Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

Francesco Calogero; Farrin Payandeh

doi:10.1080/14029251.2019.1640460

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Volume 26, Issue 4, July 2019, Pages 509 - 519

Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations

Authors

Francesco Calogero^a^{, b}^{, *}^{, †}, Farrin Payandeh^c^{, ‡}^{, §}

^aPhysics Department, University of Rome “La Sapienza”, Rome, Italy

^bINFN, Sezione di Roma 1

^cDepartment of Physics, Payame Noor University, PO BOX 19395-3697 Tehran, Iran

^*e-mail: francesco.calogero@roma1.infn.it

^†e-mail: francesco.calogero@uniroma1.it

^‡e-mail: farrinpayandeh@yahoo.com

^§e-mail: f_payandeh@pnu.ac.ir

Corresponding Authors

Francesco Calogero, Farrin Payandeh

Received 13 March 2019, Accepted 6 April 2019, Available Online 9 July 2019.

DOI: 10.1080/14029251.2019.1640460 How to use a DOI?
Abstract: In this paper we identify certain peculiar systems of 2 discrete-time evolution equations,
x˜n=F(n)(x1,x2), n=1,2,
which are algebraically solvable. Here ℓ is the “discrete-time” independent variable taking integer values (ℓ = 0, 1, 2,...), x_n ≡ x_n(ℓ) are 2 dependent variables, and x˜n≡xn(ℓ+1) are the corresponding 2 updated variables. In a previous paper the 2 functions F⁽ⁿ⁾(x₁,x₂), n = 1, 2, were defined as follows: F⁽ⁿ⁾(x₁,x₂) = P₂ (x_n,x_n₊₁), n = 1,2 mod[2], with P₂(x₁,x₂) a specific second-degree homogeneous polynomial in the 2 (indistinguishable!) dependent variables x₁(ℓ) and x₂(ℓ). In the present paper we further clarify some aspects of that model and we present its extension to the case when F(n)(x1,x2)=Qk(n)(x1,x2), n = 1, 2, with Qk(n)(x1,x2) a specific homogeneous function of arbitrary (integer) degree k (hence a polynomial of degree k when k > 0) in the 2 dependent variables x₁(ℓ) and x₂(ℓ).
Copyright: © 2019 The Authors. Published by Atlantis and Taylor & Francis
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. Introduction and main results

The results reported in this paper are a nontrivial extension of those reported in [1], to which the interested reader is referred: (i) for a terse overview of an old (see [2,3])—and recently substantially improved (see [4–10])—technique to identify solvable dynamical systems in continuous-time t; (ii) for an introduction to the extension of that approach to the case of discrete-time ℓ (see [11–13]); (iii) for a very terse review of previous results on analogous solvable discrete-time models (see [5]). To make the relevance of the present paper immediately clear we report already in this introductory section what we consider its main findings.

Notation 1.1.

Hereafter ℓ = 0, 1, 2,... denotes the discrete-time independent variable; the dependent variables are x_n ≡ x_n(ℓ) (with n = 1, 2), and the notation x˜n≡xn(ℓ+1) indicates the once-updated values of these variables. We shall also use other dependent variables, for instance y_m ≡ y_m(ℓ) (with m = 1, 2) and then of course likewise ỹ_m ≡ y_m(ℓ + 1). Variables such as x_n and y_m are generally assumed to be complex numbers (this does not exclude that they might in some cases take only real values); note that while these quantities generally depend on the discrete-time variable ℓ, only occasionally this is explicitly indicated. Parameters such as a, b, α, β, γ, B_n, C_j (with n = 1, 2; j = 1, 2, 3) are generally time-independent complex numbers, while k, q, r are real integers; and indices such as n, m, j are of course positive integers (the values they may take shall be explicitly indicated or be quite clear from the context). The quantity S denotes an arbitrarily assigned sign, S = ±: note that generally the assignment of the sign S shall depend on the discrete-time ℓ, S ≡ S(ℓ) (and of course it has the same assigned value S(ℓ) for each value of ℓ). Finally: the convention is hereafter adopted according to which ∑s=s−s+f(s)=0 and ∏s=s−s+f(s)=1 whenever s₊ < s₋.

Remark 1.1.

In this paper the term solvable generally characterizes systems of discrete-time evolution equations the initial-values problem of which is explicitly solvable by algebraic operations.

The main result of [1] is to provide the explicit solution of the initial-values problem for the system of 2 nonlinearly-coupled discrete-time evolution equations

x˜n=−(x1+x2)[a(x1+x2)+Sb(xn−xn+1)], n=1,2 ,mod[2].(1.1)

(Note here the notational changes with respect to [1]: implying α = 2a, β = 2b and the explicit introduction of the arbitrary ℓ-dependent sign S ≡ S(ℓ), which is indeed implicit in the formulas (2.11b) and (1.2a) of [1].)

Remark 1.2.

The characteristic of the discrete-time evolution of this model is that, if the sign S(ℓ) is positive, S(ℓ) = +, then

x˜n≡xn(ℓ+1)=−[x1(ℓ)+x2(ℓ)]{a[x1(ℓ)+x2(ℓ)]+b[xn(ℓ)−xn+1(ℓ)]}, n=1,2 ,mod[2];(1.2)

while if S(ℓ) = −, then

x˜n≡xn(ℓ+1)=−[x1(ℓ)+x2(ℓ)]{a[x1(ℓ)+x2(ℓ)]−b[xn(ℓ)−xn+1(ℓ)]}, n=1,2 ,mod[2].(1.3)

So it might appear that we are dealing here with a large plurality of distinct dynamical systems, as yielded by all the possible (ℓ-dependent!) assignments of the values—positive or negative sign—of S(ℓ). But this is not really the case, because it is evident that the different outcomes of these two systems (1.2) and (1.3) is merely to exchange the roles of the variables x₁(ℓ) and x₂(ℓ); hence the system (1.1) yields a well-defined, unique evolution if we consider the two variables x₁(ℓ) and x₂(ℓ) to identify 2 indistinguishable entities, such as the 2 different zeros of a generic second-degree polynomial. And it was indeed shown in [1] that the solution of the initial-values problem for the discrete-time evolution (1.1)—with the sign S(ℓ) being arbitrarily assigned for every value of ℓ—is provided by the 2 zeros x₁(ℓ) and x₂(ℓ) of a specific second-degree polynomial,

p2(z;ℓ)=z2+y1(ℓ)z+y2(ℓ)=[z−x1(ℓ)][z−x2(ℓ)],(1.4)

the 2 coefficients of which, y₁(ℓ) and y₂(ℓ), are unambiguously determined and indeed explicitly known in terms of the initial values x₁(0) and x₂(0).

This remark is reported here as an introduction to the somewhat more peculiar phenomenon associated with the more general models considered in the present paper, see below.

The first main result of the present paper is to provide (in the following Sections 2 and 3) the explicit solution of the initial-values problem for the following, more general, system of 2 nonlinearly-coupled discrete-time evolution equations

x˜n=d(B1x1+B2x2)k[αgn(B1x1+B2x2)+β(−1)nSbn+1(g2x1−g1x2)], n=1,2 ,mod[2],(1.5)

with (above and hereafter)

d=(1/2)[(B1)2C2+(B2)2C1−B1B2C3]−1,(1.6)

g1=2B1C2−B2C3, g2=2B2C1−B1C3,(1.7)

and with k an arbitrary (time-independent, integer) parameter (of course k must be chosen to be a positive integer if one prefers that the right-hand side of these equations be homogeneous polynomials of degree k + 1). The 7 parameters α, β and B_n, C_j (with n = 1, 2; j = 1, 2, 3) are arbitrary (see Notation 1.1).

Remark 1.3.

The restriction to integer values of the parameter k—and of the analogous exponents q and r, see below and Notation 1.1—is to make sure that the right-hand sides of the main discrete-time evolution equations we introduce and discuss in this paper are analytic, hence unambiguously defined, functions.

For k = 1 the system (1.5)–(1.7) can of course be re-written as follows:

x˜n=an1(x1)2+an2(x2)2+an3x1x2, n=1,2,(1.8)

with

anm=dBn[αgnBn+Sβ(−1)n+mBn+1gm+1], n=1,2, mod[2]; m=1,2, mod[2];(1.9)

an3=d[2αgnB1B2+Sβ(−1)nBn+1(B2g2−B1g1)],(1.10)

with the parameters d and g_n defined as above (see (1.5)–(1.7)).

Remark 1.4.

This case with k = 1 is sufficiently interesting to deserve this additional remark. Its equations of motion (1.8)–(1.10) are written in terms of the 6 parameters a_n_j (n = 1, 2; j = 1, 2, 3) in terms of the 7 a priori arbitrary parameters α, β and B_n, C_j (with n = 1, 2; j = 1, 2, 3). But this does not imply that these 6 parameters a_n_j can be arbitrarily assigned: the fact that the right-hand sides of the 2 recursions (1.5) feature a common zero—they both vanish when B₁x₁ + B₂x₂ vanishes—is easily seen to imply that these 6 parameters are constrained to satisfy (at least!) the following nonlinear relationship:

(a11a22−a21a12)2+(a13a21−a11a23)(a13a22−a12a23)=0(1.11)

(see, if need be, Remark 5.3 of Ref. [8]).

For k = −1 the system (1.5)–(1.7) can of course be re-written as follows:

x˜n=Dn1x1+Dn2x2B1x1+B2x2, n=1,2,(1.12)

Dnm=d[αgnBm+Sβ(−1)n+mBn+1gm+1], n=1,2, mod[2]; m=1,2, mod[2],(1.13)

again with the parameters d and g_n defined as above in (1.5)–(1.7).

The second main result of the present paper is to provide the explicit solution of the initial-values problem for the system of 2 nonlinearly-coupled discrete-time evolution equations

x˜n=(2x1+x2)k[(−1)ka(2x1+x2)+S(−1)nnb(x1−x2)], n=1,2.(1.14)

Note that in this case—differently from those reported above, see (1.1) (as well as (1.5)–(1.7) with B₁ = B₂ and C₁ = C₂)—the discrete-time evolutions of the 2 variables x₁(ℓ) and x₂(ℓ) are essentially different; hence these dependent variables are no more related to each other by just an exchange of their identities. Yet these evolution equations, (1.14), still have a somewhat analogous property to those discussed above: for any given pair of initial data, x₁(0) and x₂(0), only 2 different solutions, say the two pairs x1(+)(ℓ), x2(+)(ℓ) and x1(−)(ℓ), x2(−)(ℓ), emerge, not 2^ℓ as it might instead be inferred due to the indeterminacy of the signs S(ℓ) appearing in these evolution equations (1.14) at every step of the discrete-time evolution they yield. This is demonstrated by the explicit solution of the initial-values problem for this model, as reported below; but the interested reader may readily understand the origin of this remarkable phenomenon by noting that a simple iteration of (1.14) entails the (double-step) formula

xn(ℓ+2)=(−1)k3k+1ak[2x1(ℓ)+x2(ℓ)]k(k+2)⋅⋅{a2[2x1(ℓ)+x2(ℓ)]−S(ℓ)S(ℓ+1)(−1)nnb2[x1(ℓ)−x2(ℓ)]}, n=1,2;(1.15)

indeed the quantity S(ℓ)S(ℓ + 1) is again just a sign, i.e. it can only take the 2 values +1 or −1, as implied by the very definition of S(ℓ), see Notation 1.1. Hence this formula clearly shows that—starting from the initial values x_n(0)—also at the ℓ = 2 level (as at the ℓ = 1 level)—this evolution yields only two (not four!) alternative values for the pair x₁(2), x₂(2), say x1(+)(2), x2(+)(2) (corresponding to S(0)S(1) = +) respectively x1(−)(2), x2(−)(2) (corresponding to S(0)S(1) = −); and this phenomenology prevails—see below—at every subsequent level ℓ > 2 of the discrete-time evolution.

Additional results and proofs—including the explicit solutions of the initial-values problems for the 2 systems of 2 discrete-time evolution equations (1.5)–(1.7) respectively (1.14)—are provided in Section 2. Section 3 contains some additional developments.

2. Additional results and proofs

The starting point of our treatment in this Section 2 is the following algebraically solvable system of 2 discrete-time evolution equations in the 2 dependent variables y₁ ≡ y₁(ℓ) and y₂ ≡ y₂(ℓ):

y˜1=α(y1)1+k, y˜2=β2y2(y1)q+γ(y1)r,(2.1)

where the 6 parameters α, β, γ, k, q, r can be a priori arbitrarily assigned (see Notation 1.1; but see also below for eventual restrictions on these parameters). The fact that the initial-values problem for this system of 2 discrete-time evolution equations is solvable is demonstrated by exhibiting its solution:

y1(ℓ)=α[(1+kℓ−1)/k][y1(0)](1+k)ℓ,(2.2)

y2(ℓ)=β2ℓα(q/k2)[(1+kℓ)−kℓ−1][y1(0)](q/k)[(1+k)ℓ−1]Y2(ℓ),(2.3)

Y2(ℓ)=y2(0)+γ∑s=0ℓ−1{β−2(s+1)αu[(1+k)s−1]/k2+(q/k)s[y1(0)][u(1+k)s+q]/k},(2.4)

u≡kr−(1+k)q.(2.5)

The interested reader will verify that this solution is consistent with the initial data y₁(0), y₂(0) and that it does satisfy the system of evolution equations (2.1). Note that the assumption that the 3 parameters k, q, r be integers (see Notation 1.1) is necessary and sufficient to guarantee that all exponents in these formulas are integers, thereby excluding any nonanalyticity/indeterminacy in the equations of motion (2.1) and in their solutions (2.2)–(2.5).

For reasons that shall be clear in the following, we are also interested in the solution of the system (2.1) in the particular case when the 2 parameters q and r are related to k as follows:

q=2k, r=2(1+k).(2.6)

Note that this assignment implies u = 0 (see (2.5)). In this case the sum in the right-hand side of (2.4) becomes a geometric sum, hence it can be performed explicitly; therefore in this special case the formulas (2.2)–(2.5) are replaced by the following, more explicit, versions:

y1(ℓ)=α[(1+kℓ−1)/k][y1(0)](1+k)ℓ,(2.7)

y2(ℓ)=β2ℓα2[(1+k)ℓ−kℓ−1]/k[y1(0)]2[(1+k)ℓ−1]Y2(ℓ),(2.8)

Y2(ℓ)=y2(0)+γβ−2[y1(0)]2[(α/β)2ℓ−1(α/β)2−1].(2.9)

Our next task is to identify various systems—satisfied by 2 new dependent variables x₁ ≡ x₁(ℓ) and x₂ ≡ x₂(ℓ)—the solutions of which can be identified via the solution of the system (2.1). To this end we set, to begin with,

y1=−(x1+x2), y2=x1x2,(2.10)

which clearly implies that x₁ and x₂ are the 2 zeros of the following second-degree monic polynomial:

z2+y1z+y2=(z−x1)(z−x2),(2.11)

implying

(xn)2+y1xn+y2=0, n=1,2,(2.12)

xn(ℓ)=(1/2){−y1(ℓ)+(−1)n{[y1(ℓ)]2−4y2(ℓ)}1/2}, n=1,2.(2.13)

Likewise (replacing ℓ with ℓ + 1)

(x˜n)2+y˜1x˜n+y˜2=0, n=1,2,(2.14)

x˜n=(1/2[−y˜1+(−1)nΔ˜1]), (Δ˜1)2=(y˜1)2−4y˜2.(2.15)

We then use the evolution equations (2.1) to express, in the right-hand sides of (2.15), ỹ₁ and ỹ₂ in terms of y₁ and y₂, and then the relations (2.10) to express y₁ and y₂ in terms of x₁ and x₂; thereby getting the following system of discrete-time evolution equations for the 2 dependent variables x₁(ℓ) and x₂(ℓ):

x˜n=(1/2){−α[−(x1+x2)]k+1+(−1)nΔ1}, n=1,2,(2.16)

Δ1=S{α2[−(x1+x2)]2(k+1)−4β2x1x2[−(x1+x2)]q−4γ[−(x1+x2)]r}1/2.(2.17)

Remark 2.1.

The ± sign S ≡ S(ℓ) in this definition (2.17) of Δ₁ might be considered pleonastic in view of the sign indeterminacy of the square-root in the right-hand side of this formula (2.17). We did put it there as a reminder of the fact that, for every value of the discrete time ℓ, the assignment of the labels 1 or 2 to the solutions of the second-degree evolution equation (2.16)–(2.17) is optional. Indeed the two variables x₁(ℓ) and x₂(ℓ)—the discrete-time evolution of which is identified with the evolution of the 2 zeros of the second-degree ℓ-dependent (monic) polynomial (2.11) the 2 coefficients y₁(ℓ) and y₂(ℓ) of which evolve according to the solvable system (2.1)—should be considered indistinguishable. Note that this implies that this evolution equation is actually not quite deterministic; it is only deterministic for the couple of indistinguishable dependent variables x₁ ≡ x₁(ℓ), x₂ ≡ x₂(ℓ): a well-known phenomenon for this kind of evolution equations, as discussed above and in the past—see for instance [13] and Chapter 7 (“Discrete Time”) of the book [5] (in particular Remark 7.1.2 there).

Remark 2.2.

Of course when s is an integer the power (−z)^s can be replaced by z^s respectively −z^s for s even respectively odd.

The results obtained so far allow to formulate the following

Proposition 2.1.

The solution of the initial-values problem for the system of discrete-time evolution equations (2.16)–(2.17) is provided—up to the limitations implied by Remark 2.1—by the 2 zeros x₁ ≡ x₁(ℓ) and x₂ ≡ x₂(ℓ) of the polynomial (2.11) (see (2.13)), with its coefficients y₁ ≡ y₁(ℓ) and y₂ ≡ y₂(ℓ) given by the formulas (2.2)–(2.5) where of course (see (2.10)) y₁(0) = −x₁(0) − x₂(0) and y₂(0) = x₁(0)x₂(0).

We believe that the interest—both theoretical and applicative—of the system (2.16)–(2.17) is modest, due to the appearance of a square root in the right-hand side of its equations of motion. Hence our next step is to restrict attention to the values identified by Eq. (2.6). Indeed these assignments—beside allowing the more explicit solution of the system of recursions (2.1) characterizing the discrete-time evolution of the 2 dependent variables y₁(ℓ) and y₂(ℓ), see (2.7)–(2.9)—also allow (remarkably!) to get rid of the square-root in the right-hand side of (2.17), provided we more-over make the assignments

γ=a2−b2, α=2a, β=2b;(2.18)

obtaining thereby just the system of evolution equations (1.5)–(1.7). This allows us to prove a sub-case of our first main result, in the guise of the following:

Proposition 2.2.

The solution of the initial-values problem for the system of discrete-time evolution equations (1.5)–(1.7)—with B₁ = B₂ = −1, C₁ = C₂ = 0, C₃ = 1 (compare eq. (3.15) below with (2.10))—is provided by the 2 zeros x₁ ≡ x₁(ℓ) and x₂ ≡ x₂(ℓ) of the polynomial (2.11) (see (2.13)), with its coefficients y₁ ≡ y₁(ℓ) and y₂ ≡ y₂(ℓ) given by the formulas (2.7)–(2.9) (with the assignments (2.18) and (2.6)), where of course (see (2.10)) y₁(0) = −x₁(0)−x₂(0) and y₂(0) = x₁(0)x₂(0).

Let us again emphasize that, for each value of ℓ, the assignment of the labels 1 or 2 to the two zeros of the polynomial (2.11) is optional.

This Proposition 2.2 corresponds to the special case—with B₁ = B₂ = −1, C₁ = C₂ = 0, C₃ = 1—of the first main result reported in Section 1 (see the paragraph including the eqs. (1.5)–(1.7)); a proof of the first main result in the general case with arbitrary parameters B_n and C_j is provided at the end of the paper.

Our next step is to modify the relationship among the variables x_n(ℓ) and y_n(ℓ) by replacing the second-degree (monic) polynomial (2.11) with the following (monic) third-degree polynomial:

z2+y1z2+y2z+y3=(z−x1)2(z−x2),(2.19)

where of course now

y1=−(2x1+x2), y2=x1(x1+2x2), y3=−(x1)2x2.(2.20)

Note that—following [6] and [8]—we are now focussing on a special polynomial of degree 3 which features only 2 zeros, the zero x₁ with multiplicity 2 and the zero x₂ with multiplicity 1: this of course implies that these 2 zeros are now quite distinct, and moreover that now only 2 of the 3 coefficients y₁, y₂, y₃ can be arbitrarily assigned, the unassigned one being determined in terms of the other two by the requirement that the 3 equations (2.20) be simultaneously satisfied.

Remark 2.3.

It is for instance easy to check that the coefficient y₃ of the polynomial (2.19) is given by the following formula in terms of the other 2 coefficients y₁ and y₂:

y3={−2(y1)3+9y1y2+2S[(y1)2−3y2]3/2}/27.(2.21)

Likewise, from the first 2 of the 3 formulas (2.20) one easily obtains the following expressions of the two zeros x₁ and x₂ in terms of the 2 coefficients y₁ and y₂:

xn=(1/2){−y1+S(−1)nn[(y1)2−3y2]1/2}, n=1,2.(2.22)

Note the ambiguity in these formulas implied by the indeterminacy of the square-root sign, as evidenced by the presence of the sign S, see Notation 1.1.

It is now convenient to write again these expressions of the 2 zeros x_n ≡ x_n(ℓ), but with ℓ replaced by ℓ + 1:

x˜n=(1/2){−y˜1+S˜(−1)nn[(y˜1)2−3y˜2]1/2}, n=1,2.(2.23)

Our next step is to then assume again that the two coefficients ỹ₁ ≡ y₁(ℓ + 1) and ỹ₂ ≡ y₂(ℓ + 1) evolve according to the solvable discrete-time system (2.1). By proceeding in close analogy with the previous treatment—i.e., by replacing in the right-hand sides of (2.23) the variables ỹ₁ and ỹ₂ via the evolution equations (2.1) and then in the right-hand sides of the resulting equations y₁ and y₂ via (2.20)—we thereby obtain the following:

Proposition 2.3.

The solution of the initial-values problem for the following system of discrete-time evolution equations

x˜n=(1/2){−α[−(2x1+x2)]k+1+(−1)nSΔ}, n=1,2,(2.24)

Δ={α2[−(2x1+x2)]2(k+1)−3β2(x1)2x2[−(2x1+x2)]q−3γ[−(2x1+x2)]r}1/2,(2.25)

is provided by the 2 zeros x₁ ≡ x₁(ℓ) and x₂ ≡ x₂(ℓ) of the polynomial (2.19)–(2.20)—i.e., by the formulas (2.22)—with the coefficients y₁ ≡ y₁(ℓ) and y₂ ≡ y₂(ℓ) given by the formulas (2.2)–(2.5) where of course now (see (2.20)) y₁(0) = −2x₁(0) − x₂(0) and y₂(0) = x₁(0)[x₁(0) + 2x₂(0)].

Note that this system features the same kind of 2-fold ambiguity as discussed in the previous Section 1 (see after eq. (1.14)).

However, a “defect” of this solvable system is the appearance in the right-hand side of its discrete-time equations of motion (2.24)–(2.25) of a square root; but this “defect” can now be eliminated by restricting the parameters q and r to satisfy the condition (2.6)—just the same condition that allows to replace the solution (2.2)–(2.5) with the more explicit solution (2.7)–(2.9)—and by moreover replacing the assignments (2.18) with the following assignments

γ=3(a2−b2), α=3a, β=3b.(2.26)

It is indeed easily seen that there thereby holds the following:

Proposition 2.4.

The solution of the initial-values problem for the system of discrete-time evolution equations (1.14) is provided by the 2 distinct zeros x₁ ≡ x₁(ℓ) and x₂ ≡ x₂(ℓ) of the polynomial (2.19)–(2.20)—i.e., by the formulas (2.22)—with the coefficients y₁ ≡ y₁(ℓ) and y₂ ≡ y₂(ℓ) given by the formulas (2.7)–(2.9) with (2.26) where of course now (see (2.20)) y₁(0) = −2x₁(0) − x₂(0) and y₂(0) = x₁(0)[x₁(0) + 2x₂(0)].

Let us again emphasize that, for each value of the discrete-time ℓ, this prescription yields 2 different solutions, say the 2 different pairs x1(+)(ℓ), x2(+)(ℓ) and x1(−)(ℓ), x2(−)(ℓ).

This Proposition 2.4 corresponds to the second main result reported in Section 1 (see above the paragraph including Eq. (1.14)).

3. Additional developments

An important issue is the possibility to generalize the algebraically solvable systems treated in the previous Section 2—which feature the 2 arbitrary (possibly complex) parameters a and b—to more general analogous models involving more free parameters. Following the treatment given in [1], let us outline how this can be done for the system (1.14).

The procedure is to introduce the simple invertible change of dependent variables

z1=A11x1+A12x2, z2=A21x1+A22x2,(3.1)

x1=(A22z1−A12z2)/D, x2=(−A21z1+A11z2)/D,(3.2)

D=A11A22−A12A21,(3.3)

where of course the 4 parameters A_nm (n = 1, 2 ; m = 1, 2) are 4, a priori arbitrary, time-independent constants (the restriction to time-independent constants is because we prefer in this paper to focus on autonomous systems of discrete-time evolutions).

It is then a matter of simple algebra to obtain the—of course algebraically solvable—evolution equations satisfied by the dependent variables z₁ ≡ z₁(ℓ) and z₂ ≡ z₂(ℓ):

z˜n=D−(k+1)[(2A22−A21)z1+(A11−2A12)z2]k⋅⋅[An1f1(z1,z2;k)+An2f2(z1,z2;k)], n=1,2,(3.4)

fn(z1,z2;k)=(θk;2,n;1A22−θk;1,n;1A21)z1+(θk;1,n;0A11+θk;2,n;0A12)z2,(3.5)

θk;n1,n2;n=(−1)kn1a+(−1)nn2Sb.(3.6)

Let us also display these equations in the—possibly more relevant to applicative contexts—special cases with k = ±1.

For k = 1:

z˜n=an1(z1)2+an2(z2)2+an3z1z2, n=1,2,(3.7)

an1=D−2λ2(η1An1+η2An2), n=1,2,(3.8)

an2=D−2λ1(η3An1+η4An2), n=1,2,(3.9)

an3=D−2[λ1(An1η11+An2η21)+λ2(An1η12+An2η22)], n=1,2,(3.10)

λn=(−1)n(2An2−An1), n=1,2,(3.11)

ηn1=−θ1;1,n;nA21−θ0;2,n;n+1A22, n=1,2,(3.12)

ηn2=−θ0;1,n;nA11+θ0;2,n;n+1A12, n=1,2.(3.13)

For k = −1:

z˜n=An1f1(z1,z2;−1)+An2f2(z1,z2;−1)λ1z1+λ2z2, n=1,2,(3.14)

with f_n(z₁,z₂;−1) and λ_n (n = 1, 2) defined as above (see (3.5), (3.6) and (3.11)).

An alternative generalization is based on the replacement of the relations (2.10) and (2.20) and their generalization via (3.1)–(3.3) by the following more general relations:

y1=B1x1+B2x2, y2=C1(x1)2+C2(x2)2+C3x1x2.(3.15)

Note that these relations involve the 5 a priori arbitrary parameters B_n and C_j (n = 1, 2; j = 1, 2, 3), and that they are easily inverted:

x1=−g±Γ2f, x2=y1−B1x1B2,(3.16)

f=(B2)2C1+(B1)2C2−C3B1B2(B2)2, g=(B2C3−2B1C2)y1(B2)2,(3.17)

Γ2=g2−4fh, h=C2(y1)2−(B2)2y2(B2)2.(3.18)

Starting from these formulas, and proceeding in close analogy with the treatment provided above—which involve of course the assumption that the quantities x_n and y_n (n = 1, 2) are ℓ-dependent while the parameters B_n and C_j (n = 1, 2; j = 1, 2, 3) are ℓ-independent, and moreover that the quantities y_n evolve according to the solvable discrete-time evolution equations (2.1) with (2.6)—one arrives at the equations (1.5)–(1.7). Note again the presence—in the right-hand side of eq. (1.5)—of the sign S ≡ S(ℓ), and that these equations—involving no square roots in their right-hand sides—have been obtained thanks to the assignments (2.6) and by moreover setting (in place of (2.18) respectively (2.26))

γ=[(C3)2−4C1C2](β2−α2)4[(B1)2C2+(B2)2C1−B1B2C3].(3.19)

This concludes the proof of the first main result of this paper (see the paragraph including eqs. (1.5)–(1.7) in Section 1).

Assigning solvable evolutions to y₁ and y₃ or y₂ and y₃ (rather than to y₁ and y₂; in the case of the third-degree polynomial (2.19)) are possible further developments, but we postpone the relevant treatments to future papers.

4. Acknowledgements

Both authors like to thank Piotr Grinevich, Paolo Santini and Nadezda Zolnikova for very useful suggestions. FP likes to thank the Physics Department of the University of Rome “La Sapienza” for the hospitality from April 2018 to February 2019 (during her sabbatical), when the results reported in this paper were obtained.

References

[1]F. Calogero and F. Payandeh, Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete- Time via 2 Nonlinearly-Coupled First-Order Recursion Relations with Polynomial Right-Hand Sides, J. Nonlinear Math. Phys, Vol. 26, No. 2, 2019, pp. 273-280.

[2]F. Calogero, “Motion of Poles and Zeros of Special Solutions of Nonlinear and Linear Partial Differen- tial Equations, and Related “Solvable” Many-Body Problems, Nuovo Cimento, No. 43B, 1978, pp. 177-241.

[3]F. Calogero, Classical many-body problems amenable to exact treatments, Springer, Heidelberg, 2001, pp. 749. Lecture Notes in Physics Monograph m66

[4]F. Calogero, New solvable variants of the goldfish many-body problem, Studies Appl. Math, Vol. 137, No. 1, 2016, pp. 123-139. http://arxiv.org/abs/1806.07502

[5]F. Calogero, Zeros of Polynomials and Solvable Nonlinear Evolution Equations, Cambridge University Press, Cambridge, U. K., 2018, pp. 168.

[6]O. Bihun and F. Calogero, Time-dependent polynomials with one double root, and related new solvable systems of nonlinear evolution equations, Qual. Theory Dyn. Syst, Vol. 18, No. 1, 26 July 2018, pp. 153-181. (201);

[7]O. Bihun. Time-dependent polynomials with one multiple root and new solvable dynamical systems, arXiv:1808.00512v1 [math-ph] 1 Aug 2018.

[8]F. Calogero and F. Payandeh, Polynomials with multiple zeros and solvable dynamical systems includ- ing models in the plane with polynomial interactions, J. Math. Phys. (submitted to, 20.11.2018)

[9]F. Calogero and F. Payandeh, Solvable dynamical systems in the plane with polynomial interactions, a chapter in Vol. 1 (Integrable Systems) of the collective book dedicated to Emma Previato for her 65th birthdate, R. Donagi and T. Shaska (editors), Integrable Systems and Algebraic Geometry, Cambridge University Press, 2019. LMS Lecture Notes Series arXiv: 1904.02151v1 [math-ph] 31 Mar. 2019

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Journal: Journal of Nonlinear Mathematical Physics
Volume-Issue: 26 - 4
Pages: 509 - 519
Publication Date: 2019/07/09
ISSN (Online): 1776-0852
ISSN (Print): 1402-9251
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TY  - JOUR
AU  - Francesco Calogero
AU  - Farrin Payandeh
PY  - 2019
DA  - 2019/07/09
TI  - Two Peculiar Classes of Solvable Systems Featuring 2 Dependent Variables Evolving in Discrete-Time via 2 Nonlinearly-Coupled First-Order Recursion Relations
JO  - Journal of Nonlinear Mathematical Physics
SP  - 509
EP  - 519
VL  - 26
IS  - 4
SN  - 1776-0852
UR  - https://doi.org/10.1080/14029251.2019.1640460
DO  - 10.1080/14029251.2019.1640460
ID  - Calogero2019
ER  -

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