c}}_n,{\mathbf{u}}_{n,i}={\displaystyle \frac{{(-1)}^i}{\left[{\kappa }_i\right]!}}{\partial }_{t_0}^{i-1}{\mathbf{c}}_n,i=2,\dots ,n,$$

(21)

with

$$\begin{array}{cc}\hfill {\mathbf{c}}_n(t_0,\dots ,t_n)& \equiv \sum _{m=2}^{n-1}\sum _{r=2}^m\sum _{s=1}^{r-1}{\displaystyle \frac{{(-1)}^{r-1}\left[{\kappa }_r\right]!}{(r-s-1)!}}{\left({\tau }_{m-1}\right)}^{r-s-1}{\left({\partial }_{t_m}^{-1}\right)}^s{\mathbf{a}}_{m,r}\left(t_m\right)\hfill \\ \hfill & +\sum _{r=1}^n{\displaystyle \frac{{(-1)}^r\left[{\kappa }_r\right]!}{(r-1)!}}{\left({\tau }_{n-1}\right)}^{r-1}{\mathbf{a}}_{n,r}\left(t_n\right),n\in \mathbb{N}.\hfill \end{array}$$

(22)

Here and in what follows we use the following shorthand notations:

$${\tau }_m=t_0+t_1+\dots +t_m,\left[{\kappa }_r\right]!={\kappa }_2{\kappa }_3\cdots {\kappa }_{r-1}{\kappa }_r,r>1,\left[{\kappa }_1\right]!=1,$$

(23)

and

$${\partial }_{t_m}^{-1}f\left(t_m\right)\equiv {\int }_0^{t_m}f\left(t\right)dt.$$

Notice that, the first term in (22) disappears for ${\mathbf{c}}_1$ and ${\mathbf{c}}_2$. Also notice that, after combining (21) with (22) and simplifying, one can see that ${\mathbf{u}}_{n,i}$ are always polynomial in variables ${\kappa }_m$. This means that there is no issue with division by zero if one of (18), for some particular $\rho$, is equal to zero.

Proof. 

Direct calculation yields

$${\mathbf{c}}_n(0,\dots ,0,t_n)=-{\mathbf{a}}_{n,1}\left(t_n\right).$$

Moreover,

$$\begin{array}{cc}\hfill {\left({\mathbf{c}}_n\right)}_{t_0}& =\sum _{m=2}^{n-1}\sum _{r=3}^m\sum _{s=1}^{r-2}{\displaystyle \frac{{(-1)}^{r-1}\left[{\kappa }_r\right]!}{(r-s-2)!}}{\left({\tau }_{m-1}\right)}^{r-s-2}{\left({\partial }_{t_m}^{-1}\right)}^s{\mathbf{a}}_{m,r}\left(t_m\right)\hfill \\ \hfill & +\sum _{r=2}^n{\displaystyle \frac{{(-1)}^r\left[{\kappa }_r\right]!}{(r-2)!}}{\left({\tau }_{n-1}\right)}^{r-2}{\mathbf{a}}_{n,r}\left(t_n\right),\hfill \end{array}$$

so that

$$\left({\partial }_{t_0}{\mathbf{c}}_n\right)(0,\dots ,0,t_n)={\kappa }_2{\mathbf{a}}_{n,2}\left(t_n\right).$$

Continuing differentiation of ${\mathbf{c}}_n$ with respect to $t_0$, we obtain

$$\left({\partial }_{t_0}^{i-1}{\mathbf{c}}_n\right)(0,\dots ,0,t_n)={(-1)}^i\left[{\kappa }_i\right]!{\mathbf{a}}_{n,i}\left(t_n\right),i=2,3,\dots ,n.$$

That means that ${\mathbb{K}}_n(0,\dots ,0,t_n)={\overline{\mathbb{K}}}_n\left(t_n\right)$ and thus (20) satisfies (for each n) the initial conditions (19). Furthermore, (20) satisfies (14) if and only if

$$\begin{array}{cc}\hfill {\left({\mathbf{u}}_{n,n}\right)}_{t_j}& =0,\hfill \\ \hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}+{\kappa }_i{\mathbf{u}}_{n,i+1}& =0,j⩽i⩽n-1,\hfill \\ \hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}+{\kappa }_i\left({\mathbf{u}}_{n,i+1}-{\mathbf{u}}_{j,i+1}\right)& =0,1⩽i⩽j-1,\hfill \end{array}$$

(24)

see Appendix B with $ϵ=-1$. In Appendix C it is shown that

$${\left({\mathbf{c}}_n\right)}_{t_j}\equiv {\left({\mathbf{c}}_n\right)}_{t_0}-{\left({\mathbf{c}}_j\right)}_{t_0},1⩽j⩽n-1,$$

(25)

and hence

$${\partial }_{t_0}^{i-1}{\left({\mathbf{c}}_n\right)}_{t_j}={\partial }_{t_0}^i{\mathbf{c}}_n-{\partial }_{t_0}^i{\mathbf{c}}_j,1⩽i⩽n-1,$$

while ${\partial }_{t_0}^i{\mathbf{c}}_j=0$ for $i⩾j$. These properties and the fact that $\left[{\kappa }_{i+1}\right]!\equiv \left[{\kappa }_i\right]!{\kappa }_{i+1}$ imply that the conditions (24) are identically satisfied. So, all ${\mathbb{K}}_n$ given by (20) satisfy the IVP (14) with (14) being of the particular form (19). Since the condition (24), being linear equations, have a unique solution for any initial conditions, the solutions (20) and (15) must—for the chosen initial conditions—coincide. □

The first few non-autonomous vectors (20) for the general IVP (19) have the form

$$\begin{array}{cc}\hfill {\mathbb{K}}_0& ={\sigma }_{-1},\hfill \\ \hfill {\mathbb{K}}_1& ={\sigma }_{-1}+{\mathbf{a}}_{1,1}\left(t_1\right)K_1,\hfill \\ \hfill {\mathbb{K}}_2& ={\sigma }_{-1}+\left[{\mathbf{a}}_{2,1}\left(t_2\right)-{\kappa }_2{\tau }_1{\mathbf{a}}_{2,2}\left(t_2\right)\right]K_1+{\mathbf{a}}_{2,2}\left(t_2\right)K_2,\hfill \\ \hfill {\mathbb{K}}_3& ={\sigma }_{-1}+\left[{\kappa }_2{\partial }_{t_2}^{-1}{\mathbf{a}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,1}\left(t_3\right)-{\kappa }_2{\tau }_2{\mathbf{a}}_{3,2}\left(t_3\right)+{\displaystyle \frac{1}{2}}{\kappa }_2{\kappa }_3{\tau }_2^2{\mathbf{a}}_{3,3}\left(t_3\right)\right]K_1\hfill \\ \hfill & +\left[{\mathbf{a}}_{3,2}\left(t_3\right)-{\kappa }_3{\tau }_2{\mathbf{a}}_{3,3}\left(t_3\right)\right]K_2+{\mathbf{a}}_{3,3}\left(t_3\right)K_3,\hfill \\ \hfill {\mathbb{K}}_4& ={\sigma }_{-1}+[{\kappa }_2{\partial }_{t_2}^{-1}{\mathbf{a}}_{2,2}\left(t_2\right)+{\kappa }_2{\partial }_{t_3}^{-1}{\mathbf{a}}_{3,2}\left(t_3\right)-{\kappa }_2{\kappa }_3{\tau }_2{\partial }_{t_3}^{-1}{\mathbf{a}}_{3,3}\left(t_3\right)-{\kappa }_2{\kappa }_3{\partial }_{t_3}^{-2}{\mathbf{a}}_{3,3}\left(t_3\right)\hfill \\ \hfill & +{\mathbf{a}}_{4,1}\left(t_4\right)-{\kappa }_2{\tau }_3{\mathbf{a}}_{4,2}\left(t_4\right)+{\displaystyle \frac{1}{2}}{\kappa }_2{\kappa }_3{\tau }_3^2{\mathbf{a}}_{4,3}\left(t_4\right)-{\displaystyle \frac{1}{6}}{\kappa }_2{\kappa }_3{\kappa }_4{\tau }_3^3{\mathbf{a}}_{4,4}\left(t_4\right)]K_1\hfill \\ \hfill & +\left[{\kappa }_3{\partial }_{t_3}^{-1}{\mathbf{a}}_{3,3}\left(t_3\right)+{\mathbf{a}}_{4,2}\left(t_4\right)-{\kappa }_3{\tau }_3{\mathbf{a}}_{4,3}\left(t_4\right)+{\displaystyle \frac{1}{2}}{\kappa }_3{\kappa }_4{\tau }_3^2{\mathbf{a}}_{4,4}\left(t_4\right)\right]K_2\hfill \\ \hfill & +\left[{\mathbf{a}}_{4,3}\left(t_4\right)-{\kappa }_4{\tau }_3{\mathbf{a}}_{4,4}\left(t_4\right)\right]K_3+{\mathbf{a}}_{4,4}\left(t_4\right)K_4,\hfill \end{array}$$

where ${\kappa }_m$ are defined by (18) and ${\tau }_m$ by (23).

Example 1.

Suppose that the functions ${\mathbf{a}}_{n,i}\left(t_n\right)$ in the initial conditions (19) are given by the simple choice ${\mathbf{a}}_{n,i}\left(t_n\right)=0$ for $i=1,\dots ,n-1$, and ${\mathbf{a}}_{n,n}\left(t_n\right)=t_n$. Then the initial conditions (19) have the form

$${\overline{\mathbb{K}}}_0\left(t_0\right)\equiv {\sigma }_{-1},{\overline{\mathbb{K}}}_n\left(t_n\right)\equiv {\sigma }_{-1}+t_n{K}_i,n=1,2,...,$$

(26)

and the solution of the IVP (14) is given by (20) and (21) with ${\mathbf{c}}_n$ in the form

$${\mathbf{c}}_n(t_0,\dots ,t_n)=\sum _{m=2}^{n-1}\sum _{s=2}^m{\displaystyle \frac{{(-1)}^{m-1}\left[{\kappa }_m\right]!}{(m-s)!s!}}{\left({\tau }_{m-1}\right)}^{m-s}{t}_m^s+{\displaystyle \frac{{(-1)}^n\left[{\kappa }_n\right]!}{(n-1)!}}{\left({\tau }_{n-1}\right)}^{n-1}{t}_n.$$

(27)

Then, using (21) with (27), we obtain the first ${\mathbb{K}}_n$ in (20) in the explicit form

$$\begin{array}{cc}\hfill {\mathbb{K}}_0& ={\sigma }_{-1},\hfill \\ \hfill {\mathbb{K}}_1& ={\sigma }_{-1}+t_1{K}_1,\hfill \\ \hfill {\mathbb{K}}_2& ={\sigma }_{-1}-{\kappa }_2(t_0+t_1)t_2{K}_1+t_2{K}_2,\hfill \\ \hfill {\mathbb{K}}_3& ={\sigma }_{-1}+{\displaystyle \frac{1}{2}}{\kappa }_2\left[t_2^2+{\kappa }_3{(t_0+t_1+t_2)}^2{t}_3\right]K_1-{\kappa }_3(t_0+t_1+t_2)t_3{K}_2+t_3{K}_3,\hfill \\ \hfill {\mathbb{K}}_4& ={\sigma }_{-1}+{\displaystyle \frac{1}{2}}{\kappa }_2\left[t_2^2-{\displaystyle \frac{1}{3}}{\kappa }_3(3t_0+3t_1+3t_2+t_3)t_3^2-{\displaystyle \frac{1}{3}}{\kappa }_3{\kappa }_4{\left(t_0+t_1+t_2+t_3\right)}^3{t}_4\right]K_1\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\kappa }_3\left[t_3^2+{\kappa }_4{(t_0+t_1+t_2+t_3)}^2{t}_4\right]K_2-{\kappa }_4(t_0+t_1+t_2+t_3)t_4{K}_3+t_4{K}_4.\hfill \end{array}$$

3.2. Frobenius Integrability in the Hereditary Subalgebra ${\mathcal{A}}_0$

The second possibility arises when the initial conditions are given by ${\mathcal{A}}_0$-valued functions.

Theorem 3.

Consider the IVP (14) with the initial conditions (14) in the form

$${\mathbb{K}}_n(0,\dots ,0,t_n)={\overline{\mathbb{K}}}_n\left(t_n\right)\equiv {\sigma }_0+\sum _{i=1}^n{\mathbf{a}}_{n,i}\left(t_n\right)K_i,n=0,1,2,...,$$

(28)

where ${\mathbf{a}}_{n,i}\left(t_n\right)$ are arbitrary differentiable functions (thus ${\overline{\mathbb{K}}}_n\left(t_n\right)$ is an ${\mathcal{A}}_0$-valued function). Then, the solution (15) is unique and attains the form

$${\mathbb{K}}_n={\sigma }_0+\sum _{i=1}^n{\mathbf{u}}_{n,i}(t_0,\dots ,t_n)K_i,$$

(29)

where

$${\mathbf{u}}_{n,i}(t_0,\dots ,t_n)=\sum _{r=i}^{n-1}{\mathbf{c}}_{r,i}\left(t_r\right)e^{-{\kappa }_i{\tau }_{r-1}}+{\mathbf{a}}_{n,i}\left(t_n\right)e^{-{\kappa }_i{\tau }_{n-1}},$$

(30)

${\tau }_m$ are again given by (23) and where ${\mathbf{c}}_{r,i}\left(t_r\right)$ are functions that satisfy non-homogeneous linear IVPs

$${\mathbf{c}}_{r,i}^{\prime }\left(t_r\right)+{\kappa }_i{\mathbf{c}}_{r,i}\left(t_r\right)={\kappa }_i{\mathbf{a}}_{r,i}\left(t_r\right),{\mathbf{c}}_{r,i}\left(0\right)=0.$$

(31)

Note that the solution of (31) is

$${\mathbf{c}}_{r,i}\left(t_r\right)={\kappa }_i{e}^{-{\kappa }_i{t}_r}{\int }_0^{t_r}{\mathbf{a}}_{r,i}\left(t\right)e^{{\kappa }_{i}t}dt.$$

(32)

In particular, for the choice ${\mathbf{a}}_{r,i}\left(t_i\right)=t_i^m$, $m=0,1,\dots$,

$${\mathbf{c}}_{r,i}\left(t_r\right)=m!\left(\sum _{k=0}^m{\displaystyle \frac{{(-{\kappa }_i)}^{k-m}}{k!}}{t}_r^k-e^{-{\kappa }_i{t}_r}\right),{\kappa }_i\ne 0.$$

(33)

Proof. 

Fix $n\in \mathbb{N}$ and assume that the solution ${\mathbb{K}}_r$ given by (29) satisfy the IVP (14) for all $r<n$. Clearly, ${\mathbb{K}}_n(0,\dots ,0,t_n)={\overline{\mathbb{K}}}_n\left(t_n\right)$ so (29) satisfy (for each n) the initial conditions (28). Furthermore, (29) satisfies (14) if and only if

$$\begin{array}{cc}\hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}+{\kappa }_i{\mathbf{u}}_{n,i}& =0,j+1⩽i⩽n,\hfill \\ \hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}+{\kappa }_i\left({\mathbf{u}}_{n,i}-{\mathbf{u}}_{j,i}\right)& =0,1⩽i⩽j.\hfill \end{array}$$

(34)

For proof of (34) see Appendix B with $ϵ=0$. Note that the equations (34) are linear and thus poss solutions for all $t_j$. Consider (30), then for $j+1⩽i⩽n$ we have

$$\begin{array}{cc}\hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}& =-{\kappa }_i\sum _{r=i}^{n-1}{\mathbf{c}}_{r,i}\left(t_r\right)e^{-{\kappa }_i{\tau }_{r-1}}-{\kappa }_i{\mathbf{a}}_{n,i}\left(t_n\right)e^{-{\kappa }_i{\tau }_{n-1}}\hfill \\ \hfill & \equiv -{\kappa }_i{\mathbf{u}}_{n,i},\hfill \end{array}$$

so that the first part of (34) is identically true, while for $1⩽i⩽j$

$$\begin{array}{cc}\hfill {\left({\mathbf{u}}_{n,i}\right)}_{t_j}& ={\mathbf{c}}_{j,i}^{\prime }\left(t_j\right)e^{-{\kappa }_i{\tau }_{j-1}}-{\kappa }_i\sum _{r=j+1}^{n-1}{\mathbf{c}}_{r,i}\left(t_r\right)e^{-{\kappa }_i{\tau }_{r-1}}-{\kappa }_i{\mathbf{a}}_{n,i}\left(t_n\right)e^{-{\kappa }_i{\tau }_{n-1}}\hfill \end{array}$$

and thus

$${\left({\mathbf{u}}_{n,i}\right)}_{t_j}+{\kappa }_i\left({\mathbf{u}}_{n,i}-{\mathbf{u}}_{j,i}\right)=\left({\mathbf{c}}_{j,i}^{\prime }\left(t_j\right)+{\kappa }_i\left[{\mathbf{c}}_{j,i}\left(t_j\right)-{\mathbf{a}}_{j,i}\left(t_j\right)\right]\right)e^{-{\kappa }_i{\tau }_{j-1}},$$

so that the second part of (34) is satisfied provided that the differential equations (31) hold. As a result, all (29) satisfy the IVP (14) with (14) being of the particular form (28) and so, by uniqueness of solutions of (34) for any initial conditions, the solutions (29) and (15) must—for the chosen initial conditions—coincide. □

The first few non-autonomous vectors (29) for the general IVP (28) have the form

$$\begin{array}{cc}\hfill {\mathbb{K}}_0& ={\sigma }_0,\hfill \\ \hfill {\mathbb{K}}_1& ={\sigma }_0+{\mathbf{a}}_{1,1}\left(t_1\right)e^{-{\kappa }_1{\tau }_0}{K}_1,\hfill \\ \hfill {\mathbb{K}}_2& ={\sigma }_0+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-{\kappa }_1{\tau }_0}+{\mathbf{a}}_{2,1}\left(t_2\right)e^{-{\kappa }_1{\tau }_1}\right]K_1+{\mathbf{a}}_{2,2}\left(t_2\right)e^{-{\kappa }_2{\tau }_1}{K}_2,\hfill \\ \hfill {\mathbb{K}}_3& ={\sigma }_0+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-{\kappa }_1{\tau }_0}+{\mathbf{c}}_{2,1}\left(t_2\right)e^{-{\kappa }_1{\tau }_1}+{\mathbf{a}}_{3,1}\left(t_3\right)e^{-{\kappa }_1{\tau }_2}\right]K_1\hfill \\ \hfill & +\left[e^{-{\kappa }_2{\tau }_1}{\mathbf{c}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,2}\left(t_3\right)e^{-{\kappa }_2{\tau }_2}\right]K_2+{\mathbf{a}}_{3,3}\left(t_3\right)e^{-{\kappa }_3{\tau }_2}{K}_3,\hfill \\ \hfill {\mathbb{K}}_4& ={\sigma }_0+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-{\kappa }_1{\tau }_0}+{\mathbf{c}}_{2,1}\left(t_2\right)e^{-{\kappa }_1{\tau }_1}+{\mathbf{c}}_{3,1}\left(t_3\right)e^{-{\kappa }_1{\tau }_2}+{\mathbf{a}}_{4,1}\left(t_4\right)e^{-{\kappa }_1{\tau }_3}\right]K_1\hfill \\ \hfill & +\left[{\mathbf{c}}_{2,2}\left(t_2\right)e^{-{\kappa }_2{\tau }_1}+{\mathbf{c}}_{3,2}\left(t_3\right)e^{-{\kappa }_2{\tau }_2}+{\mathbf{a}}_{4,2}\left(t_4\right)e^{-{\kappa }_2{\tau }_3}\right]K_2\hfill \\ \hfill & +\left[{\mathbf{c}}_{3,3}\left(t_3\right)e^{-{\kappa }_3{\tau }_2}+{\mathbf{a}}_{4,3}\left(t_4\right)e^{-{\kappa }_3{\tau }_3}\right]K_3+{\mathbf{a}}_{4,4}\left(t_4\right)e^{-{\kappa }_4{\tau }_3}{K}_4,\hfill \end{array}$$

where ${\kappa }_m$ are defined by (18), ${\tau }_m$ by (23) and ${\mathbf{c}}_{r,i}\left(t_r\right)$ are given by (32).

Example 2.

Suppose that the functions ${\mathbf{a}}_{n,i}\left(t_n\right)$ in the initial conditions (28) are given by ${\mathbf{a}}_{n,i}\left(t_n\right)=0$ for $i=1,\dots ,n-1$ and ${\mathbf{a}}_{n,n}\left(t_n\right)=t_n$, so that

$${\overline{\mathbb{K}}}_0\left(t_0\right)\equiv {\sigma }_{-1},{\overline{\mathbb{K}}}_n\left(t_n\right)\equiv {\sigma }_{-1}+t_n{K}_i,n=1,2,....$$

(35)

In this case, by (32) or (33)

$${\mathbf{c}}_{n,i}\left(t_n\right)=0,i=1,\dots ,n-1,{\mathbf{c}}_{n,n}\left(t_n\right)=t_n+{\displaystyle \frac{1}{{\kappa }_n}}\left(e^{-{\kappa }_n{t}_n}-1\right),{\kappa }_n\ne 0.$$

(36)

Then, if ${\kappa }_i\ne 0$, the first ${\mathbb{K}}_n$ in (29) have the form

$$\begin{array}{cc}\hfill {\mathbb{K}}_0& ={\sigma }_0,\hfill \\ \hfill {\mathbb{K}}_1& ={\sigma }_0+t_1{e}^{-{\kappa }_1{t}_0}{K}_1,\hfill \\ \hfill {\mathbb{K}}_2& ={\sigma }_0+\left[\left(t_1-{\displaystyle \frac{1}{{\kappa }_1}}\right)e^{-{\kappa }_1{t}_0}+{\displaystyle \frac{1}{{\kappa }_1}}{e}^{-{\kappa }_1(t_0+t_1)}\right]K_1+t_2{e}^{-{\kappa }_2(t_0+t_1)}{K}_2,\hfill \\ \hfill {\mathbb{K}}_3& ={\sigma }_0+\left[\left(t_1-{\displaystyle \frac{1}{{\kappa }_1}}\right)e^{-{\kappa }_1{t}_0}+{\displaystyle \frac{1}{{\kappa }_1}}{e}^{-{\kappa }_1(t_0+t_1)}\right]K_1\hfill \\ \hfill & +\left[\left(t_2-{\displaystyle \frac{1}{{\kappa }_2}}\right)e^{-{\kappa }_2(t_0+t_1)}+{\displaystyle \frac{1}{{\kappa }_2}}{e}^{-{\kappa }_2(t_0+t_1+t_2)}\right]K_2+t_3{e}^{-{\kappa }_3(t_0+t_1+t_2)}{K}_3,\hfill \\ \hfill {\mathbb{K}}_4& ={\sigma }_0+\left[\left(t_1-{\displaystyle \frac{1}{{\kappa }_1}}\right)e^{-{\kappa }_1{t}_0}+{\displaystyle \frac{1}{{\kappa }_1}}{e}^{-{\kappa }_1(t_0+t_1)}\right]K_1\hfill \\ \hfill & +\left[\left(t_2-{\displaystyle \frac{1}{{\kappa }_2}}\right)e^{-{\kappa }_2(t_0+t_1)}+{\displaystyle \frac{1}{{\kappa }_2}}{e}^{-{\kappa }_2(t_0+t_1+t_2)}\right]K_2\hfill \\ \hfill & +\left[\left(t_3-{\displaystyle \frac{1}{{\kappa }_3}}\right)e^{-{\kappa }_3(t_0+t_1+t_2)}+{\displaystyle \frac{1}{{\kappa }_3}}{e}^{-{\kappa }_3(t_0+t_1+t_2+t_3)}\right]K_3+t_4{e}^{-{\kappa }_4(t_0+t_1+t_2+t_3)}{K}_4.\hfill \end{array}$$

4. Non-Autonomous Soliton Hierarchies and Their Deformed Isospectral Zero-Curvature Representations

From now on, we will assume that the algebraic objects like K, $\mathbb{K}$, or $\sigma$ are vector fields on some infinite-dimensional manifold $\mathcal{M}$ with the corresponding autonomous evolution equations $u_t=K\left[u\right]$ and non-autonomous evolution equations $u_t=\mathbb{K}\left(x,t,\left[u\right]\right)$, where the square bracket denotes the dependence on u and a finite number of derivatives of u w.r.t. x (so $\left[u\right]$ denotes jet coordinates on $\mathcal{M}$) and where $u={(u_1\left(x\right),\dots ,u_N\left(x\right))}^T$ denotes points on the manifold $\mathcal{M}$.

Therefore, consider an infinite hierarchy of mutually commuting autonomous evolutionary equations on $\mathcal{M}$ of the form

$$u_{s_n}=K_n\left[u\right],n=1,2\dots ,$$

(37)

as well as a hierarchy of non-commuting evolutionary equations on $\mathcal{M}$:

$$u_{{\tau }_n}={\sigma }_n\left[u\right],n=-1,0,1,\dots ,$$

(38)

such that the commutation relations (17) are valid. The members of the hierarchy (38) are called master symmetries for (37).

In this section we obtain—under Assumptions A1 and A2—the Frobenius integrable non-autonomous hierarchies $u_{t_n}={\mathbb{K}}_n\left[u\right]$, where ${\mathbb{K}}_n$ is of the form (20) or (29), from an appropriate deformation of an isospectral zero-curvature representation of (37) by a non-standard (see Remark 2) isospectral zero-curve representation of (38).

Assumption A1.

Suppose that the commuting hierarchy (37) can be obtained from the isospectral linear problem

$$\left\{\begin{array}{c}{\psi }_x=L\psi ,\hfill \\ {\psi }_{s_i}=U_i\psi ,\hfill & i=1,2,\dots ,\hfill \end{array}\right.$$

(39)

where $L=L(\lambda ,u)$, $U_i=U_i(\lambda ,\left[u\right])$ are some $2\times 2$ matrices depending on $\left[u\right]$ and the auxiliary variable λ, s.t. ${\lambda }_{s_i}=0$ for all i.

The subscript $s_i$ denotes the total derivative with respect to the evolution parameter $s_i$. The compatibility condition, that is the condition for existence of a common multi-time solution $\psi (x,s_1,s_2,\dots )$, for the problem (39) is

$${\left({\psi }_x\right)}_{s_i}={\left({\psi }_{s_i}\right)}_x,i=1,2,\dots ,$$

(40)

and

$${\left({\psi }_{s_i}\right)}_{s_j}={\left({\psi }_{s_j}\right)}_{s_i},i,j=1,2\dots .$$

(41)

The condition (40) is equivalent to

$$L_{s_i}=\left[U_i,L\right]+{\left(U_i\right)}_x\equiv L^{\prime }\left[K_i\right],i=1,2,\dots .$$

(42)

Throughout the whole article, ${\mathrm{\Omega }}^{\prime }\left[K\right]$ denotes the directional derivative of the tensor field $\mathrm{\Omega }$ along the vector field K on $\mathcal{M}$. The identity in (42) is the consequence of Assumption A1, while the condition (41) is equivalent to

$${\left(U_i\right)}_{s_j}-{\left(U_j\right)}_{s_i}+[U_i,U_j]=0,i,j=1,2,\dots .$$

(43)

Thus, Assumption A1 means that (42) is equivalent to the corresponding equation $u_{s_i}=K_i\left[u\right]$ in (37), i.e., (42) is an isospectral zero-curvature representation for (37). It also means that ${\psi }_{s_i}={\mathcal{L}}_{K_i}\psi$ where $\mathcal{L}$ is the Lie derivative on $\mathcal{M}$. Then, the Equation (43) guarantees that all $K_i$ commute, since

$${\left({\psi }_{s_i}\right)}_{s_j}-{\left({\psi }_{s_j}\right)}_{s_i}={\mathcal{L}}_{K_j}{\mathcal{L}}_{K_i}\psi -{\mathcal{L}}_{K_i}{\mathcal{L}}_{K_j}\psi ={\mathcal{L}}_{[K_j,K_i]}\psi ={\psi }^{\prime }\left[[K_j,K_i]\right]=0.$$

Note also that (43) can be written as

$$U_i^{\prime }\left[K_j\right]-U_j^{\prime }\left[K_i\right]+[U_i,U_j]=0,i,j=1,2,\dots .$$

(44)

Assumption A2.

Suppose also that the (non-commuting) hierarchy (38) of master symmetries can be obtained from the following deformed linear isospectral problem

$$\left\{\begin{array}{c}{\psi }_x=L\psi ,\hfill \\ {\psi }_{{\tau }_i}=V_i\psi -{\lambda }^{i+1}{\psi }_{\lambda },\hfill & i=-1,0,1\dots ,\hfill \end{array}\right.$$

(45)

where $L=L(\lambda ,u)$ is the same L as in (39) while $V_i=V_i(\lambda ,\left[u\right])$ are some matrices depending on $\left[u\right]$ and λ such that ${\lambda }_{{\tau }_i}=0$.

In (45) ${\psi }_{\lambda }\equiv {\displaystyle \frac{\partial \psi }{\partial \lambda }}$. Obviously, we cannot expect that (45) poses a common multi-time solution $\psi (x,{\tau }_{-1},{\tau }_0,{\tau }_1,\dots )$. Instead, the Assumption A2 means that (45) has, for each i, a solution $\psi (x,{\tau }_i)$ so that

$${\left({\psi }_x\right)}_{{\tau }_i}={\left({\psi }_{{\tau }_i}\right)}_x,i=-1,0,\dots ,$$

which is equivalent to

$$L_{{\tau }_i}=\left[V_i,L\right]+{\left(V_i\right)}_x-{\lambda }^{i+1}{L}_{\lambda }\equiv L^{\prime }\left[{\sigma }_i\right],i=-1,0,1,\dots ,$$

(46)

and the identity in (46) is the consequence of Assumption A2. This assumption means thus that each equation in (46) is equivalent with the corresponding equation $u_{{\tau }_i}={\sigma }_i\left[u\right]$ in (38). Since the fields ${\sigma }_i$ in (38) do not commute we clearly cannot expect that ${\left({\psi }_{{\tau }_i}\right)}_{{\tau }_j}={\left({\psi }_{{\tau }_j}\right)}_{{\tau }_i}$. Instead we have

$${\left({\psi }_{{\tau }_i}\right)}_{{\tau }_j}-{\left({\psi }_{{\tau }_j}\right)}_{{\tau }_i}={\psi }^{\prime }\left[[{\sigma }_j,{\sigma }_i]\right]=(i-j){\psi }^{\prime }\left[{\sigma }_{i+j}\right]=(i-j){\psi }_{{\tau }_{i+j}},i,j=-1,0,1,\dots ,$$

which is equivalent to

$${\left(V_i\right)}_{{\tau }_j}-{\left(V_j\right)}_{{\tau }_i}+[V_i,V_j]+{\lambda }^{j+1}{\left(V_i\right)}_{\lambda }-{\lambda }^{i+1}{\left(V_j\right)}_{\lambda }=(i-j)V_{i+j},i,j=-1,0,\dots ,$$

that is to

$$V_i^{\prime }\left[{\sigma }_j\right]-V_j^{\prime }\left[{\sigma }_i\right]+[V_i,V_j]+{\lambda }^{j+1}{\left(V_i\right)}_{\lambda }-{\lambda }^{i+1}{\left(V_j\right)}_{\lambda }=(i-j)V_{i+j},i,j=-1,0,\dots .$$

(47)

Remark 2.

Usually in the literature (see for example [3]), one constructs a zero-curvature representation for (38) from the non-isospectral problem

$$\left\{\begin{array}{c}{\psi }_x=L\psi ,\hfill \\ {\psi }_{{\tau }_i}=V_i\psi ,\hfill & i=1,2,\dots ,\hfill \end{array}\right.$$

with ${\lambda }_{{\tau }_i}={\lambda }^{i+1}$. The isospectral problem (45) is however equivalent (in the sense that it leads to the same zero-curvature equations (47)) with the above isospectral problem, while being better adapted to our needs.

We will now construct an isospectral zero-curvature representation of the hierarchies

$$u_{t_n}={\mathbb{K}}_n\left[u\right],$$

with ${\mathbb{K}}_n$ given in (20) or in (29) by combining the isospectral problems (39) and (45). Consider thus the deformed isospectral linear problem

$$\left\{\begin{array}{c}{\psi }_x=L\psi ,\hfill \\ {\psi }_{t_n}=W_n\psi -{\lambda }^{\epsilon +1}{\psi }_{\lambda },\hfill & n=1,2,\dots ,\hfill \end{array}\right.$$

(48)

with $\epsilon =-1$ or $\epsilon =0$ and where ${\lambda }_{t_n}=0$, with $W_n=W_n(\lambda ,\left[u\right])$ defined as

$$W_n=V_{\epsilon }+\sum _{i=1}^n{\mathbf{v}}_{n,i}(t_0,\dots ,t_n)U_i,$$

(49)

where $L,V_{\epsilon },U_i$ are given as above in this section and where ${\mathbf{v}}_{n,i}$ are so far undetermined functions of evolution parameters $t_0,\dots ,t_n$.

Theorem 4.

The compatibility condition ${\left({\psi }_x\right)}_{t_n}={\left({\psi }_{t_n}\right)}_x$ for (48) has the form

$$L_{t_n}=\left[W_n,L\right]+{\left(W_n\right)}_x-{\lambda }^{\epsilon +1}{L}_{\lambda }\equiv L^{\prime }\left[u_{t_n}\right],n=1,2,\dots ,$$

(50)

where

$$u_{t_n}={\mathbb{K}}_n\equiv {\sigma }_{\epsilon }+\sum _{i=1}^n{\mathbf{v}}_{n,i}(t_0,\dots ,t_n)K_i\left[u\right],n=1,2,\dots .$$

(51)

The identity in (50) means that (50) is equivalent with the corresponding equation $u_{t_n}={\mathbb{K}}_n\left[u\right]$ in (51). Note that so far the vector fields ${\mathbb{K}}_n\left[u\right]$ in (51) have nothing in common with ${\mathbb{K}}_n$ in (20) or (29).

Proof. 

Due to the form of $W_n$ we have

$$\begin{array}{cc}\hfill \left[W_n,L\right]+{\left(W_n\right)}_x-{\lambda }^{\epsilon +1}{L}_{\lambda }& =\left[V_{\epsilon },L\right]+{\left(V_{\epsilon }\right)}_x-{\lambda }^{\epsilon +1}{L}_{\lambda }+\sum _{i=1}^n{\mathbf{v}}_{n,i}\left(\left[U_i,L\right]+{\left(U_i\right)}_x\right)\hfill \\ \hfill & =L^{\prime }\left[{\sigma }_{\epsilon }\right]+\sum _{i=1}^n{\mathbf{v}}_{n,i}{L}^{\prime }\left[K_i\right]\equiv L^{\prime }\left[{\mathbb{K}}_n\right].\hfill \end{array}$$

□

So far the functions ${\mathit{v}}_{n,i}$ are undetermined. Let us now demand that the compatibility conditions

$${\left({\psi }_{t_m}\right)}_{t_n}={\left({\psi }_{t_n}\right)}_{t_m},m,n=1,2,\dots ,$$

(52)

for (48) are satisfied. These conditions are equivalent with the Frobenius integrability conditions (7), since

$$\begin{array}{cc}\hfill & {\left({\psi }_{t_m}\right)}_{t_n}-{\left({\psi }_{t_n}\right)}_{t_m}={\left({\psi }^{\prime }\left[{\mathbb{K}}_m\right]\right)}_{t_n}-{\left({\psi }^{\prime }\left[{\mathbb{K}}_n\right]\right)}_{t_m}\hfill \\ \hfill & ={\psi }^{\prime \prime }[{\mathbb{K}}_n;{\mathbb{K}}_m]+{\psi }^{\prime }\left[{\displaystyle \frac{\partial {\mathbb{K}}_m}{\partial t_n}}+{\mathbb{K}}_m^{\prime }\left[{\mathbb{K}}_n\right]\right]-{\psi }^{\prime \prime }[{\mathbb{K}}_m;{\mathbb{K}}_n]-{\psi }^{\prime }\left[{\displaystyle \frac{\partial {\mathbb{K}}_n}{\partial t_m}}+{\mathbb{K}}_n^{\prime }\left[{\mathbb{K}}_m\right]\right]\hfill \\ \hfill & ={\psi }^{\prime }\left[{\displaystyle \frac{\partial {\mathbb{K}}_m}{\partial t_n}}-{\displaystyle \frac{\partial {\mathbb{K}}_n}{\partial t_m}}+[{\mathbb{K}}_n,{\mathbb{K}}_m]\right],m,n=1,2,\dots ,\hfill \end{array}$$

where ${\psi }^{\prime \prime }[{\mathbb{K}}_n;{\mathbb{K}}_m]={\psi }^{\prime \prime }[{\mathbb{K}}_m;{\mathbb{K}}_n]$ is the second directional derivative. On the other hand, (52) holds if and only if

$${\left(W_n\right)}_{t_m}-{\left(W_m\right)}_{t_n}+[W_n,W_m]+{\lambda }^{\epsilon +1}{\left(W_n\right)}_{\lambda }-{\lambda }^{\epsilon +1}{\left(W_m\right)}_{\lambda }=0.$$

(53)

Let us thus investigate the conditions under which (53) hold. Assuming $1⩽m<n,$ we have ${\left({\mathbf{v}}_{m,i}\right)}_{t_n}=0$, and then the zero-curvature relations (53) reduce to

$${\displaystyle \frac{\partial W_n}{\partial t_m}}+W_n^{\prime }\left[{\mathbb{K}}_m\right]-W_m^{\prime }\left[{\mathbb{K}}_n\right]+[W_n,W_m]+{\lambda }^{\epsilon +1}{\left(W_n\right)}_{\lambda }-{\lambda }^{\epsilon +1}{\left(W_m\right)}_{\lambda }=0.$$

(54)

We can now prove the following theorem.

Theorem 5.

The zero curvature conditions (54) are equivalent with the set of equations on the functions ${\mathbf{v}}_{n,i}$ that is exactly the same as the set of equations (24) and (34), respectively for $\epsilon =-1$ and $\epsilon =0$, on the functions ${\mathbf{u}}_{n,i}$.

The proof of this theorem can be found in Appendix D. This theorem means that the functions ${\mathit{v}}_{n,i}$ and ${\mathbf{u}}_{n,i}$ pairwise coincide so that the deformed isospectral problem (48) leads exactly to the Frobenius integrable hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ with ${\mathbb{K}}_n$ given by (20) (for $\epsilon =-1$) or by (29) (for $\epsilon =0$). Hence,

$$W_n\equiv V_{\epsilon }+\sum _{i=1}^n{\mathbf{u}}_{n,i}(t_0,\dots ,t_n)U_i.$$

(55)

Remark 3.

Thus, we obtain the same non-autonomous Frobenius integrable hierarchies of PDEs starting from the deformed spectral problem (48) and starting from the non-autonomous deformations in the case of the subalgebras ${\mathcal{A}}_{\epsilon }$ of the hereditary algebra (17), which we consider in Section 3.

4.1. Hamiltonian Structure of Non-Autonomous Soliton Hierarchies

We will now focus on soliton hierarchies. Suppose we have an infinite hierarchy of vector fields $K_n$ on $\mathcal{M}$ that are bi-Hamiltonian with respect to two compatible Poisson structures ${\pi }_0$ and ${\pi }_1$

$$K_n={\pi }_0\delta H_n={\pi }_1\delta H_{n-1},n=1,2,\dots ,$$

with ${\pi }_0$ being invertible. Then the operator $N={\pi }_1{\pi }_0^{-1}$ is an operator with the vanishing Nijenhuis torsion so that for any vector field K on $\mathcal{M}$ we have

$${\mathcal{L}}_{NK}N=N{\mathcal{L}}_{K}N.$$

(56)

Any operator satisfying (56) is called a hereditary operator. Then,

$$K_n\equiv N^{n-1}{K}_1,n=2,3,\dots ,$$

(57)

and by the hereditary property (56)

$${\mathcal{L}}_{K_n}N=0,n=1,2,\dots ,$$

and

$$[K_n,K_m]=0,n,m=1,2,\dots .$$

We now define the infinite sequence of 1-forms

$${\gamma }_n\equiv \delta H_n={\left(N^{†}\right)}^i{\gamma }_0,n=0,1,\dots ,$$

(58)

where ${\gamma }_0=\delta H_0$ and where $N^{†}={\pi }_0^{-1}{\pi }_1$. By the same hereditary property of N, they are all closed, and thus there exists an infinite sequence of functionals $H_n=\int h_{n}dx$ such that ${\gamma }_n=\delta H_n$. The infinite sequence of Poisson operators is now defined:

$${\pi }_k=N^k{\pi }_0,k=2,3,\dots .$$

The operators ${\pi }_k$ are pairwise-compatible and usually non-local. Then, it follows that the field $K_n$ is $(n+1)$ Hamiltonian.

$$K_n={\pi }_0\delta H_n={\pi }_1\delta H_{n-1}=\dots ={\pi }_n\delta H_0,n=1,2,\dots .$$

Consider also a scaling vector field ${\sigma }_0$ such that

$$\left\{\begin{array}{cc}{\mathcal{L}}_{{\sigma }_0}{K}_1=\rho K_1,\hfill & \rho \in \mathbf{R},\hfill \\ {\mathcal{L}}_{{\sigma }_0}N=N,\hfill \end{array}\right.$$

and define the infinite sequences of vector fields (master symmetries) ${\sigma }_n$ on $\mathcal{M}$ through

$${\sigma }_n=N^n{\sigma }_0,n=-1,0,1,\dots .$$

(59)

Then it can be shown, using the hereditary property (56), that the vector fields (57) and (59) satisfy the commutation relations (17).

Finally, let us assume that there exists a vector field ${\sigma }_{-1}$ such that ${\sigma }_0=N{\sigma }_{-1}$ and such that it is Hamiltonian with respect to ${\pi }_0$

$${\sigma }_{-1}={\pi }_0\delta F.$$

Then, all ${\sigma }_n$ are Hamiltonian with respect to the Poisson operator ${\pi }_{n+1}$, that is

$${\sigma }_n={\pi }_{n+1}\delta F,n=1,2,...,$$

(60)

and it immediately follows that every non-autonomous vector field ${\mathbb{K}}_n$ in (20) or in (29) is also Hamiltonian (but not bi-Hamiltonian), as

$$\begin{array}{cc}\hfill {\mathbb{K}}_n& \equiv {\sigma }_{\epsilon }+\sum _{i=1}^n{\mathbf{u}}_{n,i}(t_0,\dots ,t_n)K_i\left[u\right]=\hfill \\ \hfill & ={\pi }_{\epsilon +1}\delta \left(F+\sum _{i=1}^n{\mathbf{u}}_{n,i}(t_0,\dots ,t_n)H_{i-\epsilon -1}\left[u\right]\right).\hfill \end{array}$$

5. Non-Autonomous KdV Hierarchy

In this and following sections we apply our theory, developed above, to three well-known soliton hierarchies: Korteveg-de Vries, dispersive water waves and Ablowitz–Kaup–Newell–Segur. They all have the structure exactly as described in Section 4.1.

5.1. KdV Hierarchy

As a first illustration of our theory, consider the KdV hierarchy. The KdV equation

$$u_t={\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x$$

is a member of the bi-Hamiltonian chain of nonlinear PDEs

$$u_{t_i}=K_i\left[u\right]={\pi }_0\delta H_i={\pi }_1\delta H_{i-1},i=1,2,...,$$

(61)

with two compatible Poisson operators

$${\pi }_0={\partial }_x,{\pi }_1={\displaystyle \frac{1}{4}}{\partial }_x^3+u{\partial }_x+{\displaystyle \frac{1}{2}}{u}_x.$$

The hierarchy (61) is autonomous in the sense that none of the vector fields $K_i\left[u\right]$ of the hierarchy depends explicitly on the evolution parameters $t_j$. The KdV hierarchy (61) can be generated by the recursion operator and its adjoint

$$N\equiv {\pi }_1{\pi }_0^{-1}={\displaystyle \frac{1}{4}}{\partial }_x^2+u+{\displaystyle \frac{1}{2}}{u}_x{\partial }_x^{-1},N^{†}={\displaystyle \frac{1}{4}}{\partial }_x^2+u-{\displaystyle \frac{1}{2}}{\partial }_x^{-1}{u}_x,$$

in the sense that (57) and (58) are valid. In particular, we find that the first vector fields $K_n$ have the form

$$\begin{array}{cc}\hfill K_1& =u_x,K_2={\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x,\hfill \\ \hfill K_3& ={\displaystyle \frac{1}{16}}{u}_{5x}+{\displaystyle \frac{5}{8}}u{u}_{3x}+{\displaystyle \frac{5}{4}}{u}_x{u}_{xx}+{\displaystyle \frac{15}{8}}{u}^2{u}_x,\hfill \\ \hfill K_4& ={\displaystyle \frac{1}{64}}{u}_{7x}+{\displaystyle \frac{7}{32}}u{u}_{5x}+{\displaystyle \frac{21}{32}}{u}_x{u}_{4x}+{\displaystyle \frac{35}{32}}{u}_{xx}{u}_{3x}+{\displaystyle \frac{35}{32}}{u}_x^3+{\displaystyle \frac{35}{8}}u{u}_x{u}_{xx}+{\displaystyle \frac{35}{32}}{u}^2{u}_{3x}+{\displaystyle \frac{35}{16}}{u}^3{u}_x,\hfill \end{array}$$

the first conserved one-forms (cosymmetries) ${\gamma }_n\equiv \delta H_n$ are

$$\begin{array}{cc}\hfill {\gamma }_0& =2,{\gamma }_1=u,{\gamma }_2={\displaystyle \frac{1}{4}}{u}_{xx}+{\displaystyle \frac{3}{4}}{u}^2,\hfill \\ \hfill {\gamma }_3& ={\displaystyle \frac{1}{16}}{u}_{4x}+{\displaystyle \frac{5}{8}}u{u}_{xx}+{\displaystyle \frac{5}{16}}{u}_x^2+{\displaystyle \frac{5}{8}}{u}^3,\hfill \\ \hfill {\gamma }_4& ={\displaystyle \frac{1}{64}}{u}_{6x}+{\displaystyle \frac{7}{32}}u{u}_{4x}+{\displaystyle \frac{7}{16}}{u}_x{u}_{3x}+{\displaystyle \frac{21}{64}}{u}_{xx}^2+{\displaystyle \frac{35}{32}}{u}^2{u}_{xx}+{\displaystyle \frac{35}{32}}u{u}_x^2+{\displaystyle \frac{35}{64}}{u}^4,\hfill \end{array}$$

while the first Hamiltonian densities $h_n$ of the conserved functionals $H_n=\int h_{n}dx$ are

$$\begin{array}{cc}\hfill h_0& =2u,h_1={\displaystyle \frac{1}{2}}{u}^2,h_2=-{\displaystyle \frac{1}{8}}{u}_x^2+{\displaystyle \frac{1}{4}}{u}^3,\hfill \\ \hfill h_3& ={\displaystyle \frac{1}{32}}{u}_{xx}^2+{\displaystyle \frac{5}{32}}{u}^2{u}_{xx}+{\displaystyle \frac{5}{32}}{u}^4,\hfill \\ \hfill h_4& =-{\displaystyle \frac{1}{128}}{u}_{3x}^2+{\displaystyle \frac{7}{64}}u{u}_{xx}^2-{\displaystyle \frac{35}{64}}{u}^2{u}_x^2+{\displaystyle \frac{7}{64}}{u}^5.\hfill \end{array}$$

With the KdV hierarchy (61) one can also relate the hierarchy of its master symmetries (59) with the first few ${\sigma }_n$ of the form

$$\begin{array}{cc}\hfill {\sigma }_{-1}& =1,{\sigma }_0=u+{\displaystyle \frac{1}{2}}x{u}_x,\hfill \\ \hfill {\sigma }_1& ={\displaystyle \frac{1}{2}}{u}_{xx}+{\displaystyle \frac{1}{8}}x{u}_{3x}+u^2+{\displaystyle \frac{1}{2}}xu{u}_x+{\displaystyle \frac{1}{4}}{u}_x{\partial }_x^{-1}u.\hfill \end{array}$$

The master symmetries ${\sigma }_n$ are in general non-local. They are Hamiltonian due to (60) with the conserved functional

$$F=\int xudx.$$

The symmetries $K_i$ and master symmetries ${\sigma }_j$ of the KdV equation generate the hereditary algebra (17) with

$$\rho ={\displaystyle \frac{1}{2}}\mathrm{so}\mathrm{that}{\kappa }_n=n-{\displaystyle \frac{1}{2}}.$$

(62)

The matrix Lax representation (42) for the KdV hierarchy is given by

$$L=\left(\begin{array}{cc}0& 1\\ \lambda -u& 0\end{array}\right),$$

(63)

and

$$U_n={\displaystyle \frac{1}{2}}\sum _{i=0}^{n-1}\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}{\left({\gamma }_i\right)}_x& {\gamma }_i\\ (\lambda -u){\gamma }_i-{\displaystyle \frac{1}{2}}{\left({\gamma }_i\right)}_{xx}& {\displaystyle \frac{1}{2}}{\left({\gamma }_i\right)}_x\end{array}\right){\lambda }^{n-i-1},n=1,2,\dots .$$

In particular, $U_1=L$

$$U_2=\left(\begin{array}{cc}-{\displaystyle \frac{1}{4}}{u}_x& \lambda +{\displaystyle \frac{1}{2}}u\\ {\lambda }^2-{\displaystyle \frac{1}{2}}u\lambda -{\displaystyle \frac{1}{2}}{u}^2-{\displaystyle \frac{1}{4}}{u}_{xx}& {\displaystyle \frac{1}{4}}{u}_x\end{array}\right)$$

and

$$U_3=\left(\begin{array}{cc}-{\displaystyle \frac{1}{4}}{u}_x\lambda -{\displaystyle \frac{1}{16}}(u_{3x}+6u{u}_x)& {\lambda }^2+{\displaystyle \frac{1}{2}}u\lambda +{\displaystyle \frac{1}{8}}(u_{xx}+3u^2)\\ {\lambda }^3-{\displaystyle \frac{1}{2}}u{\lambda }^2-{\displaystyle \frac{1}{8}}(u_{xx}+u^2)\lambda -{\displaystyle \frac{1}{16}}{u}_{4x}+{\displaystyle \frac{1}{2}}u{u}_{xx}+{\displaystyle \frac{3}{8}}{u}_x^2+{\displaystyle \frac{3}{8}}{u}^3& {\displaystyle \frac{1}{4}}{u}_x\lambda +{\displaystyle \frac{1}{16}}(u_{3x}+6u{u}_x)\end{array}\right).$$

The Lax formulation for the hierarchy of the KdV master symmetries ${\sigma }_n$ is given by (46) with $V_n$ of the form

$$V_n={\displaystyle \frac{1}{2}}\sum _{i=-1}^{n-1}\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}{\sigma }_i& {\partial }_x^{-1}{\sigma }_i\\ (\lambda -u){\partial }_x^{-1}{\sigma }_i-{\displaystyle \frac{1}{2}}{\left({\sigma }_i\right)}_x& {\displaystyle \frac{1}{2}}{\sigma }_i\end{array}\right){\lambda }^{n-i-1},n=-1,0,1\dots ,$$

so that

$$V_{-1}=\left(\begin{array}{cc}0& 0\\ 0& 0\end{array}\right),V_0=\left(\begin{array}{cc}-{\displaystyle \frac{1}{4}}& {\displaystyle \frac{1}{2}}x\\ {\displaystyle \frac{1}{2}}(\lambda -u)x& {\displaystyle \frac{1}{4}}\end{array}\right).$$

5.2. Non-Autonomous KdV Hierarchy in the Case ${\mathcal{A}}_{-1}$

We now present the deformed KdV hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ for the case of the hereditary subalgebra ${\mathcal{A}}_{-1}$.

For the general initial conditions (19), the first members (20) of the non-autonomous KdV hierarchy take the form

$$\begin{array}{cc}\hfill u_{t_0}& =1,\hfill \\ \hfill u_{t_1}& =1+{\mathbf{a}}_{1,1}\left(t_1\right)u_x,\hfill \\ \hfill u_{t_2}& =1+\left[{\mathbf{a}}_{2,1}\left(t_2\right)-{\displaystyle \frac{3}{2}}(t_0+t_1){\mathbf{a}}_{2,2}\left(t_2\right)\right]u_x+{\mathbf{a}}_{2,2}\left(t_2\right)\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right],\hfill \\ \hfill u_{t_3}& =1+\left[{\displaystyle \frac{3}{2}}{\partial }_{t_2}^{-1}{\mathbf{a}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,1}\left(t_3\right)-{\displaystyle \frac{3}{2}}(t_0+t_1+t_2){\mathbf{a}}_{3,2}\left(t_3\right)+{\displaystyle \frac{15}{8}}{(t_0+t_1+t_2)}^2{\mathbf{a}}_{3,3}\left(t_3\right)\right]u_x\hfill \\ \hfill & +\left[{\mathbf{a}}_{3,2}\left(t_3\right)-{\displaystyle \frac{5}{2}}(t_0+t_1+t_2){\mathbf{a}}_{3,3}\left(t_3\right)\right]\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right]\hfill \\ \hfill & +{\mathbf{a}}_{3,3}\left(t_3\right)\left[{\displaystyle \frac{1}{16}}{u}_{5x}+{\displaystyle \frac{5}{8}}u{u}_{3x}+{\displaystyle \frac{5}{4}}{u}_x{u}_{xx}+{\displaystyle \frac{15}{8}}{u}^2{u}_x\right].\hfill \end{array}$$

If we consider the initial conditions given by the choice ${\mathbf{a}}_{n,i}\left(t_n\right)={\delta }_{i,n}{t}_n$, that is (26), the deformed vector fields ${\mathbb{K}}_n$ are given by (20) with (21) specified by (27) with $\rho$ and ${\kappa }_m$ as in (62). In this case the first few non-autonomous vector fields ${\mathbb{K}}_n$ are given by formulas in Example 1 and so the first few members of our hierarchy specify to

$$\begin{array}{cc}\hfill u_{t_0}& =1,\hfill \\ \hfill u_{t_1}& =1+t_1{u}_x,\hfill \\ \hfill u_{t_2}& =1-{\displaystyle \frac{3}{2}}(t_0+t_1)t_2{u}_x+t_2\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right],\hfill \\ \hfill u_{t_3}& =1+\left[{\displaystyle \frac{3}{4}}{t}_2^2+{\displaystyle \frac{15}{8}}{(t_0+t_1+t_2)}^2{t}_3\right]u_x-{\displaystyle \frac{5}{2}}(t_0+t_1+t_2)t_3\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right]\hfill \\ \hfill & +t_3\left[{\displaystyle \frac{1}{16}}{u}_{5x}+{\displaystyle \frac{5}{8}}u{u}_{3x}+{\displaystyle \frac{5}{4}}{u}_x{u}_{xx}+{\displaystyle \frac{15}{8}}{u}^2{u}_x\right].\hfill \end{array}$$

The Lax representation of the above non-autonomous KdV hierarchy is given by (50), with L as in (63) and $W_n$ defined by (55) with $\epsilon =-1,$ where, in the general case, ${\mathbf{u}}_{n,i}$ are given by (21) and in the case of special initial conditions (26) the functions ${\mathbf{u}}_{n,i}$ are specified by (27).

5.3. Non-Autonomous KdV Hierarchy in the Case ${\mathcal{A}}_0$

The deformed KdV hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ for the case of the hereditary subalgebra ${\mathcal{A}}_0$ and with the general initial conditions (28) is given by the vector fields ${\mathbb{K}}_n$ as in (29), thus in particular

$$\begin{array}{cc}\hfill u_{t_0}& =u+{\displaystyle \frac{1}{2}}x{u}_x,\hfill \\ \hfill u_{t_1}& =u+{\displaystyle \frac{1}{2}}x{u}_x+{\mathbf{a}}_{1,1}\left(t_1\right)e^{-{\displaystyle \frac{1}{2}}{t}_0}{u}_x,\hfill \\ \hfill u_{t_2}& =u+{\displaystyle \frac{1}{2}}x{u}_x+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-{\displaystyle \frac{1}{2}}{t}_0}+{\mathbf{a}}_{2,1}\left(t_2\right)e^{-{\displaystyle \frac{1}{2}}(t_0+t_1)}\right]u_x+{\mathbf{a}}_{2,2}\left(t_2\right)e^{-{\displaystyle \frac{3}{2}}(t_0+t_1)}\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right],\hfill \\ \hfill u_{t_3}& =u+{\displaystyle \frac{1}{2}}x{u}_x+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-{\displaystyle \frac{1}{2}}{t}_0}+{\mathbf{c}}_{2,1}\left(t_2\right)e^{-{\displaystyle \frac{1}{2}}(t_0+t_1)}+{\mathbf{a}}_{3,1}\left(t_3\right)e^{-{\displaystyle \frac{1}{2}}(t_0+t_1+t_2)}\right]u_x\hfill \\ \hfill & +\left[e^{-{\displaystyle \frac{3}{2}}(t_0+t_1)}{\mathbf{c}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,2}\left(t_3\right)e^{-{\displaystyle \frac{3}{2}}(t_0+t_1+t_2)}\right]\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right]\hfill \\ \hfill & +{\mathbf{a}}_{3,3}\left(t_3\right)e^{-{\displaystyle \frac{3}{2}}(t_0+t_1+t_2)}\left[{\displaystyle \frac{1}{16}}{u}_{5x}+{\displaystyle \frac{5}{8}}u{u}_{3x}+{\displaystyle \frac{5}{4}}{u}_x{u}_{xx}+{\displaystyle \frac{15}{8}}{u}^2{u}_x\right].\hfill \end{array}$$

If we consider the initial conditions (35) the deformed vector fields ${\mathbb{K}}_n$ are given by (29) with (30) specified by (36) with $\rho$ and ${\kappa }_m$ given by (62). In this case, as in Example 2, we have

$$\begin{array}{cc}\hfill u_{t_0}& =u+{\displaystyle \frac{1}{2}}x{u}_x,\hfill \\ \hfill u_{t_1}& =u+{\displaystyle \frac{1}{2}}x{u}_x+t_1{e}^{-{\displaystyle \frac{1}{2}}{t}_0}{u}_x,\hfill \\ \hfill u_{t_2}& =u+{\displaystyle \frac{1}{2}}x{u}_x+\left(t_1+2e^{-{\displaystyle \frac{1}{2}}{t}_1}-2\right)e^{-{\displaystyle \frac{1}{2}}{t}_0}{u}_x+t_2{e}^{-{\displaystyle \frac{3}{2}}(t_0+t_1)}\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right],\hfill \\ \hfill u_{t_3}& =u+{\displaystyle \frac{1}{2}}x{u}_x+\left(t_1+2e^{-{\displaystyle \frac{1}{2}}{t}_1}-2\right)e^{-{\displaystyle \frac{1}{2}}{t}_0}{u}_x+\left(t_2+{\displaystyle \frac{2}{3}}{e}^{-{\displaystyle \frac{3}{2}}{t}_2}-{\displaystyle \frac{2}{3}}\right)e^{-{\displaystyle \frac{3}{2}}(t_0+t_1)}\left[{\displaystyle \frac{1}{4}}{u}_{xxx}+{\displaystyle \frac{3}{2}}u{u}_x\right]\hfill \\ \hfill & +t_3{e}^{-{\displaystyle \frac{3}{2}}(t_0+t_1+t_2)}\left[{\displaystyle \frac{1}{16}}{u}_{5x}+{\displaystyle \frac{5}{8}}u{u}_{3x}+{\displaystyle \frac{5}{4}}{u}_x{u}_{xx}+{\displaystyle \frac{15}{8}}{u}^2{u}_x\right].\hfill \end{array}$$

The Lax representation of the above non-autonomous KdV hierarchies is given by (50), with L given by (63) and with respective $W_n$ defined by (55) with $\epsilon =0$.

6. Non-Autonomous DWW Hierarchy

6.1. Autonomous DWW Hierarchy

Let us now apply our theory to the DWW hierarchy. It is a bi-Hamiltonian hierarchy given by

$${\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_n}=K_n={\pi }_0{\gamma }_n={\pi }_1{\gamma }_{n-1},n=1,2,...,$$

(64)

where ${\gamma }_n\equiv \delta H_n$ are exact one-forms and where

$${\pi }_0=\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}v{\partial }_x-{\displaystyle \frac{1}{2}}{\partial }_{x}v& {\partial }_x\\ {\partial }_x& 0\end{array}\right),{\pi }_1=\left(\begin{array}{cc}{\displaystyle \frac{1}{4}}{\partial }_x^3+{\displaystyle \frac{1}{2}}u{\partial }_x+{\displaystyle \frac{1}{2}}{\partial }_{x}u& 0\\ 0& {\partial }_x\end{array}\right)$$

are two compatible Poisson operators. This hierarchy is generated by (57) and (58) with the recursion operator and its adjoint given by

$$N={\pi }_1{\pi }_0^{-1}=\left(\begin{array}{cc}0& {\displaystyle \frac{1}{4}}{\partial }_x^2+u+{\displaystyle \frac{1}{2}}{u}_x{\partial }_x^{-1}\\ 1& v+{\displaystyle \frac{1}{2}}{v}_x{\partial }_x^{-1}\end{array}\right),N^{†}=\left(\begin{array}{cc}0& 1\\ {\displaystyle \frac{1}{4}}{\partial }_x^2+u-{\displaystyle \frac{1}{2}}{\partial }_x^{-1}{u}_x& v-{\displaystyle \frac{1}{2}}{\partial }_x^{-1}{v}_x\end{array}\right).$$

In particular, we have the symmetries

$$\begin{array}{cc}\hfill K_1& =\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right),K_2=\left(\begin{array}{c}{\displaystyle \frac{1}{4}}{v}_{xxx}+u{v}_x+{\displaystyle \frac{1}{2}}v{u}_x\\ u_x+{\displaystyle \frac{3}{2}}v{v}_x\end{array}\right),\hfill \\ \hfill K_3& =\left(\begin{array}{c}{\displaystyle \frac{3}{8}}{v}^2{u}_x+{\displaystyle \frac{3}{2}}uv{v}_x+{\displaystyle \frac{3}{2}}u{u}_x+{\displaystyle \frac{1}{4}}{u}_{3x}+{\displaystyle \frac{3}{8}}v{v}_{3x}+{\displaystyle \frac{9}{8}}{v}_x{v}_{2x}\\ {\displaystyle \frac{3}{2}}v{u}_x+{\displaystyle \frac{3}{2}}u{v}_x+{\displaystyle \frac{15}{8}}{v}^2{v}_x+{\displaystyle \frac{1}{4}}{v}_{3x}\end{array}\right),\hfill \end{array}$$

cosymmetries

$${\gamma }_0=\left(\begin{array}{c}2\\ v\end{array}\right),{\gamma }_1=\left(\begin{array}{c}v\\ u+{\displaystyle \frac{3}{4}}{v}^2\end{array}\right),{\gamma }_2=\left(\begin{array}{c}u+{\displaystyle \frac{3}{4}}{v}^2\\ {\displaystyle \frac{1}{4}}{v}_{xx}+{\displaystyle \frac{3}{2}}uv+{\displaystyle \frac{5}{8}}{v}^3\end{array}\right),$$

and functionals $H_n=\int h_{n}dx$, where

$$h_0=2u+{\displaystyle \frac{1}{2}}{v}^2,h_1=uv+{\displaystyle \frac{1}{4}}{v}^3,h_2=-{\displaystyle \frac{1}{8}}{v}_x^2+{\displaystyle \frac{1}{2}}{u}^2+{\displaystyle \frac{3}{4}}u{v}^2+{\displaystyle \frac{5}{32}}{v}^4.$$

With the DWW hierarchy (64) one can also relate the hierarchy of its master symmetries (59) with the first few ${\sigma }_n$ of the form

$${\sigma }_{-1}=\left(\begin{array}{c}-v\\ 2\end{array}\right),{\sigma }_0=\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right),{\sigma }_1=\left(\begin{array}{c}{\displaystyle \frac{3}{4}}{v}_{xx}+{\displaystyle \frac{1}{4}}x{v}_{xxx}+uv+xu{v}_x+{\displaystyle \frac{1}{2}}xv{u}_x\\ x{u}_x+{\displaystyle \frac{3}{2}}xv{v}_x+v^2+2u\end{array}\right).$$

The master symmetries ${\sigma }_n$ are Hamiltonian as in (60) with

$$F=\int \left(2xu+{\displaystyle \frac{1}{2}}x{v}^2\right)dx.$$

The symmetries $K_i$ and the master symmetries ${\sigma }_j$ constitute the hereditary algebra (17) with

$$\rho =1\mathrm{so}\mathrm{that}{\kappa }_n=n.$$

(65)

The zero-curvature formulation (42) for the DWW hierarchy is given by

$$L=\left(\begin{array}{cc}0& 1\\ {\lambda }^2-v\lambda -u& 0\end{array}\right)$$

(66)

and by

$$U_n={\displaystyle \frac{1}{2}}\sum _{i=0}^{n-1}\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}{\left({\gamma }_{i1}\right)}_x& {\gamma }_{i1}\\ \left({\lambda }^2-v\lambda -u\right){\gamma }_{i1}-{\displaystyle \frac{1}{2}}{\left({\gamma }_{i1}\right)}_{xx}& {\displaystyle \frac{1}{2}}{\left({\gamma }_{i1}\right)}_x\end{array}\right){\lambda }^{n-i-1},n=1,2,\dots ,$$

(67)

with ${\gamma }_{i1}$ denoting the first component of ${\gamma }_i$. In particular, $U_1=L$ and

$$U_2=\left(\begin{array}{cc}-{\displaystyle \frac{1}{4}}{v}_x& \lambda +{\displaystyle \frac{1}{2}}v\\ {\lambda }^3-{\displaystyle \frac{1}{2}}v{\lambda }^2-\left(u+{\displaystyle \frac{1}{2}}{v}^2\right)\lambda -{\displaystyle \frac{1}{2}}uv-{\displaystyle \frac{1}{4}}{v}_{2x}& {\displaystyle \frac{1}{4}}{v}_x\end{array}\right),$$

$$U_3=\left(\begin{array}{cc}-{\displaystyle \frac{1}{4}}{v}_x\lambda -{\displaystyle \frac{1}{4}}{u}_x-{\displaystyle \frac{3}{8}}v{v}_x& {\lambda }^2+{\displaystyle \frac{1}{2}}v\lambda +{\displaystyle \frac{3}{8}}{v}^2+{\displaystyle \frac{1}{2}}u\\ {\left(U_3\right)}_{21}& {\displaystyle \frac{1}{4}}{v}_x\lambda +{\displaystyle \frac{1}{4}}{u}_x+{\displaystyle \frac{3}{8}}v{v}_x\end{array}\right),$$

where

$${\left(U_3\right)}_{21}={\lambda }^4-{\displaystyle \frac{1}{2}}v{\lambda }^3-{\displaystyle \frac{1}{8}}\left(4u+v^2\right){\lambda }^2-{\displaystyle \frac{1}{8}}\left(8uv+3v^3+2v_{2x}\right)\lambda -{\displaystyle \frac{1}{8}}\left(4u^2+3u{v}^2+2u_{2x}+3v_x^2+3v{v}_{2x}\right).$$

The deformed Lax formulation for the hierarchy of the DWW master symmetries ${\sigma }_n$ is given by (46) with $V_n$ of the form

$$V_n={\displaystyle \frac{1}{2}}\sum _{i=-1}^{n-1}\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}{\sigma }_{i2}& {\partial }_x^{-1}{\sigma }_{i2}\\ \left({\lambda }^2-v\lambda -u\right){\partial }_x^{-1}{\sigma }_{i2}-{\displaystyle \frac{1}{2}}{\left({\sigma }_{i2}\right)}_x& {\displaystyle \frac{1}{2}}{\sigma }_{i2}\end{array}\right){\lambda }^{n-i-1},n=-1,0,\dots ,$$

with ${\sigma }_{i2}$ denoting the second component of ${\sigma }_i$, thus

$$V_{-1}=0,V_0=\left(\begin{array}{cc}-{\displaystyle \frac{1}{2}}& x\\ \left({\lambda }^2-v\lambda -u\right)x& {\displaystyle \frac{1}{2}}\end{array}\right).$$

(68)

6.2. Non-Autonomous DWW Hierarchy in the Case ${\mathcal{A}}_{-1}$

The first few vector fields of the deformed DWW hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ in the case ${\mathcal{A}}_{-1}$ and with the general initial conditions (19) have the form

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_0}& =\left(\begin{array}{c}-v\\ 2\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_1}& =\left(\begin{array}{c}-v\\ 2\end{array}\right)+{\mathbf{a}}_{1,1}\left(t_1\right)\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_2}& =\left(\begin{array}{c}-v\\ 2\end{array}\right)+\left[{\mathbf{a}}_{2,1}\left(t_2\right)-2(t_0+t_1){\mathbf{a}}_{2,2}\left(t_2\right)\right]\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right)+{\mathbf{a}}_{2,2}\left(t_2\right)\left(\begin{array}{c}{\displaystyle \frac{1}{4}}{v}_{xxx}+u{v}_x+{\displaystyle \frac{1}{2}}v{u}_x\\ u_x+{\displaystyle \frac{3}{2}}v{v}_x\end{array}\right).\hfill \end{array}$$

If we consider the initial conditions (26) the deformed vector fields ${\mathbb{K}}_n$ are given by (20) with (21) specified by (27) with $\rho$ and ${\kappa }_n$ as in (65). Explicitly,

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_0}& =\left(\begin{array}{c}-v\\ 2\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_1}& =\left(\begin{array}{c}-v\\ 2\end{array}\right)+t_1\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_2}& =\left(\begin{array}{c}-v\\ 2\end{array}\right)-2(t_0+t_1)t_2\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right)+t_2\left(\begin{array}{c}{\displaystyle \frac{1}{4}}{v}_{xxx}+u{v}_x+{\displaystyle \frac{1}{2}}v{u}_x\\ u_x+{\displaystyle \frac{3}{2}}v{v}_x\end{array}\right).\hfill \end{array}$$

6.3. Non-Autonomous DWW Hierarchy in the Case ${\mathcal{A}}_0$

The deformed DWW hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ in the case ${\mathcal{A}}_0$ and with the general initial conditions (28) is given by the vector fields (29), thus in particular

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_0}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_1}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right)+{\mathbf{a}}_{1,1}\left(t_1\right)e^{-t_0}\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_2}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right)+\left[{\mathbf{c}}_{1,1}\left(t_1\right)e^{-t_0}+{\mathbf{a}}_{2,1}\left(t_2\right)e^{-(t_0+t_1)}\right]\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right)\hfill \\ \hfill & +{\mathbf{a}}_{2,2}\left(t_2\right)e^{-2(t_0+t_1)}\left(\begin{array}{c}{\displaystyle \frac{1}{4}}{v}_{xxx}+u{v}_x+{\displaystyle \frac{1}{2}}v{u}_x\\ u_x+{\displaystyle \frac{3}{2}}v{v}_x\end{array}\right).\hfill \end{array}$$

For the initial conditions (35) from Example 2 the deformed vector fields ${\mathbb{K}}_n$ are given by (29) with (30) specified by (36) with $\rho$ and ${\kappa }_n$ as in (65). In this case

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_0}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_1}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right)+t_1{e}^{-t_0}\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}u\\ v\end{array}\right)}_{t_2}& =\left(\begin{array}{c}2u+x{u}_x\\ v+x{v}_x\end{array}\right)+\left(t_1+(e^{-t_1}-1)\right)e^{-t_0}\left(\begin{array}{c}{u}_x\\ v_x\end{array}\right)+t_2{e}^{-2(t_0+t_1)}\left(\begin{array}{c}{\displaystyle \frac{1}{4}}{v}_{xxx}+u{v}_x+{\displaystyle \frac{1}{2}}v{u}_x\\ u_x+{\displaystyle \frac{3}{2}}v{v}_x\end{array}\right).\hfill \end{array}$$

The Lax representation for all the above non-autonomous DWW hierarchies is given by (50) with L as in (66) and with $W_n$ as in (55) with respective ${\mathbf{u}}_{n,i}$ and $\epsilon =-1$ or 0.

7. Non-Autonomous AKNS Hierarchy

7.1. Autonomous AKNS Hierarchy

In the last example we apply our theory to the AKNS hierarchy. It is a bi-Hamiltonian hierarchy given by

$${\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_n}=K_n={\pi }_0{\gamma }_n={\pi }_1{\gamma }_{n-1},n=1,2,...,$$

where ${\gamma }_n=\delta H_n$ are exact one-forms and where

$${\pi }_0=\left(\begin{array}{cc}0& i\\ -i& 0\end{array}\right),{\pi }_1=\left(\begin{array}{cc}-q{\partial }_x^{-1}q& -{\displaystyle \frac{1}{2}}{\partial }_x+q{\partial }_x^{-1}r\\ -{\displaystyle \frac{1}{2}}{\partial }_x+r{\partial }_x^{-1}q& -r{\partial }_x^{-1}r\end{array}\right)$$

are two compatible Poisson operators ($i^2=-1$ throughout this whole section). This hierarchy is generated by (57) and (58) with the recursion operator and its adjoint given by

$$N={\pi }_1{\pi }_0^{-1}=i\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}{\partial }_x-q{\partial }_x^{-1}r& -q{\partial }_x^{-1}q\\ r{\partial }_x^{-1}r& -{\displaystyle \frac{1}{2}}{\partial }_x+r{\partial }_x^{-1}q\end{array}\right),N^{†}=-i\left(\begin{array}{cc}{\displaystyle \frac{1}{2}}{\partial }_x-r{\partial }_x^{-1}q& r{\partial }_x^{-1}r\\ -q{\partial }_x^{-1}q& -{\displaystyle \frac{1}{2}}{\partial }_x+q{\partial }_x^{-1}r\end{array}\right).$$

In particular, we have the symmetries

$$K_1=i\left(\begin{array}{c}-2q\\ 2r\end{array}\right),K_2=\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right),K_3=i\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{q}_{xx}-q^{2}r\\ -{\displaystyle \frac{1}{2}}{r}_{xx}+q{r}^2\end{array}\right),K_4=\left(\begin{array}{c}-{\displaystyle \frac{1}{4}}{q}_{xxx}+{\displaystyle \frac{3}{2}}qr{q}_x\\ -{\displaystyle \frac{1}{4}}{r}_{xxx}+{\displaystyle \frac{3}{2}}qr{r}_x\end{array}\right),$$

cosymmetries

$${\gamma }_1=\left(\begin{array}{c}-2r\\ -2q\end{array}\right),{\gamma }_2=i\left(\begin{array}{c}{r}_x\\ -q_x\end{array}\right),{\gamma }_3=\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{r}_{xx}-q{r}^2\\ {\displaystyle \frac{1}{2}}{q}_{xx}-r{q}^2\end{array}\right),{\gamma }_4=i\left(\begin{array}{c}-{\displaystyle \frac{1}{4}}{r}_{xxx}+{\displaystyle \frac{3}{2}}qr{r}_x\\ {\displaystyle \frac{1}{4}}{q}_{xxx}-{\displaystyle \frac{3}{2}}qr{q}_x\end{array}\right),$$

and respective Hamiltonians $H_n=\int h_{n}dx$, where

$$h_1=-2rq,h_2=-ir{q}_x,h_3=-{\displaystyle \frac{1}{2}}{q}_x{r}_x-{\displaystyle \frac{1}{2}}{q}^2{r}^2,h_4=i\left({\displaystyle \frac{1}{4}}r{q}_{xxx}+{\displaystyle \frac{3}{4}}{q}^{2}r{r}_x\right).$$

With the AKNS hierarchy (64) one can relate the hierarchy of its master symmetries (59) with ${\sigma }_{-1}$ and ${\sigma }_0$ given by

$${\sigma }_{-1}=i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right),{\sigma }_0=\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right).$$

One can directly check that in this case

$$\rho =0\mathrm{and}{\kappa }_n=n-1$$

(69)

in the hereditary algebra (17). Moreover, the master symmetries ${\sigma }_n$ are Hamiltonian as in (60) with

$$F=-2\int xqrdx.$$

The zero-curvature formulation (42) for the AKNS hierarchy is given by [14]

$$L=\left(\begin{array}{cc}-i\lambda & q\\ r& i\lambda \end{array}\right)$$

(70)

and by

$$U_n=i\left(\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right){\lambda }^{n-1}+{\displaystyle \frac{i}{2}}\sum _{j=1}^{n-1}\left(\begin{array}{cc}-{\partial }_x^{-1}\left[r{K}_{j1}+q{K}_{j2}\right]& K_{j1}\\ -K_{j2}& {\partial }_x^{-1}\left[r{K}_{j1}+q{K}_{j2}\right]\end{array}\right){\lambda }^{n-j-1},n=1,2,\dots .$$

(71)

In particular,

$$\begin{array}{cc}\hfill U_1& =\left(\begin{array}{cc}-i& 0\\ 0& i\end{array}\right),U_2\equiv L,U_2=\left(\begin{array}{cc}-i{\lambda }^2-{\displaystyle \frac{i}{2}}qr& q\lambda +{\displaystyle \frac{i}{2}}{q}_x\\ r\lambda -{\displaystyle \frac{i}{2}}{r}_x& i{\lambda }^2+{\displaystyle \frac{i}{2}}qr\end{array}\right),\hfill \\ \hfill U_3& =\left(\begin{array}{cc}-i{\lambda }^3-{\displaystyle \frac{i}{2}}qr\lambda +{\displaystyle \frac{1}{4}}r{q}_x-{\displaystyle \frac{1}{4}}q{r}_x& q{\lambda }^2+{\displaystyle \frac{i}{2}}{q}_x\lambda -{\displaystyle \frac{1}{4}}{q}_{xx}+{\displaystyle \frac{1}{2}}{q}^{2}r\\ r{\lambda }^2-{\displaystyle \frac{i}{2}}{r}_x\lambda -{\displaystyle \frac{1}{4}}{r}_{xx}+{\displaystyle \frac{1}{2}}q{r}^2& i{\lambda }^3+{\displaystyle \frac{i}{2}}qr\lambda -{\displaystyle \frac{1}{4}}r{q}_x+{\displaystyle \frac{1}{4}}q{r}_x\end{array}\right).\hfill \end{array}$$

The deformed Lax formulation for the hierarchy of the master symmetries ${\sigma }_n$ is given by (46) with $V_i$ of the form

$$V_n=i\left(\begin{array}{cc}-x& 0\\ 0& x\end{array}\right){\lambda }^{n+1}+{\displaystyle \frac{i}{2}}\sum _{j=-1}^{n-1}\left(\begin{array}{cc}-{\partial }_x^{-1}\left[r{\sigma }_{j1}+q{\sigma }_{j2}\right]& {\sigma }_{j1}\\ -{\sigma }_{j2}& {\partial }_x^{-1}\left[r{\sigma }_{j1}+q{\sigma }_{j2}\right]\end{array}\right){\lambda }^{n-j-1},$$

where $n=-1,0,1,...$, and thus

$$V_{-1}=\left(\begin{array}{cc}-ix& 0\\ 0& ix\end{array}\right),V_0=\left(\begin{array}{cc}-ix\lambda & xq\\ xr& ix\lambda \end{array}\right).$$

(72)

7.2. Non-Autonomous AKNS Hierarchy in the Case ${\mathcal{A}}_{-1}$

The first four systems of the deformed AKNS hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ in the case ${\mathcal{A}}_{-1}$ and with the general initial conditions (19) are

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_0}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_1}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)+i{\mathbf{a}}_{1,1}\left(t_1\right)\left(\begin{array}{c}-2q\\ 2r\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_2}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)+i\left[{\mathbf{a}}_{2,1}\left(t_2\right)-(t_0+t_1){\mathbf{a}}_{2,2}\left(t_2\right)\right]\left(\begin{array}{c}-2q\\ 2r\end{array}\right)+{\mathbf{a}}_{2,2}\left(t_2\right)\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_3}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)+i\left[{\partial }_{t_2}^{-1}{\mathbf{a}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,1}\left(t_3\right)-{\tau }_2{\mathbf{a}}_{3,2}\left(t_3\right)+{\tau }_2^2{\mathbf{a}}_{3,3}\left(t_3\right)\right]\left(\begin{array}{c}-2q\\ 2r\end{array}\right)\hfill \\ \hfill & +\left[{\mathbf{a}}_{3,2}\left(t_3\right)-2{\tau }_2{\mathbf{a}}_{3,3}\left(t_3\right)\right]\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)+i{\mathbf{a}}_{3,3}\left(t_3\right)\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{q}_{xx}-q^{2}r\\ -{\displaystyle \frac{1}{2}}{r}_{xx}+q{r}^2\end{array}\right),\hfill \end{array}$$

where ${\tau }_2=t_0+t_1+t_2$.

If we consider the initial conditions (26) the deformed vector fields ${\mathbb{K}}_n$ are given by (20) with (21) specified by (27) with $\rho$ and ${\kappa }_n$ as in (69), yielding

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_0}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_1}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)+i{t}_1\left(\begin{array}{c}-2q\\ 2r\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_2}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)-i(t_0+t_1)t_2\left(\begin{array}{c}-2q\\ 2r\end{array}\right)+t_2\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_3}& =i\left(\begin{array}{c}-2xq\\ 2xr\end{array}\right)+i\left[{\displaystyle \frac{1}{2}}{t}_2^2+{(t_0+t_1+t_2)}^2{t}_3\right]\left(\begin{array}{c}-2q\\ 2r\end{array}\right)-2(t_0+t_1+t_2)t_3\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)\hfill \\ \hfill & +i{t}_3\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{q}_{xx}-q^{2}r\\ -{\displaystyle \frac{1}{2}}{r}_{xx}+q{r}^2\end{array}\right).\hfill \end{array}$$

7.3. Non-Autonomous AKNS Hierarchy in the Case ${\mathcal{A}}_0$

The deformed AKNS hierarchy $u_{t_n}={\mathbb{K}}_n\left[u\right]$ in the case ${\mathcal{A}}_0$ and with the general initial conditions (28) is given by the vector fields (29); the first few of them read

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_0}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_1}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i{\mathbf{a}}_{1,1}\left(t_1\right)\left(\begin{array}{c}-2q\\ 2r\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_2}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i{\mathbf{a}}_{2,1}\left(t_2\right)\left(\begin{array}{c}-2q\\ 2r\end{array}\right)+{\mathbf{a}}_{2,2}\left(t_2\right)e^{-(t_0+t_1)}\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_3}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i{\mathbf{a}}_{3,1}\left(t_3\right)\left(\begin{array}{c}-2q\\ 2r\end{array}\right)\hfill \\ \hfill & +\left[e^{-(t_0+t_1)}{\mathbf{c}}_{2,2}\left(t_2\right)+{\mathbf{a}}_{3,2}\left(t_3\right)e^{-{\tau }_2}\right]\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)+i{\mathbf{a}}_{3,3}\left(t_3\right)e^{-2{\tau }_2}\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{q}_{xx}-q^{2}r\\ -{\displaystyle \frac{1}{2}}{r}_{xx}+q{r}^2\end{array}\right),\hfill \end{array}$$

where ${\tau }_2=t_0+t_1+t_2$. Notice that, in this case ${\kappa }_1=0$ and thus all ${\mathbf{c}}_{r,1}=0$ due to (32).

For the initial conditions (35) the deformed vector fields ${\mathbb{K}}_n$ are given by (29) with (30) specified by (36) with $\rho$ and ${\kappa }_n$ as in (69), which yields

$$\begin{array}{cc}\hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_0}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_1}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i{t}_1\left(\begin{array}{c}-2q\\ 2r\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_2}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i\left[\left(t_1-{\displaystyle \frac{1}{{\kappa }_1}}\right)e^{-{\kappa }_1{t}_0}+{\displaystyle \frac{1}{{\kappa }_1}}{e}^{-{\kappa }_1(t_0+t_1)}\right]\left(\begin{array}{c}-2q\\ 2r\end{array}\right)+t_2{e}^{-{\kappa }_2(t_0+t_1)}\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right),\hfill \\ \hfill {\left(\begin{array}{c}q\\ r\end{array}\right)}_{t_3}& =\left(\begin{array}{c}{\left(xq\right)}_x\\ {\left(xr\right)}_x\end{array}\right)+i\left[\left(t_1-{\displaystyle \frac{1}{{\kappa }_1}}\right)e^{-{\kappa }_1{t}_0}+{\displaystyle \frac{1}{{\kappa }_1}}{e}^{-{\kappa }_1(t_0+t_1)}\right]\left(\begin{array}{c}-2q\\ 2r\end{array}\right),\hfill \\ \hfill & +\left[\left(t_2-{\displaystyle \frac{1}{{\kappa }_2}}\right)e^{-{\kappa }_2(t_0+t_1)}+{\displaystyle \frac{1}{{\kappa }_2}}{e}^{-{\kappa }_2(t_0+t_1+t_2)}\right]\left(\begin{array}{c}{q}_x\\ r_x\end{array}\right)+i{t}_3{e}^{-{\kappa }_3(t_0+t_1+t_2)}\left(\begin{array}{c}{\displaystyle \frac{1}{2}}{q}_{xx}-q^{2}r\\ -{\displaystyle \frac{1}{2}}{r}_{xx}+q{r}^2\end{array}\right).\hfill \end{array}$$

The Lax representation for all the above non-autonomous AKNS hierarchies are given by (50) with L as in (66) and with $W_n$ as in (55) with respective ${\mathbf{u}}_{n,i}$ and $\epsilon =-1$ or 0.

8. Conclusions

In this work, we have developed a systematic theory for constructing non-autonomous infinite hierarchies of the soliton type, which are integrable in the Frobenius sense. The theory is formulated as an initial value problem for the Frobenius conditions on Lie algebras, specifically focusing on the hereditary algebras linked to soliton hierarchies. To illustrate our approach, we constructed integrable non-autonomous deformations of the KdV, DWW, and AKNS soliton hierarchies. We also provided construction of the zero-curvature representations and Hamiltonian structures of the considered non-autonomous soliton hierarchies.

Importantly, our construction is not restricted to deformations of autonomous soliton hierarchies composed of mutually commuting systems. In principle, it can be extended to infinite hierarchies that are closed under the commutator. Whether such a more general approach could yield interesting examples of integrable non-autonomous hierarchies of the soliton type remains an open question and requires further research. Furthermore, it should be noted that our theory is not limited to infinite-dimensional hierarchies.

Since our theory is based on a deformation approach, the question naturally arises whether these deformations can be simplified. A definitive answer depends on the specific Lie algebra involved. In the case of our examples involving the hereditary algebras, one can observe that, through transformations of the independent variables, finite subsets (sub-hierarchies) of the infinite non-autonomous hierarchies can be simplified in such a way that just a single equation remains non-autonomous. For the full infinite non-autonomous hierarchies, such simplification is not possible. In this case, this particular non-autonomous equation would have to consist of infinite combinations of equations from the original soliton hierarchy with time-dependent coefficients. The corresponding transformations would involve infinite sums of time-dependent terms.

The most well-known integrable non-autonomous hierarchies are finite-dimensional Hamiltonian hierarchies of the Painlevé type. Such Painlevé hierarchies naturally arise in the context of self-similar solutions of soliton hierarchies, see for instance [15,16]. The finite-dimensional reductions associated with the stationary flows of the non-autonomous soliton hierarchies constructed in this work would also be of the Painlevé type, as they would admit the so-called isomonodromic Lax representations. This is a consequence of the specific Lax representation for the master symmetries that are key to our construction. Such reductions of non-autonomous soliton hierarchies will be the focus of our future research.