Integrability of S-deformable surfaces: conservation laws, Hamiltonian structures and more

0.2 MB·English

by  I. S. Krasil'shchik

INTEGRABILITY OF S-DEFORMABLE SURFACES:
CONSERVATION LAWS, HAMILTONIAN STRUCTURES
AND MORE
′ ∗
I.S. KRASILSHCHIK ANDA. SERGYEYEV
Abstract. We present infinitely many nonlocal conservation laws, a
5 pairofcompatiblelocalHamiltonianstructuresandarecursionoperator
1 for the equations describing surfaces in three-dimensional space that
0
admit nontrivial deformations which preserve both principal directions
2
and principal curvatures(or, equivalently,the shapeoperator).

g
u
A

3 Contents

] 1. Introduction 2
I
S 2. Preliminaries 2
. 3. Infinite hierarchy of nonlocal conservation laws 3
n
i 4. Recursion operator and Hamiltonian structures 10
l
n 5. CompatibleHamiltonianstructuresinevolutionaryrepresentation 12
[
Acknowledgments 17

3 References 17
v
1
7
1
7
0
.
1
0
5
1
:
v
i
X
r
a
(IK) Independent University of Moscow, B. Vlasevsky 11, 119002 Moscow,
Russia&MathematicalInstitute,SilesianUniversityinOpava,NaRybn´ıcˇku1,
746 01 Opava, Czech Republic
(AS) Mathematical Institute, Silesian University in Opava, Na Rybn´ıcˇku 1,
746 01 Opava, Czech Republic
E-mail addresses: [email protected], [email protected].
2010 Mathematics Subject Classification. 37K05, 37K10, 53A05.
Keywords and phrases. Nonlocalconservationlaws,Hamiltonianstructures,recursion
operator, shape operator, Gauss–Codazzi equations, S-deformable surfaces.
∗
Corresponding author.
1
2 INTEGRABILITY OF S-DEFORMABLE SURFACES
1. Introduction
The class of surfaces in three-dimensional space that admit nontrivial de-
formations which simultaneously preserve principal directions and principal
curvatures (or, equivalently, the shape operator, also known as the Wein-
garten operator) has [4] a long and distinguished history: it was studied
already by Finikoff and Gambier [6, 7] more than 80 years ago but the in-
vestigation of preservation of principal directions and principal curvatures
dates back to Bonnet [2, 3]. For the sake of brevity we shall refer to the
surfaces from the class in question as to the S-deformable surfaces.
Ferapontov [4] has established integrability of the corresponding Gauss–
Codazzi equations (1) by presenting the associated Lax pair with a non-
removable spectral parameter; cf. also [12] and references therein for the
general study of integrability of the Gauss–Codazzi equations.
A natural next step in the study of the equations in question, which we
rewrite in the form (2), is to explore their geometric structures naturally
related to integrability: symmetries, conservation laws, Hamiltonian struc-
tures and recursion operators.
In what follows we implement this program. Namely, after recalling the
explicit form of the equations under study and of their Lax pair in Section 2
we proceed to construct an infinite sequence of nontrivial nonlocal conser-
vation laws for (2) in Section 3, and a recursion operator along with a pair
of compatible local Hamiltonian structures in Sections 4 and 5.
2. Preliminaries
Consider the system [4] of the Gauss–Codazzi equations describing the
S-deformable surfaces
∂ H = β H , ∂ H = β H ,
1 2 12 1 2 1 21 2
∂ β +∂ β = 0, (1)
1 12 2 21
1 1
′ ′
η ∂ β +η ∂ β + η β + η β +H H = 0,
1 1 12 2 2 21 2 1 12 2 2 21 1 2
where η = η (x) and η = η (y) are arbitrary smooth functions.
1 1 2 2
Upon expressing β via H we rewrite this system in the form
ij k
1η′v2v + 1η′uvu +(η η )uu v +u2v3
u = 2 1 x 2 2 y 1− 2 y y ,
yy
uv(η η )
1 2
− (2)
1η′uvv + 1η′u2u +(η η )vv u +u3v2
v = 2 1 x 2 2 y 2− 1 x x ,
xx
uv(η η )
2 1
−
where u = H , v = H . Its (complex) sl -valued zero-curvature representa-
1 2 2
tion reads [4]
u
y
ψ 1 i√λ+η2 u ψ
1 = · v u 1 , (3)
(cid:18)ψ2(cid:19)x 2√λ+η1  −u −i√λ+η2 · vy(cid:18)ψ2(cid:19)
 vx 
ψ i √λ+η1 v ψ
1 = − · u v 1 ,
(cid:18)ψ2(cid:19)y 2√λ+η2  v √λ+η1 · ux(cid:18)ψ2(cid:19)
 
INTEGRABILITY OF S-DEFORMABLE SURFACES 3
where λ R.
∈
For any zero-curvature representation of the form
ψ1 A1 A1 ψ
= 1 2 1 ,
ψ2 A2 A2 ψ
(cid:18) (cid:19)x (cid:18) 1 2(cid:19)(cid:18) 2(cid:19)
ψ1 B1 B1 ψ1
= 1 2
ψ2 B2 B2 ψ2
(cid:18) (cid:19)y (cid:18) 1 2(cid:19)(cid:18) (cid:19)
we can (cf. e.g. [16] and references therein) consider the associated Riccati
covering
w = A1+(A1 A2)w A2w2,
x 2 1− 2 − 1
w = B1+(B1 B2)w B2w2
y 2 1 − 2 − 1
where w = ψ1/ψ2; see e.g. [13] and references therein for more details on
(differential) coverings.
In particular, for (3) the Riccati covering is
λ+η u u
w = i 2 yw+ (1+w2), (4)
x
sλ+η1 · v 2√λ+η1
λ+η v iv
w = i 1 xw+ (1 w2).
y
− sλ+η2 · u 2√λ+η2 −
By changing the parametrization λ = 1/µ2, µ > 0, we transform (4) to
1+η µ2 u uµ
w = i 2 yw+ (1+w2), (5)
x
s1+η1µ2 · v 2 1+η1µ2
1+η µ2 v p ivµ
w = i 1 xw+ (1 w2).
y
− s1+η2µ2 · u 2 1+η2µ2 −
p
3. Infinite hierarchy of nonlocal conservation laws
Consider the formal Taylor expansions
w = w +w µ+w µ2+w µ3...,
0 1 2 3
w2 = w2+2w w µ+(2w w +w2)µ2+2(w w +w w )µ3+...,
0 0 1 0 2 1 0 3 1 2
1
= α1+α1µ2+ +α1 µ2k +...,
1+η µ2 0 2 ··· 2k
1
1
p = α2+α2µ2+ +α2 µ2k +...,
1+η µ2 0 2 ··· 2k
2
p1+η µ2
1 = α12+α12µ2+ +α12µ2k +...,
s1+η2µ2 0 2 ··· 2k
1+η µ2
2 = α21+α21µ2+ +α21µ2k +...,
s1+η1µ2 0 2 ··· 2k
where
η k
j j
α = (2k 1)!!, j = 1,2,
2k − 2 −
(cid:16) (cid:17)
4 INTEGRABILITY OF S-DEFORMABLE SURFACES
and
κ+(κ+ 1)...(κ+ a+1) κ−(κ− 1)...(κ− b+1)
αjl = − − − − ηaηb,
2k a! · b! j l
a+b=2k
X
where κ± = 1/2 and j = 1, l = 2 or j = 2, l = 1.
±
Substitutingthese expansions into (5) and equating coefficients at powers
of µ, we obtain the following infinite tower of 1-dimensional coverings:
iu
w = yα21w ,
0,x v 0 0
iu u
w = yα21w + α1(1+w2),
1,x v 0 1 2 0 0
iu
w = y(α21w +α21w )+uα1w w ,
2,x v 2 0 0 2 0 0 1
iu u
w = y(α21w +α21w )+ α1(1+w2)+α1(2w w +w2) ,
3,x v 2 1 0 3 2 2 0 0 0 2 1
... (cid:16) (cid:17)
iu
w = y(α21w +α21 w +...α21w )
2k,x v 2k 0 2k−2 2 0 2k
+u α1 w w +α1 (w w +w w )+
2k−2 0 1 2k−4 0 3 1 2
(cid:16)
···+α10(w0w2k−1+···+wk−1wk) ,
iu (cid:17)
w = y(α21w +α21 w + +α21w )
2k+1,x v 2k 1 2k−2 3 ··· 0 2k+1
u
+ α1 (1+w2)+α1 (2w w +w2)+
2 2k 0 2k−2 0 2 1
(cid:16)
···+α10(2w0w2k +···+2wk−1wk+1+wk2) ,
... (cid:17)
and
iv
w = xα12w ,
0,y − u 0 0
iv iv
w = xα12w α2(w2 1),
1,y − u 0 1− 2 0 0 −
iv
w = x(α12w +α12w ) ivα2w w ,
2,y − u 2 0 0 2 − 0 0 1
iv iv
w = x(α12w +α12w ) α2(w2 1)+α2(2w w +w2) ,
3,y − u 2 1 0 3 − 2 2 0 − 0 0 2 1
iv (cid:16) (cid:17)
w = x(α12w +α12 w + +α12w )
2k,y − u 2k 0 2k−2 2 ··· 0 2k
iv α2 w w +α2 (w w +w w )+
− 2k−2 0 1 2k−4 0 3 1 2
(cid:16)
···+α20(w0w2k−1+···+wk−1wk) ,
iv (cid:17)
w = x(α12w +α12 w +...α12w )
2k+1,y − u 2k 1 2k−2 3 0 2k+1
iv
α2 (w2 1)+α2 (2w w +w2)+
− 2 2k 0 − 2k−2 0 2 1
(cid:16)
INTEGRABILITY OF S-DEFORMABLE SURFACES 5
···+α20(2w0w2k +···+2wk−1wk+1+wk2) ,
... (cid:17)
Apply the following gauge transformation in the above covering:
w = eθ0, w = θ eθ0, k > 0.
0 k k
Then the latter transforms to
iu
θ = yα21,
0,x v 0
θ = uα1coshθ ,
1,x 0 0
iu
θ = yα21+uα1θ eθ0,
2,x v 2 0 1
iu 1
θ = yα21θ +u α1coshθ +α1(θ + θ2)eθ0 ,
3,x v 2 1 2 0 0 2 2 0
... (cid:16) (cid:17)
iu
θ = yX21+uX1 ,
2k,x v 2k 2k
iu
θ = yX21 +uX1 ,
2k+1,x v 2k+1 2k+1
...
where
X22k1 = α221k +α221k−2θ2+···+α221θ2k−2,
X21k = α12k−2θ1+α12k−4(θ3+θ1θ2)+···+α10(θ2k−1+θ1θ2k−2+
(cid:16)
+θk−1θk) eθ0,
···
X22k1+1 = α221kθ1+α22(cid:17)1k−2θ3+···+α221θ2k−1,
1 1
X1 = α1 coshθ + α1 (θ + θ2)+α1 (θ +θ θ + θ2)+
2k+1 2k 0 2k−2 2 2 1 2k−4 4 1 3 2 2
(cid:16) 1
···+α10(θ2k +θ1θ2k−1+···+θk−1θk+1+ 2θk2) eθ0,
(cid:17)
and
iv
θ = xα12,
0,y − u 0
θ = ivα2sinhθ ,
1,y − 0 0
iv
θ = xα12 ivα2θ eθ0,
2,y − u 2 − 0 1
iv 1
θ = xα12θ iv α2sinhθ +α2(θ + θ2)eθ0 ,
3,y − u 2 1− 2 0 0 2 2 1
... (cid:16) (cid:17)
iv
θ = xY12 ivY2,
2k,y − u 2k − 2k
iv
θ = xY12 ivY2 ,
2k+1,y − u 2k+1− 2k+1
6 INTEGRABILITY OF S-DEFORMABLE SURFACES
where
Y21k2 = α122k +α122k−2θ2+···+α122θ2k−2,
Y22k = α22k−2θ1+α2k−4(θ3+θ1θ2)+
(cid:16)
···+α20(θ2k−1+θ1θ2k−2+···+θk−1θk) eθ0,
Y21k2+1 = α122kθ1+α122k−2θ3+···+α122θ2k−1, (cid:17)
1 1
Y2 = α2 sinhθ + α2 (θ + θ2)+(θ +θ θ + θ2)+
2k+1 2k 0 2k−2 2 2 1 4 1 3 2 2
(cid:16) 1
···+α20(θ2k +θ1θ2k−1+···+θk−1θk+1+ 2θk2) eθ0.
(cid:17)
Remark. The above complex covering can be reduced to the real form if we
set θ = p +iq , k = 0,1,..., and consider real and imaginary parts of the
k k k
defining equations separately. In the particular case p = 0 we obtain a real
0
covering which is employed below (see Section 4) to construct a recursion
operator for symmetries of (2).
Thus, we have obtained an infinite tower of Abelian coverings
E oo E oo ... oo E oo E oo ...,
0 2k 2k+1
whereE is the initial system (2) and E is obtained by extending E with the
s
nonlocal variables θ ,...,θ . It only remains to prove that for any s N
0 s
∈
the conservation law
iu iv
ω = yX21+uX1 dx xY12+ivY2 dy (6)
s v s s − u s s
is nontrivial on Es−(cid:16)1. (cid:17) (cid:16) (cid:17)
Denote by D[l], D[l] the total derivatives on E , l 1.
x y l
≥
Proposition 1. The only solutions of the system
D[l](f)= 0, D[l](f)= 0, f C∞(E ), (7)
x y l
∈
are constants.
Proof of Proposition 1 (Part I). The total derivatives on E have the form
l
l l
iu ∂ iv ∂
D[l] = D + yX21+uX1 , D[l] = D xY12+ivY2 ,
x x v s s ∂θ y y− u s s ∂θ
s s
Xj=0(cid:16) (cid:17) Xj=0(cid:16) (cid:17)
where D and D are the total derivatives on E. Obviously, a function f
x y
C∞(E ) is a solution to (7) if and only if ∈
l
∂f ∂f ∂f
= 0, = 0, = 0,
∂x ∂y ∂u
and
l l l l
∂f ∂f ∂f ∂f
X12 = 0, Y21 = 0, X1 =0, Y2 = 0.
s ∂θ s ∂θ s∂θ s ∂θ
s s s s
s=0 s=0 s=1 s=1
X X X X
(8)
We shall proceed with the proof after establishing Lemma 1. (cid:3)
INTEGRABILITY OF S-DEFORMABLE SURFACES 7
Consider the vector fields
l l
∂ ∂
TX = X1 , TY = Y2
l s∂θ l s ∂θ
s s
s=1 s=1
X X
on E and let
l
Z = [[...[TX ,TY ],...],TY ]
l l l l
l−1 times
for l 2.
| {z }
≥
Lemma 1. For any l 2 we have
≥
∂
α1(α2)l−1 for even l,
0 0 ∂θ
Z = l
l  ∂
α10(α20)l−1cosh(θ0)∂θ for odd l.
l
Proof of Lemma 1. Consider the formal series
2
∞
1
Q = 1+ θ λj
j
2  
j=1
X
 
and denote by Q its coefficient at λj.
[j]
Assign to any monomial p = ϕ(θ )θ ...θ the weight
0 i1 ir
p = i + +i .
1 r
| | ···
Then the quantities Q are homogeneous and Q = j. In addition, one
[j] [j]
| |
has
1, if i = j,
∂Q
[j]
= θj−i if i < j, (9)
∂θ
i 0, otherwise
for any i 1. 
≥ 
Consider the vector fields
∞ ∞
∂ ∂
TX = X1 , TY = Y2
s∂θ s ∂θ
s s
s=1 s=1
X X
on the space E∞ = liminvl→∞El. Then obviously, TX E = TXl, TY E =
| l | l
TY and [TX,TY] = [TX ,TY ]. These fields can be written as follows:
l E l l
| l
∂ ∂ ∂
TX = α1 cosh(θ ) +eθ0 (Q +o(0)) +(Q +o(1)) +...
0 0 ∂θ [1] ∂θ [2] ∂θ
(cid:18) 1 (cid:18) 2 3
∂
+(Q[l−1]+o(l 2)) +... ,
··· − ∂θ
l
(cid:19)(cid:19)
∂ ∂ ∂
TY = α2 sinh(θ ) +eθ0 (Q +o(0)) +(Q +o(1)) +...
0 0 ∂θ [1] ∂θ [2] ∂θ
(cid:18) 1 (cid:18) 2 3
∂
+(Q[l−1]+o(l 2)) +... ,
··· − ∂θ
l
(cid:19)(cid:19)
8 INTEGRABILITY OF S-DEFORMABLE SURFACES
where o(j) denotes terms of weight j. To simplify notation, we shall use
≤
the short form
∂ ∂ ∂ ∂
TX = α10 cosh(θ0)∂θ +eθ0 Q[1]∂θ +Q[2]∂θ +···+Q[l−1]∂θ +... +o,
(cid:18) 1 (cid:18) 2 3 l (cid:19)(cid:19)
∂ ∂ ∂ ∂
TY = α20 sinh(θ0)∂θ +eθ0 Q[1]∂θ +Q[2]∂θ +···+Q[l−1])∂θ +... +o.
(cid:18) 1 (cid:18) 2 3 l (cid:19)(cid:19)
One readily checks that
[A+o,B +o] = [A,B]+o
in all subsequent computations.
We shall now prove, by induction on l, that the fields
Z = [[...[TX,TY],...],TY]
l
l−1 times
are of the form | {z }
∂ ∂ ∂
α1(α2)l−1 +θ + +θ +... +o
0 0 ∂θ 1∂θ ··· j∂θ
(cid:18) l l+1 l+j (cid:19)
if l is even and
∂ ∂ ∂
α1(α2)l−1 cosh(θ ) +eθ0 Q + +Q +... +o
0 0 0 ∂θ [1]∂θ ··· [j]∂θ
(cid:18) l (cid:18) l+1 l+j (cid:19)(cid:19)
for odd l. The claim of Lemma 1 readily follows from these formulas for
Z (please note, however, that the Z in the statement of Lemma 1 are not
l l
identical to those defined above).
Let l = 2. Then by virtue of (9) we have
Z = [TX,TY]
2
∞ ∞
∂ ∂ ∂ ∂
= cosh(θ ) +eθ0 Q ,sinh(θ ) +eθ0 Q +o
 0 ∂θ [j+1]∂θ 0 ∂θ [j+1]∂θ 
1 j 1 j
j=1 j=1
X X
 ∞ 
∂ ∂
= α1α2eθ0(cosh(θ ) sinh(θ )) + θ +o
0 0 0 − 0 ∂θ j∂θ 
2 j+2
j=1
X
 ∞ 
∂ ∂
= α1α2 + θ +o.
0 0∂θ j∂θ 
2 j+2
j=1
X
 
For l = 3 one has
Z = [Z ,TY]
3 2
∞ ∞
∂ ∂ ∂ ∂
= α1α2 + θ ,α2 sinh(θ ) +eθ0 Q +o
 0 0∂θ j∂θ  0 0 ∂θ [j+1]∂θ 
2 j+2 1 j
j=1 j=1
X X
  ∞   ∞ ∞ 
∂ ∂ ∂ ∂
= α1(α2)2eθ0 + θ + θ + θ
0 0 ∂θ j∂θ s∂θ j∂θ 
3 j+3 s+3 j+s+3
j=1 s=1 j=1
X X X
  
INTEGRABILITY OF S-DEFORMABLE SURFACES 9
∞
∂ ∂
α1(α2)2 sinh(θ ) +eθ0 Q +o
− 0 0  0 ∂θ [j]∂θ 
3 j+3
j=1
X
 ∞ 
∂ ∂
= α1(α2)2eθ0 +2 Q
0 0 ∂θ [j]∂θ 
3 j+3
j=1
X
 ∞ 
∂ ∂
α1(α2)2 sinh(θ ) +eθ0 Q +o
− 0 0  0 ∂θ [j]∂θ 
3 j+3
j=1
X
 ∂  ∂
= α1(α2)2 cosh(θ ) +eθ0 Q +o.
0 0 0 ∂θ [j]∂θ
(cid:18) 3 (cid:18) j+3(cid:19)(cid:19)
The induction step also uses property (9) of the quantities Q and is
[j]
accomplished by the computations quite similar to those given above. (cid:3)
Let us complete the proof of Proposition 1.
Proof of Proposition 1 (Part II). Weprovebyinductiononlthatsystem(7)
possesses constant solutions only.
For l = 1, the equations are
iu ∂f ∂f
D (f)+α21 y +α1cosh(θ )u = 0,
x 0 v ∂θ 0 0 ∂θ
0 1
iv ∂f ∂f
D (f) α12 x α2isinh(θ )v = 0,
y − 0 u ∂θ − 0 0 ∂θ
0 1
where f is a function on E , i.e., it may depend on jet variables as well
1
as on θ and θ . Analyzing the coefficients of the fields D and D , one
0 1 x y
immediately sees that f can depend on x, y, θ , and θ only and from the
0 1
above equations it readily follows that f = const.
Assume now that the result is valid for some l > 1 and let f C∞(E )
l+1
∈
[l+1] [l+1]
be such that D (f)= D (f)= 0. Then, by (8),
x y
TX (f)= 0, TY (f)= 0.
l+1 l+1
Consequently, Z (f) = 0. Using Lemma 1, we see that
l+1
∂f
= 0.
∂θ
l+1
Thus,f isafunctiononE anditisconstantbytheinductionhypothesis. (cid:3)
l
Corollary 1. The conservation law ω given by (6) is nontrivial on E .
s s
[s] [s]
Proof. Indeed, if ω = d f, whered = dx D +dy D is the horizontal
s h h x y
∧ ∧
deRham differential on E then D[s+1](f θ )= 0 and D[s+1](f θ ) =
s x s+1 x s+1
0, which contradicts to Proposition 1. − − (cid:3)
Thus we have constructed an infinite hierarchy of nonlocal conservation
laws ω , s N, for (2).
s
∈
10 INTEGRABILITY OF S-DEFORMABLE SURFACES
4. Recursion operator and Hamiltonian structures
It is readily checked that (2) possesses an Abelian covering S with the
fiber coordinates s defined by the formulas
i
u v
y x
(s ) = , (s ) = ,
0 x 0 y
v − u
(s ) = cos(s )u, (s ) = vsin(s ),
1 x 0 1 y 0
(s ) = sin(s )u, (s ) = vcos(s ).
2 x 0 2 y 0
−
Note that the conservation laws associated with s , i = 1,2, are potential
i
conservation laws in the terminology of [14] and that we have θ = is and
0 0
θ = α1s modulo the addition of arbitrary constants.
1 0 1
The following result is readily verified by straightforward computation.
Proposition 2. Suppose thatU,V andS arefibercoordinates ofthetangent
i
covering VS.
Thenthetangentcoveringover(2)admitsaB¨acklundauto-transformation
(i.e., a recursion operator for (2)) of the form
1 sin(s )(η η )
U˜ = η U + η′ cos(s )+ 0 2 − 1 u S
1 2 1 0 v y 1
(cid:18) (cid:19)
1 cos(s )(η η )
′ 0 2 1
+ η sin(s )+ − u S ,
−2 1 0 v y 2
(cid:18) (cid:19) (10)
1 cos(s )(η η )
V˜ = η V + η′ sin(s ) 0 2 − 1 v S
2 2 2 0 − u x 1
(cid:18) (cid:19)
1 sin(s )(η η )
′ 0 2 1
+ η cos(s )+ − v S .
2 2 0 u x 2
(cid:18) (cid:19)
Equations(10)definearecursionoperatorfor(15)inthefollowingfashion
(see e.g. [15, 17, 18, 23] and references therein for details).
Suppose (U,V)T is a symmetry shadow for (15) in a covering C over S
(here and below the superscript T indicates the transposed matrix). Then
we have a (possibly trivial) covering C′ over C arising from substituting
our U and V into the equations defining S . Under these assumptions (10)
i
defines a new symmetry shadow (U˜,V˜)T for (15) in C′, i.e., we have a
recursion operator for (15).
Note that (10) was found using the method from [17]; cf. e.g. [15, 11, 19,
20, 23] for some other related techniques.
Starting with a simple seed symmetry like (0,0)T yields, through the
repeated application of (10), an infinite hierarchy of shadows of nonlocal
symmetries for (2). It is an interesting open problem to find the minimal
covering in which it is possible to lift all these shadows to full-fledged non-
local symmetries of (2) and to find the commutation relations among these
nonlocal symmetries, cf. e.g. [21].
Moreover, one can readily establish the following result:
Proposition 3. System (2) admits (in a generalized sense of [9, 11, 13]) a
pair of compatible local Hamiltonian structures P of the form
i
P = f D D +D f +D f +f , i= 1,2,
i i,3 x y y i,2 x i,1 i,0
◦ ◦