Deautonomizations of integrable equations and their reductions

Abstract

1. Introduction

There exist several criteria and methods to test the integrability of a given difference equation and to derive discrete integrable systems. Multidimensional consistency [1], algebraic entropy [2], singularity confinement [3] and integrability conditions (existence of infinite hierarchies of generalized symmetries and canonical conservation laws) [4, 5] are some of these criteria and, for instance, multidimensional consistency has been used to classify integrable equations in [6, 7], and singularity confinement has been employed for the deautonomization of known autonomous equations [3, 8–10].

To answer this question, we are going to employ the integrability conditions [4, 5] which not only test the integrability of the equation under consideration but also serve as determining equations for its generalized symmetries. So this approach allows us not only to derive integrable non-autonomous partial difference equations but also to construct their symmetries (differential-difference equations) and conservation laws which will also depend explicitly on $$n$$ and $$m$$.

Here, $${\cal{S}}$$ and $${\cal{T}}$$ are the shift operators in the first and second direction, respectively, $$Q_{n,m}$$ stands for the defining function of the $$d$$-quad equation under consideration, and $$R$$ is a function of $$n$$, $$m$$ and variables $$u_{n-N,m}$$, $$\ldots$$, $$u_{n+N,m}$$.

As $$N$$ depends on the order of the symmetry (i.e. the number of forward and backwards shifted values of $$u$$ involved in the symmetry generator) and there is no way to choose $$N$$ a priori, we start our investigations with $$N=1$$ (first order symmetries) and then proceed to second order ones ($$N=2$$). In every step $$N$$, we separate the integrable equations as the ones admitting symmetries of order $$N$$ and none of lower order (when $$N>1$$). This procedure determines completely the introduced functions of $$n$$ and $$m$$. Finally, employing more of the integrability conditions, we construct the lowest order symmetries which are non-autonomous differential-difference equations of order two.

The article is organized as follows. Section 2 presents the non-autonomous quadratic quad equation and its autonomous two-quad counterpart and some of their integrability aspects. Section 3 deals with the procedure of deautonomization using Hirota’s and the discrete potential KdV equations as illustrative examples and presents the relative results for all the ABS equations, the deformed H list, as well as for the discrete sine-Gordon and the Hietarinta-Viallet equations. Section 4 presents reductions for some of the derived non-autonomous differential-difference equations and their relations to alternating QRT maps. Section 5 deals with the $$(k,-1)$$ periodic reductions of the deautonomized KdV equations. The final section gives an overall evaluation of our results and the appendix contains outcomes from our analysis in Section 3 and some canonical conservation laws we employ in section 5.

2. Two difference equations and their symmetries

In this section, we present two difference equations, derive their relation and discuss their integrability properties. In particular, we present their lowest order generalized symmetries in the first lattice direction (of order two) and compute the first two canonical conserved densities. The importance of these two equations stems from the fact that both of them are related to the deautonomized equations we are discussing in the next section.

The first equation is non-autonomous, quadratic and defined on an elementary quadrilateral of the lattice and has the following form.

This is an integrable equation, as we explain below, and can be solved uniquely with respect to any corner value of $$z$$ (hence, we can solve uniquely the initial value problem if initial data are given for instance along one horizontal line and two consecutive vertical lines).

The symmetries in the vertical direction for the quadratic equation (6) can be derived by exploiting its relation to the quad equations presented in the next section. However, for equation (8) we do not know any symmetries in the $$m$$ direction.

3. Deautonomizations

Integrability conditions provide us the means not only to test if a given equation is integrable but also to find its generalized symmetries [5]. In this context, we employ equation (2) to find non-autonomous integrable generalizations of known partial difference equations defined on an elementary quadrilateral on the lattice. We describe our approach by using Hirota’s KdV and discrete potential KdV equations as illustrative examples and then present some results about the ABS equations [6] and their deformed counterparts [22].

3.1 Hirota’s KdV equation

To find the equations in the family (13) the lowest generalized symmetries of which are of order two (at least in the $$n$$ direction), we consider the determining equation (14) with $$N=2$$ along with the requirement $$a_{n+1,m} \ne a_{n,m}$$ to exclude any symmetries of order one. A linear system of differential equations for $$R$$ can be derived which is consistent provided that $$a_{n+2,m} =a_{n,m}$$. Moving to symmetries in the $$m$$ direction and assuming that they are of order one, then our previous analysis implies that $$a_{n,m+1}=a_{n,m}$$. However, assuming that the equation admits symmetries of order two in the $$m$$ direction, our analysis yields two constraints, namely $$a_{n,m+2}= a_{n,m}$$ and $$a_{n+1,m+1} + a_{n,m} = a_{n+1,m}+a_{n,m+1}$$, which imply that $$a_{n,m}$$ must separate variables and be periodic with period two with respect to each variable.

We can summarize the above analysis for the symmetries of equation (13) in the following statement which can be proven by direct computations.

Specifically it maps equation (13) to $$W_{n,m}(a_{n+1,m},a_{n,m},0)=0$$ in (6), and symmetries (17) and (18) to (9) with $$c_n=1$$ and $$c_n =(-1)^n$$, respectively.

Finally, at the limit $$a_{n,m} \rightarrow \alpha$$, equation (13) reduces to (15) and in the same way (17)–(19) reduce to symmetries of the latter equation. Specifically, (19) reduces to the $$n$$ dependent symmetry in (16), (17) becomes the second order symmetry of (15) and (18) after multiplying it with the constant $$(a_{n,m}-a_{n+1,m}) (-1)^n$$ leads to the first order autonomous symmetry in (16).

3.2 The discrete potential KdV equation

Assuming that equation (21) admits a first order formal recursion operator, condition (22) with $$N=1$$ leads to a system of partial differential equations which is consistent provided that either $$R=0$$ or $$ \alpha_{n+1} = \alpha_n$$. Hence equation (21) admits symmetries of order one only if $$\alpha_{n+1}=\alpha_{n}$$ and they can be found in [23]. To find the equations in family (21) admitting symmetries of order two but not of order one, we consider (22) with $$N=2$$, along with $$ \alpha_{n+1} \ne \alpha_n$$ to exclude symmetries of lower order. The analysis of this equation implies that $$\alpha_n$$ must be a function of period two, $$\alpha_{n+2} = \alpha_{n}$$, and for the corresponding symmetries we can state the following result which can be shown by direct computations.

Symmetries (23), as well as (28) below, were first given in an equivalent form in [14].

3.3 Deautonomization of the ABS equations

We can go through the same analysis starting with any of the ABS equations H1–H3, Q1–Q3¹ [6] or the deformed H equations dH1–dH3 [22] and derive corresponding integrable equations depending on a periodic function of $$n$$. We summarize these results in the following statement which can be verified by direct computations.

In the $$m$$ direction, the lowest order symmetries of equation (25) are of order one if $$\beta_{m+1} = \beta_{m}$$, and of order two if $$\beta_{m+2}=\beta_{m}$$ and $$\beta_{m+1} \ne \beta_{m}$$. The formulae for latter symmetries follow from (26), (28) and (29) by applying the changes $$(u_{n+i,m},\alpha_{n+i}) \rightarrow (u_{n,m+i},\beta_{m+i})$$.

At the limit $$\alpha_n \rightarrow \alpha$$, equation (25) reduces to the original autonomous equation and (26) yields the first two symmetries of the latter equation. Indeed, symmetry (26) with $$g_n=1$$ reduces to the second order symmetry of the corresponding autonomous equation [4], whereas with $$g_n$$ such that $$g_n-g_{n+1}= k(\alpha_{n},\alpha_{n+1})$$ it reduces to a linear combination of the first and the second order symmetry of the original equation. Finally, at the limit $$\alpha_n \rightarrow \alpha$$, (28) and (29) lead to the $$n$$ dependent first order symmetries of the autonomous equations H1 and H2, respectively, given in [23].

3.4 Other deautonomized partial difference equations

We can apply the same procedure to other equations and derive their non-autonomous integrable counterparts which admit higher order symmetries. Herein, we present the discrete sine-Gordon and non-autonomous Hietarinta–Viallet equations, as well as the deautonomization of H1 admitting symmetries of order three in the $$n$$ direction.

3.4.1 Discrete sine-Gordon equation

3.4.2 Hietarinta–Viallet equation

3.4.3 Discrete potential KdV equation and third order symmetries

In all our previous examples, we have considered equations admitting symmetries of order one or two. We can continue and consider deautonomized equations admitting symmetries of order three in the $$n$$ direction. But this is a very difficult computational task as the calculations are lengthy and cumbersome. We were able to partially analyse the discrete potential KdV equation, and we can present the outcome of our analysis in the following form.

Symmetry (36) yields the first three generalized symmetries of H1 at the limit $$\alpha_{n} \rightarrow \alpha$$. In particular, (36) with $$g_n=1$$ reduces to third order symmetry of H1, with $$g_n= \alpha_{n+2}$$ leads to a linear combination of the third and the second order symmetries, and with $$g_{n}=\alpha_{n} \alpha_{n+1}$$ yields a linear combination of the first three symmetries of H1. On the other hand, (37) reduces to the first order $$n$$ dependent symmetry of the original potential KdV equation [23]. It should be noted that equation (35) was also discussed in [14] and a third order symmetry was given there (with some misprints) which corresponds to (36) with $$g_n = \alpha_{n+1} (\alpha_{n+1}-\alpha_{n+2}) (\alpha_{n}-\alpha_{n+2})$$.

4. Differential-difference equations and reductions

Exact reductions of a differential-difference equation lead to ordinary difference equations and in particular, as it was conjectured in [25], every autonomous difference equation obtained by an exact reduction of the differential-difference equation is an integrable mapping. In this section, we present exact reductions of the non-autonomous equations (9), (17), (23), and of their conserved forms, which lead to non-autonomous ordinary difference equations and corresponding first integrals. As some of these reductions lead to alternating QRT maps [18, 19], these examples suggest that in certain cases the QRT conjecture is also valid for non-autonomous equations.

5. Difference equations and periodic reductions

The difference equations we derived in the previous section can also be interpreted as defining certain group invariant solutions of the related partial difference equations. In this section, we consider some particular periodic reductions of the latter equations and show how they lead to families of non-autonomous ordinary difference equations. Since the periodic functions appearing in the equations are specific, our approach differs from similar considerations in [15, 26] where the reduction determines also the form of the arbitrary functions involved in the equations.

The former is (13) with $$a_{n,m} = \alpha_{n}$$ and the latter is (21) with $$\beta_{m} =0$$ where in both cases $$\alpha_{n}$$ is a non-constant and periodic with period two, i.e. $$\alpha_{n} = a (-1)^n + b$$. According to our previous analysis (see Theorems 3.1 and 3.2), these equations admit symmetries of order one and two in the $$m$$ and $$n$$ direction, respectively, and corresponding canonical conservation laws can be found in the Appendix. Moreover, solutions of these two equations are related according to

In both equations, we have suppressed the second index of functions $$v$$ and $$u$$ since it does not vary.

Moreover, we can replace shift operator $$\cal{T}$$ with $${\cal{S}}^k$$ in any conserved form of the original equations and potentially derive first integrals of the resulting maps. If $$({\cal{S}}-1)(p_n) = ({\cal{T}}-1)(q_n)$$ is a conserved form of the equation, then the corresponding candidate for a first integral will be $$p_n -q_{n}-\ldots-q_{n+k-1}$$. In the same way, we may derive a Lax pair for the reduced equations. More precisely, if system $$\Psi_{n+1,m} = L_{n,m} \Psi_{n,m}$$, $$\Psi_{n,m+1} = M_{n,m} \Psi_{n,m} $$ is the Lax pair of the original equation and $$L_{n,m+1} M_{n,m} = M_{n+1,m} L_{n,m}$$ is its compatibility condition, then in view of the $$(k,-1)$$ periodic reduction the latter relation becomes $$L_{n+k} M_{n} = M_{n+1} L_{n}$$, where we have omitted the second index. We may now define matrices $${\cal{M}}_n = L_{n+k-1} L_{n+k-2} \cdots L_{n} M_n^{-1}$$ and $${\cal{L}}_n = L_{n+k}$$, and then write the reduced compatibility condition as $${\cal{M}}_{n+1} = {\cal{L}}_n {\cal{M}}_n {\cal{L}}_n^{-1}$$.

It should be noted though that we can construct first integrals and Lax pairs in the way we described above only when $$k$$ is a multiple of two. This is because the equations are non-autonomous and function $$\alpha_{n}$$ is periodic with period $$2$$. As a consequence we separate the two cases below and first we discuss the even reductions $$(k=2 p)$$ for both equations and present their connections, and then we analyse the odd order reductions $$(k=2 p +1)$$.

5.1 The even order reductions

5.1.1 Hirota’s KdV equation

The trace of matrix $${\cal{M}}_n$$ is a polynomial of the form $$\sum_{i=0}^{p} F^{(i)}_n \lambda^{2 i-2}$$, and the coefficients of $$\lambda$$ are first integrals of the map. For the cases, we have checked we found $$F^{(p)}_n = J_n$$ and $$F^{(0)}_n = I_n$$, where $$J_n$$ and $$I_n$$ are given in (52) and (A.3), respectively.

5.1.2 The potential KdV equation

First integrals follow from the reduction of the conserved forms of equation (47) and can be found in the Appendix, see relations (A.9–A.11).

5.1.3 Bäcklund transformations

One can derive connections among the maps constructed previously by applying the same periodic reduction to Proposition 5.1. It is not difficult to see that transformation (48) is compatible only with the even order reductions because of the periodicity of function $$\alpha_{n}$$. So we state the following

5.2 The odd order reductions

5.2.1 Hirota’s KdV equation

5.2.2 The potential KdV equation

6. Concluding remarks

We presented a method for the deautonomization of partial difference and differential-difference equations using the existence of infinite hierarchies of symmetries in both directions as integrability detector. In this way, we derived integrable non-autonomous difference equations along with compatible differential-difference equations (symmetries) and conservation laws. Our results include non-autonomous versions of various known quad equations and their second and third order non-autonomous symmetries which, as far as we are aware, are new and deviate from previous works [22, 28, 29] in which the first symmetries are of order one and related to Yamilov’s discrete non-autonomous Krichever–Novikov equation [30]. The most interesting phenomenon is that most of the deautonomized equations and their symmetries are related via Miura transformations to the two-quad equation (8) and its symmetries (9). Our results along with the ones in [16, 17, 31] justify the importance of equation (9) as it was already pointed in [17].

Reductions of the derived differential-difference equations led to integrable non-autonomous ordinary difference equations and in certain cases to alternating QRT maps, whereas the reductions of the corresponding conservation laws yielded integrals for the resulting equations. And the periodic reductions of equations (46) and (47) led to families of non-autonomous maps and discrete Painlevé type equations.

It would be interesting to apply this method for the derivation of non-autonomous differential-difference equations of order higher than two (with one such example provided by (36)), but also to extend it to systems of difference equations and their symmetries.

Acknowledgements

The author would like to thank Frank Nijhoff and Andy Hone for useful discussions and suggestions.

Appendix

In this section, we have collected three canonical conservation laws and their reductions for equations (46) and (47).

• i) Canonical conservation laws for the non-autonomous Hirota KdV equation (46).  

  \begin{eqnarray} \left( \rho,\sigma\right) &=& \left( \log v_{n,m},\,\, \log \frac{v_{n,m} v_{n+1,m}}{\omega_{n+1,m}}\right)\quad {\mbox{with }}\,\, ({\cal{S}}^2-1)(\rho) = ({\cal{T}}-1)(\sigma), \\ \end{eqnarray}

  (A.1)

   

  \begin{eqnarray} \left( \varrho^{(0)},\varphi^{(0)} \right) &=& \left( \log \frac{v_{n,m}}{F_{(+)}}, \log v_{n,m}\right)\!,\quad \left( \varrho^{(1)},\varphi^{(1)} \right) = \left(\frac{2 v_{n,m} v_{n,m-1}}{F_{(+)} F_{(-)}}, \frac{v_{n,m} v_{n,m-1}-\alpha_{n}}{\alpha_{n} F_{(-)}} \right)\!, \end{eqnarray}

  (A.2)

  where $$F_{(\pm)}= v_{n,m\pm 1} v_{n,m} + \alpha_{n}$$ and $$({\cal{S}}-1)(\varrho^{(i)}) = ({\cal{T}}-1)(\varphi^{(i)})$$, $$i= 0,1$$.
  Considering even order periodic reductions $$(2 p,-1)$$, the reduction of the first conservation law in (A.2) yields the first integral  

  \begin{equation} I_n = (\alpha_{n} +v_n v_{n+2 p}) \prod_{i=1}^{2 p-1} v_{n+i}, \end{equation}

  (A.3)

  and the second pair in (A.2) becomes  

  \begin{equation} K_n = \frac{2 v_n v_{n-2 p}}{(v_n v_{n + 2 p } + \alpha_{n})(v_n v_{n -2 p } + \alpha_{n})} - \sum_{i=0}^{2 p -1} \frac{1}{\alpha_{n+i}}{\cal{S}}^i \left( \frac{v_n v_{n-2 p} - \alpha_{n}}{v_n v_{n-2 p} + \alpha_{n}} \right)\!, \end{equation}

  (A.4)

  where one has to use successively the map to eliminate the negative shifts of $$v_{n}$$ and derive the corresponding integral. Moreover, conservation law (A.1) leads to the two-integral $$(T_{n+2}=T_n)$$ 

  \begin{equation} T_n =v_{n} \prod_{i=0}^{p -1} {\cal{S}}^{2 i} \left(\frac{v_{n+2} v_{n+1} (v_{n+1} v_{n}- \alpha_n)- \alpha_n (v_{n+1} v_{n}- \alpha_{n+1})}{v_{n} v_{n+1}} \right)\!. \end{equation}

  (A.5)
• ii) Canonical conservation laws for the non-autonomous H1 equation (47).  

  \begin{eqnarray} \left(\tilde{\rho},\tilde{\sigma} \right) &=& \left(\log (u_{n,m+1}-u_{n-1,m}),\,\, \log ( h_{n,m})\right),\quad {\mbox{with }}\,\, ({\cal{S}}^2-1)(\tilde{\rho}) = ({\cal{T}}-1)(\tilde{\sigma}), \\ \end{eqnarray}

  (A.6)

   

  \begin{eqnarray} \tilde{\varrho}^{(0)} &=& \log G_{n,m},\quad \tilde{\varphi}^{(0)} = \log (u_{n+1,m}-u_{n,m-1}), \\ \end{eqnarray}

  (A.7)

   

  \begin{eqnarray} \tilde{\varrho}^{(1)} &=& \frac{-2}{G_{n,m} G_{n,m-1}},\quad \tilde{\varphi}^{(1)} = \frac{1}{G_{n,m-1}^2} \left( \frac{u_{n+1,m-1}-u_{n,m-2}}{u_{n,m-1}-u_{n+1,m-2}} + \frac{u_{n,m}-u_{n+1,m-1}}{u_{n,m-1}-u_{n+1,m}}\right)\!, \end{eqnarray}

  (A.8)

  where $$G_{n,m} := u_{n,m+1}- u_{n,m-1}$$ and $$({\cal{S}}-1)(\tilde{\varrho}^{(i)}) = ({\cal{T}}-1)(\tilde{\varphi}^{(i)})$$, $$i= 0,1$$.
  The even order periodic reductions $$(2 p,-1)$$ of the conservation laws (A.6) and (A.7) yield the two-integral  

  \begin{equation} T_n = \left(\prod_{i=0}^{p-1} \left((w_{n+2 i +2} +w_{n+2 i +1})(w_{n+2 i +1}+w_{n+2 i}) + \alpha_{n+1}-\alpha_{n} \right)\right) \sum_{j=1}^{2 p-1}w_{n+j} \end{equation}

  (A.9)

  and the first integral  

  \begin{equation} I_n = \prod_{i=0}^{2p-1} {\cal{S}}^i \left(\sum_{j=0}^{2 p} w_{n+j}\right){\Big{/}}\sum_{\ell=0}^{4 p-1}w_{n+\ell} \end{equation}

  (A.10)

  for the reduced map (55). Finally the pair (A.8) generates the first integral  

  \begin{equation} K_n = \frac{2}{\left(\sum_{i=0}^{4 p-1} w_{n+i}\right)\left(\sum_{i=2p}^{6 p-1} w_{n+i}\right) } + \sum_{i=0}^{2p-1} {\cal{S}}^i \left(\left(\sum_{j=0}^{4p-1} w_{n+j}\right)^{-2} \left(\frac{\sum_{s=0}^{2p} w_{n+s}}{\sum_{r=1}^{2p-1} w_{n+r}}- \frac{\sum_{s=2p+1}^{4p-1} w_{n+s}}{\sum_{r=2p}^{4p} w_{n+r}}\right)\right). \end{equation}

  (A.11)

  In the first integrals (A.9)–(A.11) we must use equation (55) and its shifts to eliminate the values $$w_{n+\ell}$$ with $$\ell> 2p$$.

Footnotes

References