YAMILOV’S THEOREM FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Уфимский математический журнал. Том 13. № 2 (2021). С. 158-165.

Dedicated to our 'master Alexei Shabat and our colleague and good friend Ravil Yamilov

YAMILOV'S THEOREM FOR DIFFERENTIAL AND DIFFERENCE EQUATIONS

DECIO LEVI, MIGUEL A. RODRÍGUEZ

Abstract. S-integrable scalar evolutionary differential difference equations in 1+1 dimensions have a very particular form described by Yamilov's theorem. We look for similar results in the case of S-integrable 2-dimensional partial difference equations and 2-dimensional partial differential equations. To do so, on one side we discuss the semi-continuous limit of S-integrable quad equations and on the other, we semi-discretize partial differential equations. For partial differential equations, we show that any equation can be semi-discretized in such a way to satisfy Yamilov's theorem. In the case of partial difference equations, we are not able to find a form of the equation such that its semi-continuous limit always satisfies Yamilov's theorem. So we just present a few examples, in which to get evolutionary equations, we need to carry out a skew limit. We also consider an S-integrable quad equation with non-constant coefficients which in the skew limit satisfies an extended Yamilov's theorem as it has non-constant coefficients. This equation turns out to be a subcase of the Yamilov discretization of the Krichever-Novikov equation with non-constant coefficient, an equation suggested to be integrable by Levi and Yamilov in 1997 and whose integrabilitv has been proved only recently by algebraic entropy. If we do a strait limit, we get non-local evolutionary equations, which show that an extension of Yamilov's theorem may exist in this case.

Keywords: differential difference equations, continuous and discrete integrable systems, Yamilov's theorem.

Mathematics Subject Classification: 39A14, 35Q53

1. Introduction

Integrabilitv properties of differential difference (DAEs) or partial difference equations (PAEs) have been thoroughly studied along the last decades, following several approaches and criteria to characterize these equations (see, among many others, references [1], [6], [9], [13]). In the case of 1 + 1 DAEs, Yamilov's theorem [10], [11], [13] provides a necessary condition for the existence of a large number of conserved quantities of S-integrable equations [5]. This is reflected in a certain symmetry of the points involved in the equation.

By this work, we start a project aiming on characterizing S-integrable equations in total, i.e. AA

As a part of this project, we plan to extend Yamilov's result to other classes of equations. To do so, in this work we provide some examples of integrable quad equations whose partial continuous limits satisfy Yamilov's condition. Moreover, we show that each PDE can be partially A

D. Levi, M.A. Rodriguez, Yamilov's theorem for differential and difference equations.

© D. Levi, M.A. Rodriguez. 2021.

MAR was partially supported by Spain's Ministerio de Ciencia, Innovación y Universidades under grant PGC2018-094898-B-I00, as well as by Universidad Complutense de Madrid under grant G/6400100/3000.

Submitted March 11, 2021.

In Section 2 we present a brief description of Yamilov's theorem and in the following Section its consequences for PDEs, Section 4 is devoted to the partial continuous limit for quad-graph equations, PAEs defined on a square, and in Section 5 we summarize our results and present some plans for future works on this project.

2. Yamilov's theorem

A

admit higher order conservation laws. Following [10], [11], [13], the search for these equations yields the conclusion that only symmetrical (in a sense to be detailed below) equations have this property. Let us consider a one-dimensional lattice, labeled by n G Z, and a function un (t) of a continuous variable t (real or complex), which takes values at the lattice points. We are A

Un = f (Un+N, Un+N_1, . . . , Un+M) = fn, (2.1)

where the dot above un denotes the derivative with respect to t, N > M and the function f truly depends on un+N and un+M. A conservation law for this equation is a relation of the form:

Dtpn = (T - l)qn,

where pn, the conserved density, and qn are functions of (un, un+i,..., un+k) (k is a finite integer number called the order),

T is the shift operator, (T$n = 0n+i) and Dt the total derivative with respect to t:

k Q

Dt=£ fn+% ^.

Yamilov's theorem provides a condition on the form of equation (2,1) ensuring that it has conservation laws.

Theorem 2.1. If (2,1) possesses a conservation law of order m > min{|^|, |M|}, then N = -M, N > 0.

The proof of this theorem is based on a detailed studying of the variational derivative of the conserved density pn of equation (2,1), which exists due to its S-integrabilitv, see [10], [11], [13],

3. Discussion of PDEs from the point of view of Theorem 2.1

According Theorem 2.1, if (2.1) is an S-integrable DAE and if the function fn contains the highest shift un+N, then it should also contain a lowest shift un-N. Defining

Vt) = Un+k ± Un-k, (3.1)

we can rewrite (2.1) in the S-integrable case as

Un = f (vN+], yN-, v^li, v{-]_i, ••• , v1+], v1-), Un). (3.2)

To perform the continuous limit, we define x = nh and, as h ^ 0, we write the following Taylor expansion

Un±j = w(x ± jh) = w(x) ± jhwx + 2 f h2w2x ± f h3w3x + 1 j4 h4w4x +----, (3.3)

where wnx is the n-th derivative of w(x) with respect to x. In this way, we can write a list of Taylor expansions for , For the lowest values of j we have:

<>'r

V2

, , 2 h --2w(x) + h W2x + 12x +----,

1 3

-2hwx + 3h w3x +----,

--2w(x) + 4h2 W2x + 4 h4w4x + ■ ■ ■ , (3-4)

=4hWx + 3 hmx + ■ ■ ■ ,

27

=2w(x) + 9h2 w2x + — h4w4x + ■ ■ ■ .

Consequently, we can express all derivatives of the function w(x) in terms of the Taylor expansions of up to order h2. Here we present the leading terms:

Wx + °(h2)'

1

W2X = ^ [^2+) - ^S+)] + 0(h2),

Wsx =2hh3 [^2-) - + 0(h2),

(3.5)

wix [3^3+) - 8^(+) + 5^(+)] + 0(h2).

As all continuous derivatives can be expressed in terms of symmetric differences, by Theorem 2.1, we do have no constraints on the form of an S-integrable PDE. This result agrees with what we know about S-integrable PDEs.

4. Discussion of PAEs from the point of view of Theorem 2.1

In this section we discuss bv examples the semi-continuous limit of S-integrable PAEs.

4.1. Lattice potential Korteweg-de Vries equation. As an introduction to the procedure necessary to do the semi-continuous limit, we consider the well known example of the lattice potential Korteweg-de Vries (lpKdV) equation [2], [6], [12]:

{un,m+1 — Un+\,m )(un,m — Un+!,m+i) = p2 — g2, (4-1)

where un,m are the values of the variable u at the points of a two dimensional lattice labeled by the integers n,m E Z. Equation (4.1) is nothing else but the nonlinear superposition formula for the KdV, The parameters p, q are obtained in the construction of the lpKdV as the parameters of the sequence of two Baeklund transformations which give the nonlinear superposition formula. The lpKdV satisfies the integrabilitv property provided by the compatibility around the cube as shown in [2].

On (4.1) we will carry out a procedure to transform the discrete index m into a continuous variable t in such a way that equation (4.1) becomes an evolutionary DAE for the unknown function uk (t) depending on a continuous variable t and a discrete index k.

The base of the method is the Taylor expansion in a parameter e of un,m at a particular solution Mo of the equation. Both the index m and the corresponding parameter q will depend on e. As a particular solution, it is convenient to use a simple function which, in this case, is given by uo = pn + qm. By the change of variables

Un,m — Vn,m Uo (4-2)

equation (4,1) becomes

(p - q + vn,m+\ - vn+i,m)(p + q + vn,m - vn+i,m+i) = p2 - q2, (4.3)

which has as a solution v0 = 0.

A simple approach to the continuous limit when m goes to infinity and e goes to zero in such a way that me is finite is the following. We define a new function Vn{t) = vn,m and a continuous variable t = t0 + me such that

Vn,m+3 = Vn{t) + jeVn{t) + 0{e2). (4.4)

q = e-1. (4.5)

Substituting (4.4), (4.5) into (4.3), we see that coefficient at e-1 vanishes, while the zero order term yields the equation

Vn + Vn+i = {Vn+i - Vn)[2p - {Vn+i - Vn)]. (4.6)

Equation (4.6) is a nonlocal DAE which has derivatives with respect to t at two different points of the lattice n and n + 1. To obtain an evolutionary DAE with a derivative just at one point, we have to mix the indices in the lattice, a skew limit as is called in [6]. We let k = n + m and introduce a new variable wk,m such that

(4.7)

Then (4.3) is transformed into

{p - q + Wk+I}m+1 - wk+1,m){p + q + Wk,m - wk+2,m+l) = p2 - ^. (4.8)

In this case, defining as above Uk{t) = wk}m with q = p + e, the e° term vanishes and at first order we get:

• = Uk-1 - Uk+1

Uk =Uk-i -Uk+i + 2p. (4'9j

Equation (4.9) is an evolutionary DAE, with terms at points k - 1k and k + 1, which thus satisfies Yamilov's condition of S-integrabilitv,

A

+ C2{u

+ Un+1

) +

(4.10)

+ C5Un

an example taken from [7], [11]. Equation (4.10) has been proven to be integrable for all values of constants Ci by checking its algebraic entropy.

In the particular case c\_ = c5 and c6 = c3, the equation is invariant under the swapping of n m

4-2.1. Continuous limit t = em and un,m = vn{t). We have:

c5 v2n + ce v2n+i + {ci + 2 C2 + C3) vnvn+i

+ e{c5vnvn + cevn+ivn+i + {C2 + c3){vn+ivn + vnvn+i)) + 0{e ) = 0.

The terms of zero order in e do not involve the derivatives in t. Hence, in order to get a DAE, we have to suppose that the coefficients q depend on e. We have the following possibilities:

1. c5 = a5e, Ce = aee, ci + 2c2 + c3 = 0,

d

{°i + C2)~dt{VnVn+i) = a5 V™ + ae v"-+i = 0, (4-12)

2. C5 = ce = 0 ci + 2c2 + C3 = ai23£

d

{°i + c2)~dt{VnVn+i) = ai23vnvn+i. (4.13)

In the first ease, the obtained equation is nonlocal. In the second case we have an ODE for VnVn+i- Then, as for the lpKdV equation, we pass to a rhombic lattice.

Under the change of variables k = n + m + 1 and wk+i+j,m+j = un+i,m+j, equation (4,10) becomes

(4.14)

(4.16)

C1 Wk-1,m,Wk,m + C2(wk + Wk-1,mWk+1,m+l) + C3Wk,m+lWk+1,m+1

+ C5Wk-1,mWk,m+1 + C6Wk,mWk+1,m+1 = 0.

Letting

t = ne, wk>n = Uk (t), (4.15)

we transform equation (4.14) into

(C1 + c5)Uk-1Uk + (C3 + c6)Uk Uk+1 + C2(Uk-1 Uk+1 + Ul)

+ e((c2tffc-1 + (C3 + ce)^fc )Uk+1 + (05^-1 + C2Uk + C3Uk+1)Uk) + 0(e2) = 0. To get a DAE, at the lowest order in e we need to set

C1 + C5 = ta, C3 + Co = eft, C2 = £7, (4-17) and choose c1 and c6 of ^^^^r In this way, at the low est order in e we get

(coUk+1 + dUk-1)Uk - (ftUk+1 + >yUk + aUk-1)Uk - 7^fc-1^fc+1 = 0. (4.18)

Equation (4.18) satisfies Yamilov S-integrabilitv theorem. Thus, for all values of q, equation (4.10) is an S-integrable and the result obtained by the algebraic entropy is shown to be correct.

4.3. H2 equation: a more complicate equation of the ABS classification. The H2

equation

(^n,m ^ra+1,m+1)(^ra+1,m ^n,m+1)

+ (ft - a)(un,m + un+1,m + un,m+1 + un+1,m+1) - a2 + ft2 = 0,

is another of the discrete integrable equations in the Adler-Bobenko-Suris list, see [2] and [11].

As in other cases, we admit that the constants a and ft depend on the parameter e, in which we carry out the limiting process. We follow the approach, which uses as starting point a background solution provided by equation (4.7a) in [8]. Following [8], we redefine the parameters of H2 (4.19):

22 p = r - a , q = r - b ,

and get the equation:

(Un,m ^ra+1,m+1)(^ra+1,m '^n,m+1)

+ (a2 - b2)(un,m + un+1,m + un,m+1 + un+1,m+1 + 2r - a2 - b2) = 0 Among others, equation (4.20) has the solution

uo = (an + brn + j)2 - 1 r. (4-21)

Making the change (4.2) in (4.21), we can rewrite (4.20) as:

)(vn+1,rn - Vn,m+1)

+ 2(a - b)((a(n + 1) + b(m + 1) + j)vn,m - (an + brn + 7)^+1,^+1) (4.22)

- 2(a + b)((an + b(m + 1) + 7)vn+1,m - (a(n + 1) + bm + 7)^,^+1) = 0,

which has the solution vo = 0.

2 , «2 n (4-19)

(4.20)

In order to take the skew limit in one of the indices, as in the previous section we define k = n + m and a new field wk,m according to (4,7), Then (4,22) is transformed into

(wk,m - wk+2,m+i)(wk+i,m - vk+i,m+i) + 2(a - b)((a(k -m + 1) + b(m + 1) + j)wk,m

- (a(k -m) + bm + r))wk+2,m+i) - 2(a + b)((a(k - m) + b(m + 1) + j)wk+i¡m (4,23)

- (a(k -m + 1) + bm + ^)wk+i,m+i) = 0.

In order to take the continuous limit in the index m, we replace a and b by m and S and we define

wk+i,m = Uk+i(t), wk+i,m+j ^ Uk+i(t) + ej Uk+i(t) + 0(e2). Then, the lower order terms of (4,23) are

( Uk - Uk+2 - £ Uk+2)(Uk+i - Uk+i - £ Ufc+i)

+ 2e(a - b)((e a(k -m + 1) + e b(m + 1) + 7)Uk

- (t a(k - m) + e bm + 7)(Uk+2 + e Uk+2)) (4.24)

- 2t(a + b)((ta(k -m) + tb(m + 1) + ^)Uk+i

- (e a(k -m + 1) + e bm + j )Uk+i + eUk+i)) + ••• = 0, which has no order zero term. The first order term of (4,24) is

Uk = 2-i(a - b)(Uk-i -Uk+i), (4.25)

and this is an equation satisfying Yamilov's theorem.

4.4. Equation rH¡ from ABS extended classification. We consider the following equation:

rHl =(a - [) (e2

+ ( - )( - )=0 J

+ (Un,m Un+i,m+i)(Un+i,m Un,m+i) °

where a, [, e are three parameters, which could depend on the steps of the lattice and

X™ =1(1 + (-1)m). (4-27)

Equation (4.26) is an ^-integrable equation of Boll classification [4] appearing in the ABS list [3], [11].

As in the previous case, we transform the variable un,m ^ vn,m in such a way that the resulting equation has v0 = 0 as a particular solution:

Un,m Un,m + f,

(a -[)(e2f2 - 1) = 0. (4.28)

Choosing one of the two signs for f in (4.28) with t = 0, we get:

Un,m un,m + —. (4,29)

The equation for vn,m is

(a - [)(xn+m+i(1 + evn,m)(1 + evn+i,m+i) + Xn+m(1 + tvn,m+i)(1 + evn+i,m) - 1)

+ (Un+i,m Un,m+i)( Vn,m Vn+i,m+i) 0

and v0 = 0 is a solution of (4,30),

In order to take a semi-continuous skew limit, we first modify the lattice from a rectangular one to an oblique lattice by defining a new index k = n + m and leaving m as it is. Then we

(4.30)

take the continuous limit in the m direction of the new lattice. The equation with wkm defined as in (4.7) now reads as

(a - ft)(Xfc+i(l + twk,m)(l + ewk+2,m+i) + Xk(1 + ewk+i,m+i)(l + ewk+i,m) - 1)

(4-31)

+ (^k+1,m — Wk+1,m+1)(Wk,m - Wk+2,m+i) = 0.

Taking the continuous limit in the m direction and assuming that the order parameter is e, we find that

Wk+i,m = Uk+i, Wk+i,m+i = Uk+i + dJk+i + 0(e2), (4.32)

and substituting (4.32) in (4.31), we get

(a - ft)(Xk+i(1 + tUk)(1 + t(Uk+2 + tUk+2 + 0(t2)))

+ Xk(1 + t(Uk+i + tUk+i + 0(e2)))(1 + tUk+i) - 1) (4.33)

+ (Uk+i - Uk+i - t(Jk+iO(t2))(Uk - Uk+2 - tUk+2 + 0(t2)) = 0.

Taking into consideration the definition (4.27) of Xm, we see that the zero order terms in (4.33) vanish. The first order terms are

(a - ft)(Xk+i(Uk + Uk+2) + 2XkUk+i) - (Uk - Uk+2)Uk+i = 0. (4.34)

Equation (4.34) is a DAE satisfying an extension of Yamilov's theorem when the function f appearing in (2.1) depends explicitly on the index n:

TT / o\2Xk-i Uk + Xk (Uk-i + Uk+i) or\

Uk = (a - ft)-----. (4.35)

Uk-i - Uk+i

Equation (4.35) is a subcase of the fc-dependent generalization of the Yamilov discretization of the Kriehever-Novikov equation postulated in [9] and discussed in [11].

5. Conclusions

A

A

this theorem have no implication for the S-integrabilitv of PDEs.

A

A

A

questions are

• Is there an extension of Theorem 2.1 when the function f depends explictly on n?

• Is (4.6) also S-integrable?

•

PDEs?

We plan to continue this research by answering these questions and looking for a compulsory

A

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3. V.E. Adler, A.I. Bobenko, Yu.B. Suris. Discrete nonlinear hyperbolic equations: classification of integrable cases // Funct. Anal. Appl. 43:1, 3-17 (2009). [arXiv:0705.1663],

4. R. Boll. Classification of 3D consistent quad-equations //J- Nonlin. Math. Phvs. 18:3, 337-365 (2011).

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10. D. Levi, R.I. Yamilov. Generalized Lie symmetries for difference equations //in Symmetries and integrability of difference equations, ed. by D. Levi, P. Olver, Z. Thomova, P. Winternitz, Cambridge University Press, Cambridge, 160-190 (2011).

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Decio Levi,

Mathematical and Physical Department

Roma Tre University

Via della Vasca Navale, 84,

100146 Roma, Italy

E-mail: [email protected]

Miguel A. Rodriguez,

Dept. Física Teórica

Universidad Complutense de Madrid

Pza. de las Ciencias, 1,

28040 Madrid, Spain

E-mail: [email protected]