Integrability properties of some symmetry reductions

INTEGRABILITY PROPERTIES OF SOME EQUATIONS OBTAINED BY SYMMETRY REDUCTIONS

arXiv:1412.6461v1 [nlin.SI] 19 Dec 2014

ˇ AK ´ H. BARAN, I.S. KRASIL′ SHCHIK, O.I. MOROZOV, AND P. VOJC Abstract. In our recent paper [1], we gave a complete description of symmetry reduction of four Lax-integrable (i.e., possessing a zero-curvature representation with a non-removable parameter) 3-dimensional equations. Here we study the behavior of the integrability features of the initial equations under the reduction procedure. We show that the ZCRs are transformed to nonlinear differential coverings of the resulting 2D-systems similar to the one found for the Gibbons-Tsarev equation in [12]. Using these coverings we construct infinite series of (nonlocal) conservation laws and prove their nontriviality. We also show that the recursion operators are not preserved under reductions.

Contents Introduction 1. Reduction of the Lax pairs 1.1. Equation (1) 1.2. Equation (2) 1.3. Equation (3) 2. Local symmetries and cosymmetries of the reduction equations 2.1. Equation (1) 2.2. Equation (2) 2.3. Equation (3) 3. Hierarchies of nonlocal conservation laws 3.1. A general construction 3.2. Equation (1) 3.3. Equation (2) 3.4. Equation (3) 3.5. Proof of nontriviality 4. On reductions of the recursion operators 4.1. A general construction 4.2. Recursion operators for symmetries of 3D systems 4.3. The negative result 5. Discussion 6. Appendix: Conservation laws Acknowledgements References

2 2 2 3 4 5 5 5 6 6 6 9 10 11 11 12 12 13 14 14 15 18 18

Date: December 22, 2014. 2010 Mathematics Subject Classification. 35B06. Key words and phrases. Partial differential equations, symmetry reductions, solutions, the Gibbons-Tsarev equation, Lax-integrable equations. 1

2

ˇ AK ´ H. BARAN, I.S. KRASIL′ SHCHIK, O.I. MOROZOV, AND P. VOJC

Introduction In [1] we gave a complete description of symmetry reductions for four three dimensional systems: the universal hierarchy equation, the 3D rdDym equation, the modified Veronese web equation, and Pavlov’s equation. The result comprised more than 30 equations, but the majority of them were either exactly solvable or linearized by the generalized Legendre transformations. Nevertheless, there were 10 ‘interesting’ reductions, among which two well-known equations, i.e., the Liouville and Gibbons-Tsarev equations, [3, 4]. The rest eight can be divided in two groups by their symmetry properties: five equations admit infinite-dimensional Lie algebras of contact symmetries (with functional parameters) and three others possess finitedimensional symmetry algebras. These are uy uxy − ux uyy = ey uxx

(1)

(reduction of the universal hierarchy equation), uyy = (ux + x)uxy − uy (uxx + 2)

(2)

(reduction of the 3D rdDym equation), and uxx = (x − uy )uxy + (2y + ux )uyy − uy

(3)

(reduction of the Pavlov equation)1. These equations are pair-wise inequaivalent (see Section 5). We deal with this three equations below and study how the integrability properties of the initial 3D systems behave under reduction. More precisely, we construct (Section 1) the reductions of the zero-curvature representations for Equations (1)– (2) and show that they result in differential coverings of the form a2 w 2 + a1 w + a0 b2 w 2 + b1 w + b0 , wy = , 2 w + c1 w + c0 w 2 + c1 w + c0 where ai , bi , ci are functions in x, y, u, ux , and uy . These coverings are similar to the one found in [12] for the Gibbons-Tsarev equation and this resemblance, by all means, reflects the relations between generalized Gibbons-Tsarev equations and integrable 3D-systems [13]. In Section 3, for every nonlinear covering we construct an infinite series of conservation laws and prove their nontriviality. We also study the behavior of the recursion operators for symmetries of threedimensional systems and show that these operators do not survive under reduction (Section 4). In Section 2 local symmetries and cosymmetries of the reduction equations are described. The corresponding conservation laws are presented in the Appendix. Throughout the text the notion of (differential) covering is understood in the sense of [7]. wx =

1. Reduction of the Lax pairs Using Lax representations of the 3D equations, whose reductions are the equations at hand, we construct here nonlinear coverings of Equations (1)–(3). 1.1. Equation (1). This equation is obtained as the reduction of the universal hierarchy equation2 uyy = uz uxy − uy uxz (4) 1All the reductions of the modified Veronese web equation were either exactly solvable or linearizable. 2To save the notation here and below, we denote by u the dependent and by x, y the dependent variables. These are not the same as in the initial equation; see the details in [1].

INTEGRABILITY PROPERTIES OF SOME SYMMETRY REDUCTIONS

3

with respect to the symmetry ϕ = uz + ux + yuy + u.

(5)

Equivalently, this reduction may be written in the form uyy = uy uxx − (ux + u)uxy + ux uy

(6)

and Equation (1) transforms to (6) by the change of variables x 7→ y, y 7→ x, u 7→ −ey u. Equation (4) admits the following Lax representation wz wy

= (wuz − uy )w−2 wx , = uy w−1 wx .

(7)

The symmetry ϕ can be extended to a symmetry Φ = (ϕ, χ) of (7), where χ = wz + wx + ywy + w and the corresponding reduction leads to the covering w3 , w2 − (ux + u)w − uy uy w 2 =− 2 w − (ux + u)w − uy

wx

=−

wy

(8)

of Equation (6). Remark 1. Equation (1) can be written in the potential form y uy e = , ux y ux x the corresponding Abelian covering being uy ey vx = , vy = . ux ux Then v enjoys the equation vy − vyy = vy vxx − vx vxy ,

(9)

(10)

which also admits the rational covering wvx − xvx + vy wx = 2 , w + (−2x + vx )w + x2 − xvx + vy wvy − xvy . wy = 2 w + (−2x + vx )w + x2 − xvx + vy of the same type.

1.2. Equation (2). This equation was obtained as the reduction of the 3D rdDym equation uty = ux uxy − uy uxx (11) with respect to the symmetry ϕ = ut − xux − uy + 2u.

(12)

The Lax representation for Equation (11) is wt wy

= (ux + w)wx , = −uy w−1 wx .

The symmetry ϕ extends to the one of (13): Φ = (ϕ, χ), where χ = wt − xwx − wy + u.

(13)

4


Reduction of the covering (13) with respect to Φ leads to the covering wx wy

w2 , w2 + (ux − x)w + uy uy w = 2 . w + (ux − x)w + uy =−

(14)

over Equation (2). 1.3. Equation (3). Finally, Equation (3) is the reduction of the Pavlov equation uyy = utx + uy uxx − ux uxy

(15)

with respect to the symmetry ϕ = ut − 2xux − yuy + 3u.

(16)

The Pavlov equation possesses the Lax pair wt wy

= (w2 − wux − uy )wx , = (w − ux )wx .

(17)

The symmetry ϕ lifts to the symmetry Φ = (ϕ, χ) of (17), where χ = wt − 2xwx − ywy + w. Reduction of the covering (17) with respect to this symmetry results in the nonlinear covering w(w − uy ) wx = − 2 , w − (uy + x)w + xuy − ux − 2y (18) w wy = − 2 w − (uy + x)w + xuy − ux − 2y of Equation (3). Remark 2. Equation (3) has a close relative. Namely, if we accomplish reduction of the Pavlov equation using another symmetry ϕ′ = ut − yux + 2x the resulting equation will be uyy = (uy + y)uxx − ux uxy − 2.

(19)

The symmetry ϕ′ can also be lifted to (17) by Φ′ = (ϕ′ , χ′ ), where χ′ = wt − ywx + 1, and the reduction of (17) will be wx wy

1 , w 2 − ux w − uy − y w − ux =− 2 . w − ux w − uy − y =−

(20)

By the change of variables u 7→ u − y 2 /2, Equation (19) transforms to the GibbonsTsarev equation uyy = uy uxx − ux uxy − 1, while (18) becomes 1 wx = − 2 , w − ux w − uy w − ux , wy = − 2 w − ux w − uy cf. [12].


5

2. Local symmetries and cosymmetries of the reduction equations We present here computational results on classical symmetries and cosymmetries of Equations (1)–(3), i.e., solutions of the equations ℓE (ϕ) = 0 and ℓ∗E (ψ) = 0, where ℓE is the linearization of the equation at hand and ℓ∗E is its formally adjoint and ϕ and ψ depend on x, y, u, ux , uy (see, e.g., [6]). The conservation laws corresponding to classical cosymmetries are presented in the Appendix below. The spaces of solutions are denoted by symc (E ) and cosymc (E ), respectively. All the equations under consideration happen to possess a scaling symmetry and thus admit weights (which we denote by |· |) with respect to which they become homogeneous. 2.1. Equation (1). We consider this equation in the form (6), i.e., uyy = uy uxx − (ux + u)uxy + ux uy . The weights are |x| = 0,

|y| = 1,

|u| = −1,

|ux | = −1,

|uy | = −2.

Symmetries. The defining equation for symmetries is3 Dy2 (ϕ) = uy Dx2 (ϕ) − (ux + u)Dx Dy (ϕ) + (uy − uxy )Dx (ϕ) + (uxx + ux )Dy (ϕ) − uxy ϕ. The space symc (E ) spans the symmetries ϕ−1 = uy ,

ϕ′0 = ux ,

ϕ0 = yuy + u,

ϕ1 = e−x ,

where the subscripts coincide with the weights4. Cosymmetries. The defining equation for cosymmetries of Equation (1) is Dy2 (ψ) = uy Dx2 (ψ)−(ux +u)Dx Dy (ψ)+2(uxy +uy )Dx (ψ)−2(uxx +ux )Dy (ψ)−3uxy ψ. The space cosymc (E ) is 6-dimensional and spans the following cosymmetries: ψ−3 = e4x (3u2x + 8u2 + 10uux + 2uy ),

ψ−2 = e3x (3u + 2ux ),

ψ−1 = e2x

and ψ3 = ψ5 =

1 , u2y

ψ4 =

2ux − yuy + 2u , u3y

−4uxyuy + 6uux + 3u2x − 4yuuy + 3u2 + 2uy + y 2 u2y , u4y

where superscript coincides with the weight5. 2.2. Equation (2). The weights are |x| = 1,

|y| = 0,

|u| = 2,

|ux | = 1,

|uy | = 2.

Symmetries. The linearized equation is Dy2 (ϕ) = (ux + x)Dx Dy (ϕ) − uy Dx2 (ϕ) + uxy Dx (ϕ) − (uxx + 2)Dy (ϕ). The space symc (E ) is generated by the symmetries ϕ−2 = 1,

ϕ−1 = ux + x,

1 ϕ0 = u − xux , 2

ϕ′0 = uy .

3Here and below D and D denote the total derivatives with respect to x and y. x y

4To a symmetry ϕ we assign the weight of the corresponding evolutionary vector field E . ϕ 5To every cosymmetry we assign the weight of the corresponding variational form, see [8]


6

Cosymmetries. The defining equation for cosymmetries reads Dy2 (ψ) = (ux + x)Dx Dy (ψ) − uy Dx2 (ψ) − 2uxy Dx (ψ) + (2uxx + 3)Dy (ψ). The space cosymc (E ) is generated by the cosymmetries ψ−3 =

e−2y (ux + x) , u3y

ψ2 = 1,

ψ−2 =

e−y , u2y

ψ3 = ux + 2x.

2.3. Equation (3). The weights of variables are |x| = 1,

|y| = 2,

|u| = 3,

|ux | = 2,

|uy | = 1.

in this case. Symmetries. The symmetries are defined by the equation Dx2 (ϕ) = (x − uy )Dx Dy (ϕ) + (2y + ux )Dy2 (ϕ) − Dy (ϕ) and the space symc (E ) spans the symmetries 2 1 ϕ0 = − xux − yuy + u, 3 3 ϕ−2 = uy + 2x,

1 ϕ−1 = ux − xuy + y − x2 , 2 ϕ−3 = 1.

Cosymmetries. The defining equation for cosymmetries is of the form Dx2 (ψ) = (x − uy )Dx Dy (ψ) + (2y + ux )Dy2 − uyy Dx + 3(2 − uxy )Dy . The space cosymc (E ) is 6-dimensional and spans the elements 54 164 256 2 4 12 36 xux uy + xuy y + x y + 2xu + uuy + u2y ux + 4yux + u2y y 5 5 5 5 5 5 512 3 32 3 96 2 2 32 2 512 4 3 2 82 2 4 x uy + xuy + x uy + y + x + ux + uy , + x ux + 5 15 5 5 5 15 5 3 9 49 2 21 2 343 3 1 49 xy + 4xux + uy ux + uy y + x uy + xuy + x + u + u3y , ψ6 = 4 2 2 4 4 24 4 2 2 2 ψ5 = 4xuy + 6x + 2y + ux + uy , 3 5 ψ4 = x + uy , 2 ψ3 = 1, 1 . ψ−1 = (−xuy + ux + 2y)2 ψ7 =

3. Hierarchies of nonlocal conservation laws Using the nonlinear coverings presented in Section 1 we construct here infinite hierarchies of nonlocal conservation laws for Equations (1)–(1). 3.1. A general construction. The initial step of the construction is the so-called Pavlov reversing, [14] (see [5] for the invariant geometrical interpretation). Let E be an equation in two independent variables x and y and unknown function u and wx = X(x, y, [u], w),

wy = Y (x, y, [u], w)

be a differential covering over E , where [u] denotes u itself and a collection of its derivatives up to some finite order. Then the system ψx = −X(x, y, [u], λ)ψλ ,

ψy = −Y (x, y, [u], λ)ψλ

(21)


7

is also compatible modulo E (thus, the nonlocal variable w turns into a formal parameter in the new setting). Assume now that X1 Xi X = X−1 λ + X0 + + ···+ i + ..., λ λ Yi Y1 + ··· + i + ..., Y = Y−1 λ + Y0 + λ λ where Xi , Yi , i ≥ −1, are functions in x, y and [u], and also expand ψ in formal Laurent series ψ1 ψi ψ = ψ−1 λ + ψ0 + + ··· + i + ... λ λ Then (21) implies X X ψi,x = − kXj ψk , ψi,y = − kYj ψk , j+k=i+1

j+k=i+1

or

ψ−1,x = −X−1 ψ−1 ,

ψ−1,y = −Y−1 ψ−1 ;

ψ0,x = −X0 ψ−1 , ψ1,x = X−1 − X1 ψ−1 ,

ψ0,y = −Y0 ψ−1 ; ψ1,y = Y−1 − Y1 ψ−1 ;

ψ2,x = 2X−1 ψ2 + X0 ψ1 − X2 ψ−1 ,

ψ2,y = 2Y−1 ψ2 + Y0 ψ1 − Y2 ψ−1 ;

...

...

and ψk,x = kX−1 ψk + (k − 1)X0 ψi−1 + · · · + Xk−2 ψ1 − Xk ψ−1 , ψk,y = kY−1 ψk + (k − 1)Y0 ψi−1 + · · · + Yk−2 ψ1 − Yk ψ−1 for all k > 2. In general, this system defines an infinite-dimensional non-Abelian covering (which may be trivial generally) over the base equation E , but in the particular case X−1 = Y−1 = 0 the covering becomes Abelian, i.e., transforms to an infinite series of (nonlocal) conservation laws. Indeed, the first pair of equations reads ψ−1,x = 0,

ψ−1,y = 0

in this case and without loss of generality we may set ψ−1 = 1. The rest equations read ψ0,x = −X0 ,

ψ0,y = −Y0 ;

ψ1,x = −X1 , ψ2,x = X0 ψ1 − X2 ,

ψ1,y = −Y1 ; ψ2,y = Y0 ψ1 − Y2 ;

ψ3,x = 2X0 ψ2 + X1 ψ1 − X3 ,

ψ3,x = 2Y0 ψ2 + Y1 ψ1 − Y3 ;

...

...

and

ψk,x = (k − 1)X0 ψk−1 + (k − 2)X1 ψk−2 + · · · + Xk−2 ψ1 − Xk , ψk,y = (k − 1)Y0 ψk−1 + (k − 2)Y1 ψk−2 + · · · + Yk−2 ψ1 − Yk for all k > 3.

(22)

Remark 3. The first two pairs of equations define local conservation laws (probably, trivial) and the potential ψ0 does not enter the other equations. This means that the obtained covering is the Whitney product of the one-dimensional Abelian covering τ0 associated to ψ0 and the infinite-dimensional τ∗ related to ψ1 , ψ2 , . . . We shall deal with τ∗ below.


8

We now confine ourselves to the case a2 w 2 + a1 w + a0 b2 w 2 + b1 w + b0 X= , Y = , (23) w 2 + c1 w + c0 w 2 + c1 w + c0 where ai , bi , and ci are functions in x, y, and [u], and deduce the needed Laurent expansions. One has ! a2 λ2 + a1 λ + a0 a1 a0 1 = a2 + + 2 · 0 λ2 + c1 λ + c0 λ λ 1 + c1 λ+c λ2 i a1 c1 λ + c0 a0 X = a2 + − . + 2 · λ λ λ2 i≥0

Let us present temporally the second factor in the form X c1 λ + c0 i X di = . − λ2 λi i≥0

i≥0

Then

a1 a0 X di a2 λ2 + a1 λ + a0 · = a + + 2 λ2 + c1 λ + c0 λ λ2 λi i≥0

a2 d1 + a1 d0 a2 d2 + a1 d1 + a0 d0 a2 di + a1 di−1 + a0 di−2 = a2 d0 + +· · ·+ +... + 2 λ λ λi Compute the coefficients di now. One has i i j i−j X c1 λ + c0 i c1 c0 i − = (−1) , j λ2 λ2i−j j=0 from where it follows that

d0 = 1, and

d1 = −c1

 k  X k−j k + j   c0k−j c2j (−1)  1  2j di = j=0 k X   k−j+1 k + j + 1   c0k−j c2j+1 (−1) 1   2j + 1 j=0

if i = 2k, (24) if i = 2k + 1

for i > 1, Or, in shorter notation [i/2]

di =

X

[i/2]−j+p(i)

(−1)

j=0

[i/2] + j + p(i) [i/2]−j 2j+p(i) c0 c1 , 2j + p(i)

(25)

where p(i) = i mod 2 is the parity of i and [k/2] is the integer part. Gathering together the results of the above computations, one obtains that in the case of coverings (23) we have X−1 = Y−1 = 0, while other coefficients are X 0 = a2 ,

Y0 = b2 ;

X 1 = a1 − a2 c1 ,

Y1 = b1 − b2 c1 ;

X2 = a0 − a1 c1 + a2 (c21 − c0 ),

Y2 = b0 − b1 c1 + b2 (c21 − c0 );

...

...

Xi = a0 di−2 + a1 di−1 + a2 di , ...

Yi = b0 di−2 + b1 di−1 + b2 di ; ...,

where the functions di are given by (24).


9

Let us now show how these general constructions look like in the particular cases of the equations under consideration. 3.2. Equation (1). Note first that the covering (8) is not of the form (23). Nevertheless, it can be transformed to the needed form by the gauge transformation w 7→ we−x . Then (8) acquires the form wx =

(ux + u)ex w2 − uy e2x w , w2 − (ux + u)ex w − uy e2x

wy = −

w2

uy ex w 2 . − (ux + u)ex w − uy e2x

We have |w| = −1. Thus, a0 = 0,

a1 = −uy e2x

a2 = (ux + u)ex ,

b0 = 0,

b1 = 0,

b2 = −uy ex ,

c0 = −uy e2x ,

c1 = −(ux + u)ex .

Let us compute the coefficients di . By (24), we have d2k =

k X

(−1)k−j

j=0

k−j k+j 2j (−(ux + u)ex ) −uy e2x 2j = e2kx

and d2k+1 =

k X

k−j+1

(−1)

j=0

k−j k+j+1 2j+1 (−(ux + u)ex ) −uy e2x 2j + 1 =e

(2k+1)x

k X k+j+1 2j + 1

j=0

or di = eix Hence,

uyk−j (ux + u)2j+1 ,

[i/2]

X [i/2] + j + p(i) u[i/2]−j (ux + u)2j+p(i) . y 2j + p(i) j=0

X0 = (ux + u)ex , X1 = (ux + u)2 − uy e2x , and Xi = e(i+1)x (ux + u)i+1 +

(26)

Y0 = −uy ex ; Y1 = (ux + u)uy e2x



k X k + j k−j uy (ux + u)2j 2j j=0

[(i+1)/2]

i−j i − 2j

X j=1

 i−j − ujy (ux + u)i−2j+1  , i − 2j + 1

[i/2] X [i/2] + j + p(i) (i+1)x u[i/2]−j+1 (ux + u)2j+p(i) Yi = −e y 2j + p(i) j=0 for i > 1 (we assume α β = 0 for β < 0). Obviously,

|Xi | = −i − 1,

|Yi | = −i − 2.

The functions Xi , Yi define, by Equations (22), the infinite number of nonlocal variables ψi for Equation (1) with |ψi | = −i − 1.


10

The corresponding conservation laws have the same weights and the first three of them coincide (up to equivalence) with the local conservation laws ω−2 , ω−3 , ω−4 described in Section 2.1. The first essentially nonlocal one is associated to ψ3 . 3.3. Equation (2). Due to Equations (14), one has a0 = 0,

a1 = 0

a2 = −1,

b0 = 0, c0 = u y ,

b1 = u y , c1 = ux − x.

b2 = 0,

Hence, X0 = −1,

Y0 = 0;

X1 = ux − x,

Y1 = uy ; 2

X2 = −(ux − x) + uy ,

Y2 = −uy (ux − x)

and [i/2]

Xi = −di =

X

(−1)[i/2]−j+p(i)+1

j=0

[i/2] + j + p(i) [i/2]−j (ux − x)2j+p(i) , uy 2j + p(i)

[(i−1)/2]

Yi = uy di−1 =

X

(−1)[(i−1)/2]−j+p(i−1) ×

j=0

[(i − 1)/2] + j + p(i − 1) [(i−1)/2]−j+1 × uy (ux − x)2j+p(i−1) 2j + p(i − 1) for i > 2. Consequently, ψ0,x = −X0 = 1,

ψ0,y = −Y0 = 0;

ψ1,x = −X1 = −ux + x,

ψ1,y = −Y1 = −uy

and one may set ψ0 = x,

ψ1 = −u +

x2 , 2

while ψ2,x = (ux − x)2 + uy + u −

x2 , 2

ψ2,y = (ux − x)uy

and for i > 2

x2 − u Xi−2 − Xi , 2

ψi,x = −(i − 1)ψi−1 + (i − 2)X1 ψi−2 + · · · + Xi−3 ψ2 + 2 x ψi,y = (i − 2)Y1 ψi−2 + · · · + Yi−3 ψ2 + − u Yi−2 − Yi , 2 where Xk , Yk are given by the above formulas. One has |Xi | = i,

|Yi | = i + 1,

|ψi | = i + 1.

The conservation law corresponding to ψi is of the weight i + 1 and the first two ones, up to equivalence coincide with those described in Section 2.2, while all the others are essentially nonlocal.


11

3.4. Equation (3). By Equation (18), we have a0 = 0,

a1 = u y

a2 = −1,

b0 = 0,

b1 = −1,

b2 = 0,

c0 = xuy − ux − 2y,

c1 = −(uy + x).

Consequently, X0 = −1,

Y0 = 0;

X1 = −x,

Y1 = −1;

X2 = −ux − x2 − 2y,

Y2 = −uy − x

Xi = uy di−1 − di ,

Yi = −di−1

and

for i > 2, where [i/2]

di =

X

[i/2]−j

(−1)

j=0

One has

[i/2] + j + p(i) (xuy − ux − 2y)[i/2]−j (uy + x)2j+p(i) . 2j + p(i) |Xi | = i,

|Yi | = i − 1.

Thus we have ψ1,x = x,

ψ1,y = 1; 2

ψ2,x = ux +

x + y, 2

ψ2,y = uy + x

and we may set x2 x3 + y, ψ2 = u + xy + . 2 6 Then the other potentials are defined by ψ1 =

ψi,x = −(i − 1)ψi−1 − (i − 2)ψi−2 (i − 3)X2 ψi−3 + . . . 2 x3 x · · · + 3Xi−4 ψ3 + 2u + 2xy + Xi−3 + + y Xi−2 − Xi , 3 2 ψi,y = −(i − 2)ψi−2 (i − 3)Y2 ψi−3 + . . . 2 x3 x · · · + 3Yi−4 ψ3 + 2u + 2xy + Yi−3 + + y Yi−2 − Yi , 3 2

i > 2. We have

|ψi | = i + 1. The conservation laws associated with ψ3 , . . . , ψ7 are equivalent to ω4 , . . . , ω8 introduced in Section 2.3. The first essentially nonlocal conservation law corresponds to ψ8 . 3.5. Proof of nontriviality. We shall now prove that the above constructed conservation laws are nontrivial. To this end, introduce the notation Eα , α = 1, 2, 3, for Equations (1), (2) and (3), respectively, and τi,α : Ei,α → Eα for the coverings defined by the nonlocal variables ψα , . . . , ψi . Let Dxi,α , be the total derivatives on Ei,α .

Dyi,α

12


Proposition 1. For all i ≥ α, the only solutions of the system Dxi,α (f ) = 0,

Dyi,α (f ) = 0

(27)

are constants. Proof. Let us present the total derivatives in the form Dxi,α = Dxα + X i,α ,

Dyi,α = Dyα + Y i,α ,

where Dxα , Dyα are the total derivatives on Eα and X i,α , Y i,α are the ‘nonlocal tails’: i i X X ∂ ∂ Xji,α X i,α = Yji,α , Y i,α = , ∂ψ ∂ψ j j j=α j=α Xji,α , Yji,α being the right-hand sides of the defining equations (22) for the potentials ψ. From the constructions of Sections 3.2–3.4 one readily sees that the quantities Xji,α and Yji,α are polynomials in ux and uy and, moreover, X i,1 = ±e(i+1)x ui+1 x

∂ + o, ∂ψi

∂ + o; ∂ψi ∂ = ±ui−2 + o, y ux ∂ψi

Y i,1 = ±e(i+1)x uix uy

∂ + o; ∂ψi

∂ + o; ∂ψi ∂ = ±ui−1 + o, y ∂ψi

X i,2 = ±uix

Y i,2 = ±ui−1 x

X i,3

Y i,3

where o denotes terms of lower degree. Now, the proof goes by induction. For small i’s the result follows from the fact that the cosymmetries corresponding to the local conservation laws do not vanish and these conservation laws are of different weights. Assume now that the statement is valid for all k < i and consider Equation (27). Then from the above estimates it follows that ∂f /∂ψi = 0. Evidently, nontriviality of the constructed conservation laws is a direct consequence of the Proposition 1. 4. On reductions of the recursion operators We show here that symmetry reductions of Equations (4), (11), and (15) are incompatible with their recursion operators and thus the latter are not inherited by Equations (1), (2), and (3), respectively. 4.1. A general construction. We treat here recursion operators for symmetries as Bäcklund transformations of the tangent coverings, cf. [9]. More precisely, let E be a differential equation given by the system E = {F = 0},

F = (F 1 (x, y, [u]), . . . , F s (x, y, [u])),

F j being functions on some jet space, [6]. Here, as above, [u] denotes the collection of u and its derivatives. The tangent covering t = tE : T E → E is the projection (x, y, [u], [q]) 7→ (x, y, [u]) of the system T E = {F (x, y, [u]) = 0, ℓF (x, y, [u], [q]) = 0} to E . The characteristic property of t is that its sections that preserve the Cartan (higher contact) distribution are identified with symmetries of E .


13

A Bäcklund transformation between equations E1 and E2 is a diagram B ⑦ ❅❅❅ ❅❅τ2 ⑦⑦ ⑦ ❅❅ ⑦⑦ ❅ ⑦ ~~ ⑦ τ1

E1

E2 ,

where τ1 and τ2 are coverings. It relates solutions of E1 and E2 to each other. A recursion operator between symmetries of E1 and E2 is a Bäcklund transformation of the form tE1 // E1 T