On the (Non)Removability of Spectral Parameters in $\mathbb{Z}_{2}$-Graded Zero-Curvature Representations and Its Applications

Abstract

We generalise to the $\mathbb{Z}_{2}$-graded set-up a practical method for inspecting the (non)removability of parameters in zero-curvature representations for partial differential equations (PDEs) under the action of smooth families of gauge transformations. We illustrate the generation and elimination of parameters in the flat structures over $\mathbb{Z}_{2}$-graded PDEs by analysing the link between deformation of zero-curvature representations via infinitesimal gauge transformations and, on the other hand, propagation of linear coverings over PDEs using the Frölicher–Nijenhuis bracket.

Appendices

Appendix A: Proof of Proposition 2

To verify the claim, let us first introduce some helpful notation. The number $k$ is called the differential order of a differential function $f(x,t,[u^{j},\xi ^{l}])$ with respect to the variable $u^{j}$ if following conditions hold:

• (1) the function $f$ essentially depends on the $k$th order derivative of $u ^{j}$ with respect to $x$:

  $$ \frac{\partial f}{\partial u^{j}_{\sigma }} \neq 0, \quad \sigma = (x\dots x),\ |\sigma | = k; $$

• (2) the function $f$ does not depend on derivatives of $u^{j}$ of order higher than $k$ with respect to $x$:

  $$ \forall p >k,\quad \frac{\partial f}{\partial u^{j}_{\sigma }} = 0, \quad \sigma = (x\dots x),\ |\sigma |= p. $$

We denote by $\operatorname{dord}_{x}^{u^{j}} (f)$ the differential order of a given function $f$ with respect to $u^{j}$. In the same manner we define the differential order $\operatorname{dord}_{x}^{\xi ^{l}}(f)$ of differential function $f(x,t, [u^{j},\xi ^{l}])$ with respect to the odd variables $\xi ^{l}$.

For $N=2$, $a=4$-SKdV equation (6a)–(6d) we have that ${u^{1}=u_{0}}$, ${u^{2} = u_{12}}$, ${\xi ^{1}=u_{1}}$, ${\xi ^{2}=u _{2}}$. The maximum of four numbers $\operatorname{dord}_{x}^{u_{0}} (f)$, $\operatorname{dord}_{x}^{u_{1}} (f)$, $\operatorname{dord}_{x} ^{u_{2}} (f)$, and $\operatorname{dord}_{x}^{u_{12}} (f)$ is called the differential order of the function $f(x,t,[u_{0},u_{1},u_{2},u _{12}])$, denoted by $\operatorname{dord}_{x} (f)$. The maximum of $\operatorname{dord}_{x} (a_{ij})$, where $a_{ij}$’s are the entries of a given matrix $A$ with differential-functional coefficients, is the differential order of the matrix $A$; it is denoted by $\operatorname{dord}_{x} (A)$. Obviously, the following formulas hold:

$$\begin{aligned} 0 \leqslant \operatorname{dord}_{x} (f + g) \leqslant & \max \bigl\{ \operatorname{dord}(f), \operatorname{dord}(g)\bigr\} , \\ 0 \leqslant \operatorname{dord}_{x} (f\cdot g) \leqslant & \max \bigl\{ \operatorname{dord}(f), \operatorname{dord}(g)\bigr\} , \\ \operatorname{dord}_{x} (\bar{D}_{x} f) =& \textstyle\begin{cases} \operatorname{dord}_{x} (f) + 1 &\text{if } f = f(x,t,[u_{0},u_{1},u _{2},u_{12}]), \\ 0 &\text{if } f = f(x,t). \end{cases}\displaystyle \end{aligned}$$

Now, for the matrix (7) we have that

$$ \operatorname{dord}_{x} A = 0, \qquad \operatorname{dord}_{x} \frac{\partial }{\partial \varepsilon } A = 0. $$

Let us calculate the maximal differential order of left-hand side of Eq. (8a),

$$ \operatorname{dord}_{x} \biggl( \frac{\partial }{\partial \varepsilon } A + \underline{[}A, Q \underline{{]}} \biggr) \leqslant \operatorname{dord}_{x} (Q). $$

The differential orders of the right-hand side and the left-hand side of Eq. (8a) must coincide. Therefore, we have that

$$ \operatorname{dord}_{x} (\bar{D}_{x} Q) = \operatorname{dord}_{x} (Q). $$

This equality holds only in the case when all entries of the matrix $Q$ do not depend on $u_{0}$, $u_{1}$, $u_{2}$, and $u_{12}$. This implies that the total derivative $\bar{D}_{x}$ in Eq. (8a) amounts to the partial derivative ${\partial }/{\partial x}$. We finally obtain the following system of equations for the entries $q_{ij}$ of the matrix $Q$:

$$\begin{aligned} \frac{\partial }{\partial x} q_{11} =& - u_{2}q_{31} \varepsilon ^{-1} - \boldsymbol{i}u_{1}q_{31} \varepsilon ^{-1} + u_{0}^{2}q_{21} \varepsilon ^{-1} - \boldsymbol{i}u_{0} q_{21} \varepsilon ^{-2} + u_{12} q_{21}\varepsilon ^{-1} + q_{12}\varepsilon , \end{aligned}$$

(22a)

$$\begin{aligned} \frac{\partial }{\partial x} q_{12} =& - u_{2}(q_{32} + q_{13} \varepsilon )\varepsilon ^{-1} + \boldsymbol{i}u_{1}( - q_{32} + q_{13} \varepsilon )\varepsilon ^{-1} + u_{0}^{2}( - \varepsilon q_{11} + \varepsilon q_{22} - 1)\varepsilon ^{-2} \\ & {} + \boldsymbol{i}u_{0} (\varepsilon q_{11} - \varepsilon q _{22} + 2)\varepsilon ^{-3} + u_{12} ( - \varepsilon q_{11} + \varepsilon q_{22} - 1)\varepsilon ^{-2} + q_{12}\varepsilon ^{-1}, \end{aligned}$$

(22b)

$$\begin{aligned} \frac{\partial }{\partial x} q_{13} =& u_{2}( - \varepsilon q _{22} + 1)\varepsilon ^{-2} + \boldsymbol{i}u_{1}( - \varepsilon q_{22} + 1 )\varepsilon ^{-2} + u_{0}^{2}q_{23}\varepsilon ^{-1} + \boldsymbol{i}u_{0} \bigl( - q_{23} + q_{13} \varepsilon ^{2}\bigr)\varepsilon ^{-2} \\ & {} + u_{12} q_{23}\varepsilon ^{-1} + q_{13}\varepsilon ^{-1}, \end{aligned}$$

(22c)

$$\begin{aligned} \frac{\partial }{\partial x} q_{21} =& - q_{11}\varepsilon + q _{22}\varepsilon - 1 - q_{21}\varepsilon ^{-1}, \end{aligned}$$

(22d)

$$\begin{aligned} \frac{\partial }{\partial x} q_{22} =& - u_{2}q_{23} + \boldsymbol{i}u_{1}q_{23} - u_{0}^{2}q_{21} \varepsilon ^{-1} + \boldsymbol{i}u_{0} q_{21} \varepsilon ^{-2} - u_{12} q_{21}\varepsilon ^{-1} - \varepsilon q_{12} + \varepsilon ^{-2}, \end{aligned}$$

(22e)

$$\begin{aligned} \frac{\partial }{\partial x} q_{23} =& q_{21}u_{2} \varepsilon ^{-1} + \boldsymbol{i}q_{21}u_{1} \varepsilon ^{-1} + \boldsymbol{i}u _{0} q_{23} - \varepsilon q_{13}, \end{aligned}$$

(22f)

$$\begin{aligned} \frac{\partial }{\partial x} q_{31} =& - q_{21}u_{2} + \boldsymbol{i}q_{21}u_{1} - \boldsymbol{i}u_{0} q_{31} + q_{32}\varepsilon - q_{31}\varepsilon ^{-1}, \end{aligned}$$

(22g)

$$\begin{aligned} \frac{\partial }{\partial x} q_{32} =& q_{11}u_{2} - \boldsymbol{i}q_{11}u_{1} - u_{0}^{2}q_{31} \varepsilon ^{-1} + \boldsymbol{i}u_{0} \bigl( - q_{32}\varepsilon ^{2} + q_{31}\bigr)\varepsilon ^{-2} - u_{12} q_{31} \varepsilon ^{-1}. \end{aligned}$$

(22h)

Since every $q_{ij}$ does not depend on $u_{0}$, $u_{1}$, $u_{2}$, and $u_{12}$, it follows that the coefficients of nonzero powers of $u_{0}$, $u_{1}$, $u_{2}$, and $u_{12}$ in (22a)–(22h) are equal to zero. We obtain the system

$$\begin{aligned} \textstyle\begin{array}{rl@{\qquad}rl@{\qquad}rl} q_{31} &= 0, & q_{32} + q_{13}\varepsilon &= 0, & - q_{32} + q_{13} \varepsilon &= 0, \\ q_{23} & = 0 , & - q_{11}\varepsilon + q_{22}\varepsilon - 1 &= 0, & - q_{32} \varepsilon ^{2} + q_{31} & = 0 , \\ q_{11} \varepsilon - q_{22}\varepsilon + 2 &= 0 , & - q_{22}\varepsilon + 1 &= 0, & q_{11} &= 0 . \end{array}\displaystyle \end{aligned}$$

(23)

By adding the last equation in the first column, $q_{11} \varepsilon - q_{22}\varepsilon + 2 = 0$, to the second equation in the other column, $- q_{11}\varepsilon + q_{22}\varepsilon - 1 = 0$, we obtain the contradiction $1=0$. Therefore, system (23) is not compatible. This proves Proposition 2: there is no $\mathfrak{sl}(2| 1)$-matrix $Q$ satisfying Eqs. (8a)–(8b) at $\varepsilon >0$.

Appendix B: Two Descriptions of One Elimination Procedure: An Example

We analyse the following tautological construction: by re-addressing Sasaki,¹² see [37], we first track how the scaling symmetry of KdV equation (14) acts on its standard matrix Lax pair; on the other hand, we reveal how these objects are phrased in the language of coverings.

2.1 B.1 The Sasaki Construction

Recall that the Korteweg–de Vries equation is

$$ \mathcal{E}= \{ u_{t} = - u_{xxx} - 6uu_{x} \} . $$

(14)

Consider the family of coverings $\tau _{\eta }\colon \tilde{\mathcal{E}}_{\eta }\to \mathcal{E}$ over it,

$$\begin{aligned} v_{x} &= 2v\eta - \bigl(v^{2} + u\bigr), \end{aligned}$$

(24a)

$$\begin{aligned} v_{t} &= -8\eta ^{3} v + 4 \eta ^{2} \bigl(v^{2} + u\bigr) + 2 \eta (-2 v u + u _{x}) + 2 v^{2} u - 2 v u_{x} + 2 u^{2} + u_{xx}; \end{aligned}$$

(24b)

these formulas are obtained from the $\mathfrak{sl}_{2}$-valued zero-curvature representation (see [37]),

$$ \alpha _{\eta }= \begin{pmatrix} \eta & u \\ -1 & -\eta \end{pmatrix} { \mathrm{d}}x + \begin{pmatrix} - (4\eta ^{3} + 2\eta u + u_{x}) & - (u_{xx} + 2\eta u_{x} + 4\eta ^{2}u + 2u^{2}) \\ 4\eta ^{2} + 2u & 4\eta ^{3} + 2\eta u + u_{x} \end{pmatrix} {\mathrm{d}}t. $$

Let us recall that the parameter $\eta$ cannot be removed from the zero-curvature representations $\alpha _{\eta }$ by using gauge transformations. However, it can be eliminated by using a wider class of transformations. Namely, consider the scaling symmetry of Eq. (14),

$$ x \mapsto \eta x, \quad t \mapsto \eta ^{3} t, \quad u \mapsto \eta ^{-2}u, \qquad \eta \in \mathbb{R}. $$

Using it, one transforms the zero-curvature representation $\alpha _{\eta }$ into

$$ \alpha '_{\eta }= \begin{pmatrix} 1 & \eta u \\ -\eta ^{-1} & -1 \end{pmatrix} {\mathrm{d}}x + \begin{pmatrix} - (4 + 2 u + u_{x}) & - \eta (u_{xx} + 2u_{x} + 4u + 2u^{2}) \\ \eta ^{-1}(4 + 2u) & 4 + 2u + u_{x} \end{pmatrix} { \mathrm{d}}t. $$

The parameter $\eta$ in $\alpha '_{\eta }$ is removable under the gauge transformation

$$ g = \begin{pmatrix} \eta ^{-1/2} & 0 \\ 0 & \eta ^{1/2} \end{pmatrix} \in C^{\infty }\bigl( \mathcal{E}^{\infty }, GL_{2}(\mathbb{C})\bigr), $$

that is, we have that $(\alpha '_{\eta })^{g} = \alpha '_{\eta } |_{\eta = 1} = \alpha _{\eta } |_{\eta = 1}$.

2.2 B.2 How the Elimination Works in Terms of the Structure Element

Let us now address the removability of parameter $\eta$ in coverings (24a)–(24b) in terms of the formalism of Cartan’s structural element.

For a vector field

$$ X = a\otimes \frac{\partial }{\partial x} + b\otimes \frac{\partial }{ \partial t} + \omega _{\sigma }\otimes \frac{\partial }{\partial u_{ \sigma }} + \varphi \otimes \frac{\partial }{\partial v}, $$

the equation for evolution of Cartan’s structural element,

$$ \frac{\mathrm{d}}{\mathrm{d}\eta } U_{\eta }= [X, U_{\eta }]^{\text{FN}}, $$

(12)

splits into the system

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}\eta } v_{x} =& \tilde{D}_{x} \varphi - \varphi \frac{\partial v_{x}}{\partial v} - \omega _{\sigma }\frac{ \partial v_{x}}{\partial u_{\sigma }} + b \biggl( \frac{\partial v_{x}}{ \partial u_{\sigma }}u_{\sigma t} + \frac{\partial v_{x}}{\partial v}v _{t} - \tilde{D}_{x}v_{t} \biggr) - v_{t}\frac{\partial b}{x} \\ &{}+ a \biggl( - \tilde{D}_{x}v_{x} + \frac{\partial v_{x}}{\partial u_{\sigma }}u_{\sigma x} + \frac{\partial v_{x}}{\partial v}v_{x} \biggr) - v_{x}\frac{\partial a}{\partial x}, \end{aligned}$$

(25a)

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}\eta } v_{t} =& \tilde{D}_{t} \varphi - \varphi \frac{\partial v_{t}}{\partial v} - \omega _{\sigma }\frac{\partial v_{t}}{\partial u_{\sigma }} + b \biggl( \frac{\partial v_{t}}{\partial u_{\sigma }}u_{\sigma t} + \frac{\partial v_{t}}{ \partial v}v_{t} - \tilde{D}_{t}v_{t} \biggr) - v_{t} \frac{\partial b}{ \partial t} \\ &{}+ a \biggl( - \tilde{D}_{t} v_{x} + \frac{\partial v_{t}}{\partial u_{\sigma }}u_{\sigma x} + \frac{\partial v_{t}}{\partial v}v_{x} \biggr) - v_{x}\frac{\partial a}{\partial t}, \end{aligned}$$

(25b)

$$\begin{aligned} \omega _{\sigma x} =& \tilde{D}_{x} \omega _{\sigma }- u_{\sigma t}\frac{ \partial b}{\partial x} - u_{\sigma x}\frac{\partial a}{\partial x}, \end{aligned}$$

(25c)

$$\begin{aligned} \omega _{\sigma t} =& \tilde{D}_{t} \omega _{\sigma }- u_{\sigma t}\frac{ \partial b}{\partial t} - u_{\sigma x}\frac{\partial a}{\partial t}. \end{aligned}$$

(25d)

Suppose now that the vector field is vertical: $X^{\mathrm{v}} = \omega ^{\mathrm{v}}_{\sigma }\otimes {\partial }/{\partial u_{\sigma }} + \varphi ^{\mathrm{v}}\otimes {\partial }/{\partial v}$. This simplifies system (25a)–(25d); it then becomes

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}\eta } v_{x} =& \tilde{D}_{x} \varphi ^{\mathrm{v}} - \varphi ^{\mathrm{v}}\frac{\partial v_{x}}{ \partial v} - \omega ^{\mathrm{v}}_{\sigma }\frac{\partial v_{x}}{ \partial u_{\sigma }}, \end{aligned}$$

(26a)

$$\begin{aligned} -\frac{\mathrm{d}}{\mathrm{d}\eta } v_{t} =& \tilde{D}_{t} \varphi ^{\mathrm{v}} - \varphi ^{\mathrm{v}}\frac{\partial v_{t}}{ \partial v} - \omega ^{\mathrm{v}}_{\sigma }\frac{\partial v_{t}}{ \partial u_{\sigma }}, \end{aligned}$$

(26b)

$$\begin{aligned} \omega ^{\mathrm{v}}_{\sigma x} =& \tilde{D}_{x} \omega ^{ \mathrm{v}}_{\sigma }, \end{aligned}$$

(26c)

$$\begin{aligned} \omega ^{\mathrm{v}}_{\sigma t} =& \tilde{D}_{t} \omega ^{ \mathrm{v}}_{\sigma }. \end{aligned}$$

(26d)

Let us use the Ansatz

$$ \omega ^{\mathrm{v}} = \omega - au_{x} - bu_{t}, \qquad \varphi ^{ \mathrm{v}} = \varphi - av_{x} -bu_{t}, $$

assuming that $a=a(x, t, \eta )$, $b=b(x, t, \eta )$, $\varphi = \varphi (\eta , u, v)$, and $\omega = \omega (\eta , u, v, u_{x}, u _{xx})$. By construction, the unknowns $\omega ^{\mathrm{v}}$ and $\varphi ^{\mathrm{v}}$ satisfy system (26a)–(26d). Using the analytic software Jets [69] and Crack [58], we find the solution

$$\begin{aligned} a =& 24c_{4}t\eta ^{3} + 2c_{4}x\eta + \frac{1}{\eta }(c_{6}+x), \\ b =& 6c_{4}t\eta + \frac{1}{\eta }(-c_{7} + 3t), \\ \omega =& 4c_{4}\eta ^{3} - 4c_{4}u\eta + u_{x}c_{4} + \frac{1}{ \eta }\biggl(- \frac{1}{2}u_{x}c_{3} - 2u\biggr) + \frac{1}{2\eta ^{2}}u_{x}, \\ \varphi =& 2c_{4}\eta ^{2} - c_{4}v^{2} - c_{3}v - c_{4}u + \frac{c _{3}}{2\eta } \bigl(v^{2} + u\bigr) - \frac{1}{2\eta ^{2}} \bigl(v^{2} + u \bigr), \end{aligned}$$

which contains four arbitrary constants $c_{3}$, $c_{4}$, $c_{6}$, and $c _{7}$.

Let us set $c_{3}=0$, $c_{4}=-1/(2\eta ^{2})$ at $\eta \neq 0$, $c_{6}=0$, and $c_{7}=0$. This determines the solution which corresponds to the lift of Galilean symmetry of (14):

$$ X_{2} = -2\eta ( 6t {\partial }/{\partial x} + {\partial }/{ \partial u} + \cdots ) - {\partial }/{\partial v}. $$

On the other hand, set $c_{3}=1/\eta$ if $\eta \neq 0$ and let $c_{4}=0$, $c_{6}=0$, and $c_{7}=0$. This yields the solution which corresponds to the lift of scaling symmetry of (14); namely, we have that

$$ X_{1} = \eta ^{-2}( - x {\partial }/{ \partial x} - 3t {\partial }/ {\partial t} + 2u {\partial }/{\partial u} + \cdots + v {\partial }/{\partial v} ). $$

(27)

The integral curves of vector field (27) encode the transformation

$$ x \mapsto \eta x, \qquad t \mapsto \eta ^{3} t, \qquad u \mapsto \eta ^{-2} u, \qquad v \mapsto \eta ^{-1} v. $$

(28)

Its action on the covering $\tau _{\eta }$ in (24a)–(24b) results in the covering $\tau ' = \tau _{\eta } |_{\eta =1}$, which is described by the formulas

$$\begin{aligned} v_{x} &= 2v - \bigl(v^{2} + u\bigr), \\ v_{t} &= -8 v + 4 v^{2} + 4u -4 v u + 2u_{x} + 2 v^{2} u - 2 v u_{x} + 2 u^{2} + u_{xx}. \end{aligned}$$

It is readily seen that the covering $\tau '$ is the image of zero-curvature representation $(\alpha '_{\eta })^{g}$ under a swapping of representations for the Lie algebra at hand. This is shown in the following diagram:

(29)

We conclude that the problem of finding transformations (which are possibly not gauge) that eliminate the parameter in a given family of zero-curvature representations can be approached via a solution of Eq. (12) in the family of coverings which are the $(\rho \rightleftarrows \boldsymbol{\varrho })$-avatars of those zero-curvature representations.

2.3 B.3 Overview: Taxonomy of the Parameters

Depending on their elimination scenario, “removable” parameters in zero-curvature representations are classified as follows:

• (1) There are parameters which are truly removable under the action of smooth families of gauge transformations (see [22, 23] by Marvan and [24, 25] by Sakovich).

• (2) There could be zero-curvature representations $\alpha _{\lambda }$ which are (piecewise-)smooth in the parameter $\lambda \in \mathcal{I}\subseteq \mathbb{C}$ but such that the families $S_{ \lambda }$ of gauge transformations removing the parameter are not smooth at all points $\lambda \in \mathcal{J}\subseteq \mathcal{I}$, where the set $\mathcal{J}$ is

  • (a) finite,

  • (b) countable,

  • (c) everywhere dense in ℐ but not amounting to it, or

  • (d) equal to the entire set ℐ of admissible values of the parameter $\lambda$.

  This analytic curiosity would be the threshold limit of the preceding case.

• (3) Next, there are parameters which cannot be removed by using gauge transformations but which indicate the presence of conserved currents in zero-curvature representations and the reducibility of such representations¹³ (see [70] and [48, §12]).

• (4) There are parameters which vanish under the action of those symmetries of the underlying differential equation which cannot be lifted to the covering Maurer–Cartan equation (see [37, 71]).

• (5) Finally, there are parameters which can be eliminated by the same procedure as in the preceding case but by using shadows of nonlocal symmetries in some auxiliary covering over the equation at hand (namely, not in the covering which grasps the ZCR geometry but in an extension of the equation’s geometry by a set of “nonlocalities”), see [72–74].