Metin Gürses, Ismagil Habibullin, and Kostyantyn Zheltukhin
Citation: J. Math. Phys. 48, 102702 (2007); doi: 10.1063/1.2799256 View online: http://dx.doi.org/10.1063/1.2799256
View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v48/i10 Published by the American Institute of Physics.
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Integrable boundary value problems for elliptic type Toda
lattice in a disk
Metin Gürsesa兲and Ismagil Habibullinb兲
Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey
Kostyantyn Zheltukhinc兲
Department of Mathematics, Faculty of Sciences, Middle East Technical University, 06531 Ankara, Turkey
共Received 16 June 2007; accepted 25 September 2007; published online 25 October 2007兲 The concept of integrable boundary value problems for soliton equations onR and R+is extended to regions enclosed by smooth curves. Classes of integrable
bound-ary conditions in a disk for the Toda lattice and its reductions are found. © 2007
American Institute of Physics. 关DOI:10.1063/1.2799256兴
I. INTRODUCTION
The inverse scattering transform method 共ISM兲 discovered in 1967 has proved to be a pow-erful tool to construct exact solutions and to solve the Cauchy problem for a large variety of nonlinear integrable models of mathematical physics. However, real physical applications are usually related to mathematical models with boundary conditions. For this reason, the problem of adopting the ISM to a boundary value problem as well as to an initial boundary value共mixed兲 problem is very important. During the last two decades, this field of research has been intensively studied. It becomes clear that only special kinds of boundary conditions preserve the integrability property of the equation given. Different approaches were worked out to look for such classes of boundary conditions based on Hamiltonian structures,1 on higher symmetries,2–4 and the Lax representation.5,6 Integrable initial boundary value problems on a half-line 共in 1+1 case兲 or a half-plane 共in 1+2 case兲 for soliton equations nowadays is a rather studied subject. Analytical aspects have been developed in Refs. 7–11 where large classes of solutions were constructed. However, boundary value problem for the elliptic soliton equations or initial boundary value problem for regions with more complicated boundary is still much less investigated 共see, Refs.
12–14兲.
If the boundary conditions are not consistent with the integrability property of the equation, then the standard version of the inverse scattering transform method cannot be applied to the corresponding boundary value problem. The method requires a very essential modification. Vari-ous ideas to extend the ISM to the initial boundary value problems are suggested in Refs. 12,
15–17and20,21.
In Refs.5and6, an effective tool to search integrable boundary conditions has been proposed based on some special involutions of the auxiliary linear problem. This method共below for the sake of convenience, we refer it as the method of involutions兲 can be applied to integrable equations in both共1+1兲- and 共1+2兲-dimensional cases. Some examples of application of the inverse scattering transform method for such kind of boundary value problems were considered in Ref.18.
a兲Electronic mail: gurses@fen.bilkent.edu.tr
b兲On leave from Ufa Institute of Mathematics, Russian Academy of Science, Chernyshevskii Str., 112, Ufa 450077, Russia.
Electronic mail: habibullinគi@mail.rb.ru and habib@fen.bilkent.edu.tr
c兲Electronic mail: zheltukh@metu.edu.tr
48, 102702-1
In this article, we show that the method of involutions allows one to extend the concept of integrability to boundary value problems on bounded共unbounded兲 regions enclosed by any closed smooth curve.
Let us explain briefly the approach we use. We call boundary value problem integrable if it admits a Lax pair. Because of this reason, we look for a boundary condition simultaneously with its Lax representation. The starting point is to make a correct assumption about the possible form of the Lax pair of the boundary value problem. Actually, this Lax pair is made up from several different Lax pairs of the original equation itself by gluing the eigenfunctions along the boundary by properly chosen additional boundary conditions. As examples, we take the Liouville equation and the two-dimensional Toda lattice equation. To generate new Lax pairs, we use point symme-tries共involutions兲 which leave invariant the nonlinear equation under consideration but change its Lax pair.
In Sec. II, as a trial example, we consider the Liouville equation. We remind the definition of integrable boundary conditions and find an example of integrable boundary conditions on a circle with its Lax representation共see the list at the end of the second section兲.
In Sec. III, we study the two-dimensional Toda lattice equation on a circular cylinder: r ⬍a,0艋艋2, −⬁⬍n⬍⬁. Several types of integrable boundary value problems for this lattice and Lax representations of the boundary value problems are found by using the method of invo-lutions共see the list at the end of the third section兲.
In Sec. IV, we consider periodicity closure constraints reducing the Toda lattice to the sinh-Gordon and Tzitzeica equations.
In Sec. V, we give a class of exact solutions of the Toda lattice on a circle with a nonhomo-geneous Neumann-type boundary condition in a disk.
II. LIOUVILLE EQUATION
In this section, we concentrate on boundary value problems for elliptic equations. Suppose that the boundary⌫ of a domain D is parametrized by the equation x⬘= f共t⬘兲 that introduces a local system of coordinates by taking the t axis along the tangent direction and the x axis along the normal direction to the curve⌫.
Suppose that the differential equation under consideration,
E共u兲 = 0, 共1兲
admits two different Lax representations. For the sake of simplicity, we take them rewritten in terms of the new coordinates
Yx= U共,u,ux, . . .兲Y共兲, Yt= V共,u,ux, . . .兲Y共兲 共2兲 and Y ˜ x= U˜ 共˜,u,ux, . . .兲Y˜共˜兲, Y ˜ t= V˜ 共˜,u,ux, . . .兲Y˜共˜兲, 共3兲
where,˜ are spectral parameters. Now, the equation of the boundary is of the form x=0. We are looking for conditions that allow to relate the equations for t evolution along the boundary since
x is fixed. More precisely, we have the following definition. Definition 1: A boundary condition
⍀共t,u,ut,ux, . . .兲 = 0 共4兲
x = 0, the function Y = F共,t,u, ...兲Y˜共˜兲 is a solution of the equation Yt= VY for any solution Y˜ of
the equation Y˜ =V˜Y˜ with ˜=h共兲, provided that the boundary condition holds.t
If a boundary condition is integrable in the sense of the definition above, this means that the corresponding boundary value problem admits the Lax representation consisting of the two Lax pairs共2兲and共3兲defined on the domain D such that the eigenfunctions Y and Y˜ satisfy along the boundary an additional boundary condition 兩共Y −FY˜兲兩⌫= 0.
To consider a circle as a boundary, we use polar coordinates共r,兲. So, the boundary is r=a. In polar coordinates, the Liouville equation is
urr+ 1 rur+ 1 r2u= 8e u . 共5兲
It admits the Lax pair
Yr= LY, Y= AY , 共6兲 where x = r, t =, U = L, V = A, and L =
冢
eu+i 2 +e−i − eu+i 2 + 1 4ur+ i 4ru eu+i 2 + 1 4ur+ i 4ru − eu+i 2 −e−i冣
, 共7兲 A = ir冢
eu+i 2 −e −i −eu+i 2 − 1 4ur− i 4ru eu+i 2 − 1 4ur− i 4ru − eu+i 2 +e−i冣
. 共8兲To obtain a second Lax representation, we use the Kelvin transformation. Equation共5兲 is invariant under the Kelvin transformation
r ¯ =a 2 r, u¯ = u + 4 ln a r. 共9兲
Under such transformation, the Lax pair共6兲 takes the form
Y ¯
r= L¯ Y¯, Y¯= A¯ Y¯ , 共10兲
where L ¯ =
冢
r4eu+i 2a4˜ +˜e−i − r 4eu+i 2a4 − r2u r 4a2 − r a2+ ir 4a2u r4eu+i 2a2˜ −r 2u r 4a2 − r a2+ ir 4a2u − r4eu+i 2a4˜ −˜e−i冣
, 共11兲A ¯ =ia2 r
冢
r4eu+i 2a4˜ −˜e−i −r 4eu+i 2a4˜ +r 2u r 4a2 + r a2− iru 4a2 r4eu+i 2a4˜ +r 2u r 4a2 + r a2− iru 4a2 − r4eu+i 2a4˜ +˜e−i冣
. 共12兲We will look for a boundary condition under which there exists a transformation˜=h共兲 and a nondegenerate matrix F共,, u兲 such that Y共兲=FY˜共h共兲兲 will solve the 兩Y共兲=A兩r=aY共兲 for
every solution Y˜ 共˜兲 of the equation 兩Y˜共˜兲=A˜兩r=aY˜ 共˜兲.
Lemma 1: The integrable boundary condition is given by
兩ur兩r=a=
− 2
a , 共13兲
and there are two choices for the matrix F and the function h, (i) h共兲 = , F =
冉
1 0 0 1冊
, 共14兲 (ii) h共兲 = − , F =冉
0 − 1 − 1 0冊
. 共15兲Proof: Let Y¯ 共兲 satisfy the equation Y¯= A¯ Y¯. On the boundary r=a, Y =FY¯共h共兲兲 has to satisfy
Y= AY. Substituting Y = FY¯ 共h共兲兲 into Y= AY and using Y¯= A¯ Y¯ for Y¯共h共兲兲, we obtain
冉
ddF − A共兲F + FA¯共h共兲兲
冊
Y¯ 共h共兲兲 = 0. 共16兲 The above equality holds ifd
dF = A共兲F − FA¯共h共兲兲. 共17兲
We have an equation for the unknown matrix F and function h共兲. To solve the boundary condi-tion 共4兲 with respect to ur, we let ur= G共, u , u兲. Assuming that F does not depend on u and
differentiating 共17兲 twice with respect to u, we obtain 2ur/u2= 0. That is, ur= g1共u,兲u
+ g2共u,兲. We substitute the above expression for urinto共17兲and let
F =
冉
f11共u,,兲 f12共u,,兲 f21共u,,兲 f22共u,,兲冊
. 共18兲
Separating terms with uand without uin共17兲, we obtain two sets of equations. We write the first set of equations, terms with u, as
uf = Pf , 共19兲
ia
冢
0 −冉
g1 4 − i 4a冊 冉
− g1 4 − i 4a冊
0 −冉
g1 4 − i 4a冊
0 0冉
− g1 4 − i 4a冊
冉
−g1 4 − i 4a冊
0 0 −冉
g1 4 − i 4a冊
0冉
−g1 4 − i 4a冊
−冉
g1 4 − i 4a冊
0冣
. 共20兲We write the second set of equations, terms without u, as f = Qf , 共21兲 where Q is a matrix ia
冢
− −冉
␦+ e u+i 2h共兲冊 冉
− g2 4 − eu+i 2冊
0 −冉
␦− e u+i 2h共兲冊
+ 0冉
− g2 4 − eu+i 2冊
冉
eu+i 2 − g2 4冊
0 −−冉
␦+ eu+i 2h共兲冊
0冉
e u+i 2 − g2 4冊
−冉
␦− eu+i 2h共兲冊
−+冣
, 共22兲with= eu+i/ 2−e−i,= eu+i/ 2h共兲−h共兲e−i, and ␦= g
2/ 4 + 1 / a.
Equations共19兲and共21兲must be compatible. This leads to the following compatibility condi-tion:
共P− Qu+关P,Q兴兲f = 0, 共23兲
where关P,Q兴 is a commutator of P and Q. The matrix 共P− Qu+关P,Q兴兲 is nonzero. To have a
nonzero solution f, the determinant of 共P− Qu+关P,Q兴兲 must be zero. It gives the following
equality:
a6e−4i
16 共h
2共兲共ag
1共u,兲 − i兲2−共ag1共u,兲 + i兲22兲2= 0. 共24兲
The above equality holds if either
共1兲 h共兲= and g1= i共1+兲/a共1−兲, where苸R\兵−1,1其 or
共2兲 h共兲= and g1= 0 or
共3兲 h共兲=− and g1= 0.
One can show that in case共1兲, there is no vector f to satisfy Eqs.共19兲and共21兲. In case共2兲, one has the only solution f = q共兲共1,0,0,1兲T if g
2= −2 / a. This gives the boundary condition 共13兲,
function h, and matrix F given by共14兲.
Case 共3兲 is similar to Case 共2兲 and gives the same boundary condition共13兲, function h, and
matrix F given by共15兲. 䊐
From the above lemma, we have the following integrable boundary value problem with corresponding Lax pairs共we have two Lax pairs for the problem兲:
共1兲 r⬍ a, urr+ 1 rur+ 1 r2u= 8e u ,
Yr共兲 = L共兲Y共兲, Y¯r共兲 = L¯共兲Y¯共兲,
Y共兲 = A共兲Y共兲, Y¯共兲 = A¯共兲Y¯共兲,
r = a, 兩ur兩r=a= −
2
a,
Y = FY¯ ,
where F is given by共14兲, L is given by共7兲, A is given by共8兲, L¯ is given by共11兲, and A¯ is given by共12兲. 共2兲 r⬍ a, urr+ 1 rur+ 1 r2u= 8e u ,
Yr共兲 = L共兲Y共兲, Y¯r共− 兲 = L¯共− 兲Y¯共− 兲,
Y共兲 = A共兲Y共兲, Y¯共− 兲 = A¯共− 兲Y¯共− 兲,
r = a, 兩ur兩r=a= −
2
a,
Y共兲 = FY¯共− 兲,
where F is given by共15兲, L is given by共7兲, A is given by共8兲, L¯ is given by共11兲, and A¯ is given by共12兲.
Remark 1: The above boundary value problem admits infinitely many explicit solutions of the
form
u = ln
冉
n2共␣2+2兲
4r2共␣共cos n+sin n兲2兲
冊
, 共25兲for any␣,, n and of the form
u = − 2 ln共k共r2+ a2兲 + r共␣cos+sin兲兲, 共26兲 where␣2+2= 4 + 4k2a2. We note that all these solutions have a singularity inside the region r
⬍a. Unfortunately, we failed to find regular solutions to the above boundary value problem.
III. TWO-DIMENSIONAL TODA LATTICE
We make the same assumption, as in the case of the Liouville equation, for the coordinates. Hence, boundary is given by x = 0. Again, we suppose that the differential equation under consid-eration admits two different Lax representations,
Yx= UY, Y˜x= U˜ Y˜ ,
Yt= VY, Y˜t= V˜ Y˜ . 共27兲
For the two-dimensional Toda lattice equation, U, V, U˜ , and V˜ in共27兲are linear operators.
Definition 2: A boundary condition
⍀共u兲 = 0 共28兲
is integrable if there exists a linear differential operator A such that on the boundary x = 0, we have that Y˜ =AY is a solution of Y˜t= V˜ Y˜ for any solution Y of Yt= VY, provided that the boundary
condition holds.
To consider a circle as a boundary, we use polar coordinates共r,兲. So, the boundary is r=a. The two-dimensional Toda lattice equation in polar coordinates becomes
urr+
1
rur+
1
r2u=共n − 1兲 −共n兲, 共29兲
where共n兲=exp共u共n兲−u共n+1兲兲. The above equation admits a Lax pair
1,r共n兲 = ei 2 1共n + 1兲 − 1 2
冉
ur共n兲 − i ru共n兲冊
1共n兲 − e−i 2 共n − 1兲1共n − 1兲, 共30兲 1,共n兲 =ire i 2 1共n + 1兲 − ir 2冉
ur共n兲 − i ru共n兲冊
1共n兲 + ire−i 2 共n − 1兲1共n − 1兲. 共31兲To obtain other Lax representations, we use symmetries of Eq.共29兲. 共1兲 Reflection on,
˜ = −. 共32兲
共2兲 The Kelvin transformation,
r ˜ =a r, ˜ = u + 4n lnu a r. 共33兲 共3兲 Reflection on n, u ˜ = − u共− n兲. 共34兲
Using the transformation共32兲, we obtain the following Lax representation: 2,r共n兲 = e−i 2 2共n + 1兲 − 1 2
冉
ur共n兲 + i ru共n兲冊
2共n兲 − ei 2 共n − 1兲2共n − 1兲, 共35兲 2,共n兲 = ire−i 2 2共n + 1兲 − ir 2冉
ur共n兲 + i ru共n兲冊
2共n兲 + irei 2 共n − 1兲2共n − 1兲. 共36兲Using the Kelvin transformation共33兲, we obtain the following Lax representation: 3,r共n兲 = ei 2 3共n + 1兲 − 1 2
冉
− r2 a2 ur共n兲 + 4n r a2− ir a2u冊
4共n兲 − r4e−i 2a4 共n − 1兲3共n − 1兲, 共37兲3,共n兲 =ia 2ei 2r 3共n + 1兲 − ia2 2r
冉
− r2 a2 ur共n兲 + 4n r a2− ir a2u冊
3共n兲 + ir3e−i 2a2 共n − 1兲3共n − 1兲. 共38兲 Using the transformations共34兲, we obtain the following Lax representation:4,r共n兲 = ei 2 4共n − 1兲 − 1 2
冉
− ur共n兲 + i ru共n兲冊
4共n兲 − e−i 2 共n兲4共n + 1兲, 共39兲 4,共n兲 =ire i 2 4共n − 1兲 − ir 2冉
− ur共n兲 + i ru共n兲冊
4共n兲 + ire−i 2 共n兲4共n + 1兲. 共40兲According to Definition 2, to obtain the integrable boundary conditions, we relate the equations for evolution of the above Lax representations, on the boundary r = a. We consider the case when the eigenfunctions are related by the multiplication operatori= A共, n , u , . . .兲·j.
It turns out共see Lemma共7兲兲 that Lax pairs corresponding to the Kelvin transformation共33兲 and the symmetry 共34兲are gauge equivalent. A solution of共38兲transforms to a solution of 共40兲 without any boundary conditions. So, some boundary value problems have two possible Lax pairs. In Lemma 2, we derive the first boundary value problem in the list at the end of this section.
Lemma 2: Let1共n兲 be a solution of Eq. (31), then on the boundary r = a, a function2共n兲
= A ·1共n兲, where
A = e2in+g共兲, g共兲 is an arbitrary function of , 共41兲
is a solution of Eq.(36), provided, that the following boundary condition:
u共n兲 = 2in+ g共兲 + k共n兲, k共n兲 is an arbitrary function of n, 共42兲
holds for all n.
Proof: On the boundary r = a, we substitute2共n兲=A共n,, u , . . .兲·1共n兲 into Eq.共36兲and use
共31兲 for 1,共n兲. The resulting equation holds if the coefficients of 1共n+1兲, 1共n兲, and 1共n
− 1兲 are zero. Thus, we obtain
iaei 2 A共n兲 = iae−i 2 A共n + 1兲, 共43兲 A共n兲 −ia 2
冉
ur共n兲 − iu共n兲 a冊
A共n兲 = − ia 2冉
ur共n兲 + iu共n兲 a冊
A共n兲, 共44兲 iae−i 2 共n − 1兲A共n兲 = iaei 2 共n − 1兲A共n − 1兲. 共45兲From Eqs.共43兲and共45兲, we have that A共n兲=e2iA共n−1兲. Hence, A共n兲=e2inb共兲, where b共兲 is a
function of only. Substituting A共n兲=e2inb共兲 into Eq.共44兲, we obtain
b+共2in − u共n兲兲b = 0. 共46兲
Since the function b does not depend on n, we have that the coefficient of b in the above equation does not depend on n, so u共n兲=2in+h共兲. Integrating with respect to, we obtain the boundary condition共42兲, where k is an arbitrary function of n and g共兲=兰h共兲d. Then, solving Eq.共44兲, assuming that the found boundary condition holds, we obtain A = e2in+兰g共兲d, the expression共41兲
for A. 䊐
In Lemma 3 and Lemma 4, we derive the Lax pair for the second boundary value problem in the list.
Lemma 3: Let1共n兲 be a solution of Eq. (31), then on the boundary r = a, a function3共n兲
= A ·1共n兲, where
A = eia兰g共兲d, g共兲 is an arbitrary function of , 共47兲
is a solution of Eq.(38), provided that the following boundary condition:
ur共n兲 =
2n
a + g共兲, 共48兲
holds for all n.
Proof: On the boundary r = a, we substitute3共n兲=A共n兲·1共n兲 into Eq.共38兲and use共31兲for 1,共n兲. The resulting equation holds if the coefficients of1共n+1兲,1共n兲, and1共n−1兲 are zero.
Thus, we obtain iaei 2 A共n兲 = iaei 2 A共n + 1兲, 共49兲 A共n兲 −ia 2
冉
ur共n兲 − iu共n兲 a冊
A共n兲 = ia 2冉
ur共n兲 − 4n a + iu共n兲 a冊
A共n兲, 共50兲 iae−i 2 共n − 1兲A共n兲 = iae−i 2 共n − 1兲A共n − 1兲. 共51兲From Eqs.共49兲and共51兲, we have that A does not depend on n. Hence, the coefficient of A in
共50兲must be a function ofonly. This gives the boundary condition共48兲. Then, solving Eq.共50兲,
assuming that共48兲holds, we obtain the expression共47兲for A. 䊏
Lemma 4: Let1共n兲 be a solution of Eq. (31), then on the boundary r = a, a function4共n兲
= A ·1共n兲, where
A = e2in+u共n兲+ia兰g共兲d,g共兲 is an arbitrary function of , 共52兲
is a solution of Eq.(38), provided that the following boundary condition:
ur共n兲 =
2n
a + g共兲, 共53兲
holds for all n.
Proof: On the boundary r = a, we substitute4共n兲=A共n,, u , . . .兲·1共n兲 into Eq.共40兲and共38兲
and use共38兲for 1,共n兲. The resulting equation holds if the coefficients of 2共n+1兲, 2共n兲, and 2共n−1兲 are zero. Thus, we obtain
iaei 2 A共n兲 = iae−i 2 共n兲A共n + 1兲, 共54兲 A共n兲 −ia 2
冉
ur共n兲 − iu共n兲 a冊
A共n兲 = − ia 2冉
− ur共n兲 + iu共n兲 a冊
A共n兲, 共55兲iae−i
2 共n − 1兲A共n兲 =
iaei
2 A共n − 1兲. 共56兲
From Eqs.共54兲and共56兲, we have that A共n兲=e−2i共n兲A共n+1兲. Hence, A共n兲=e2in+u共n兲b共兲, where F共兲 is a function of only. Substituting A共n兲=e−2inb共兲 into Eq.共55兲, we obtain
b−共2in − iaur共n兲兲b = 0. 共57兲
Since the function b does not depend on n, we have that the coefficient of b in the above equation does not depend on n. This gives the boundary condition共53兲. Then, solving Eq.共55兲, assuming
that the found boundary condition holds, we obtain the expression共52兲for A. 䊏
In Lemma 5 and Lemma 6, we derive the Lax pair for the third boundary value problem in the list.
Lemma 5: Let2共n兲 be a solution of Eq. (36), then on the boundary r = a, a function3共n兲
= A ·2共n兲, where
A = e−2in+ia兰g共兲d, g共兲 is an arbitrary function of , 共58兲
is a solution of Eq.(38), provided that the following boundary condition: ur共n兲 = −
i
au共n兲 + g共兲, 共59兲
holds for all n.
Proof: On the boundary r = a, we substitute3共n兲=A共n,, u , . . .兲·2共n兲 into Eq.共38兲and use
共36兲 for 2,共n兲. The resulting equation holds if the coefficients of 2共n+1兲, 2共n兲, and 2共n
− 1兲 are zero. Thus, we obtain
iae−i 2 A共n兲 = iaei 2 A共n + 1兲, 共60兲 A共n兲 −ia 2
冉
ur共n兲 + iu共n兲 a冊
A共n兲 = − ia 2冉
− ur共n兲 + 4n r − iu共n兲 a冊
A共n兲, 共61兲 iaei 2 A共n兲 = iae−i 2 A共n − 1兲. 共62兲From Eqs.共60兲and共62兲, we have that A共n兲=e−2iA共n−1兲. Hence, A共n兲=e−2inb共兲, where b共兲 is
a function of only. Substituting A共n兲=e−2inb共兲 into Eq.共61兲, we obtain
b− ia
冉
ur共n兲 +i
au共n兲
冊
b = 0. 共63兲Since the function b does not depend on n, we have that the coefficient of b in the above equation does not depend on n. This gives the boundary condition共59兲. Then, solving Eq.共61兲, assuming
that the found boundary condition holds, we obtain the expression共58兲for A. 䊏
Lemma 6: Let2共n兲 be a solution of Eq. (36), then on the boundary r = a, a function4共n兲
= A ·2共n兲, where
A = e2u共n兲+兰g共兲d, g共兲 is an arbitrary function of , 共64兲
is a solution of Eq.(40), provided that the following boundary condition: ur共n兲 = −
i
holds for all n.
Proof: On the boundary r = a, we substitute4共n兲=A共n,, u , . . .兲·2共n兲 into Eq.共40兲and use
共36兲 for 2,共n兲. The resulting equation holds if the coefficients of 2共n+1兲, 2共n兲, and 2共n
− 1兲 are zero. Thus, we obtain
iae−i 2 A共n兲 = iae−i 2 共n兲A共n + 1兲, 共66兲 A共n兲 −ia 2
冉
ur共n兲 + iu共n兲 a冊
A共n兲 = − ia 2冉
− ur共n兲 + iu共n兲 a冊
A共n兲, 共67兲 iaei 2 共n − 1兲A共n兲 = iaei 2 A共n − 1兲. 共68兲From Eqs.共66兲and共68兲, we have that A共n兲=共n兲A共n+1兲. Hence, A共n兲=eu共n兲F共兲, where F共兲 is
a function of only. Substituting A共n兲=eu共n兲b共兲 into Eq.共67兲, we obtain
b+共− iaur共n兲 + u共n兲兲b = 0. 共69兲
Since the function b does not depend on n, we have that the coefficient of b in the above equation does not depend on n. We obtain the boundary condition共65兲. Solving Eq.共67兲and assuming that
the found boundary condition holds, we obtain the expression共64兲for A. 䊏
In Lemma 7, we show that the Lax representations corresponding to the Kelvin transformation
共33兲and the symmetry共34兲are equivalent.
Lemma 7: Let3共n兲 be a solution of Eq. (38), then on the boundary r = a, a function4共n兲
= A ·2共n兲, where
A = e2in+u共n兲 共70兲
is a solution of Eq.(40).
Proof: On the boundary r = a, we substitute4共n兲=A共n,, u , . . .兲·3共n兲 into Eq.共40兲and use
共36兲 for 3,共n兲. The resulting equation holds if the coefficients of 3共n+1兲, 3共n兲, and 3共n − 1兲 are zero. Thus, we obtain
iaei 2 A共n兲 = iae−i 2 共n兲A共n + 1兲, 共71兲 A共n兲 −ia 2
冉
− ur共n兲 + 4n a − iu共n兲 a冊
A共n兲 = − ia 2冉
− ur共n兲 + iu共n兲 a冊
A共n兲, 共72兲 iae−i 2 共n − 1兲A共n兲 = iaei 2 A共n − 1兲. 共73兲From Eqs.共66兲and共68兲, we have that A共n兲=e−2i共n兲A共n+1兲. Hence, A共n兲=e2in+u共n兲b共兲, where b共兲 is a function of only. Substituting A共n兲=e2in+u共n兲b共兲 into Eq.共67兲, we obtain
b= 0. 共74兲
Hence, the function b is a constant. This gives us the expression共64兲for A. 䊏
From the above lemmas, we have the following list of integrable boundary value problems with corresponding Lax pairs. Some of the integrable boundary value problems admit two differ-ent Lax pairs. We give both Lax pairs in the list.
The list of integrable boundary value problems for the two-dimensional Toda lattice and corresponding Lax pairs is as follows.
共1兲 r⬍ a, urr+ 1 rur+ 1 r2u=共n − 1兲 −共n兲, 1,r共n兲 = U11共n兲, 2,r共n兲 = U22共n兲, 1,共n兲 = V11共n兲, 2,共n兲 = V22共n兲, r = a, u共n兲 = 2in+ g共兲 + k共n兲, 2= e2in+g共兲1,
where the action of operator U1is given by共30兲, V1is given by共31兲, U˜2is given by共35兲, and V ˜ 2 is given by共36兲. 共2兲 r⬍ a, urr+ 1 rur+ 1 r2u=共n − 1兲 −共n兲, 1,r共n兲 = U11共n兲, 3,r共n兲 = U33共n兲, 1,共n兲 = V11共n兲, 3,共n兲 = V33共n兲, r = a, ur共n兲 = 2n a + g共兲, 3= eia兰g共兲d1,
where the action of operator U1is given by共30兲, V1is given by共31兲, U3is given by共37兲, and V3 is given by共38兲. 共3兲 r⬍ a, urr+ 1 rur+ 1 r2u=共n − 1兲 −共n兲, 1,r共n兲 = U11共n兲, 4,r共n兲 = U44共n兲, 1,共n兲 = V11共n兲, 4,共n兲 = V44共n兲, r = a, ur共n兲 = 2n a + g共兲, 4= e2in+u共n兲+ia兰g共兲d1,
where the action of operator U1is given by共30兲, V1is given by共31兲, U4is given by共39兲, and V4 is given by共40兲.
共4兲 r⬍ a, urr+ 1 rur+ 1 r2u=共n − 1兲 −共n兲, 2,r共n兲 = U22共n兲, 3,r共n兲 = U33共n兲, 2,共n兲 = V22共n兲, 3,共n兲 = V33共n兲, r = a, ur共n兲 = − i au共n兲 + g共兲, 3= e−2in+ia兰g共兲d2,
where the action of operator U2is given by共35兲, V2is given by共36兲, U3is given by共37兲, and V3 is given by共38兲. 共5兲 r⬍ a, urr+ 1 rur+ 1 r2u=共n − 1兲 −共n兲, 2,r共n兲 = U22共n兲, 4,r共n兲 = U44共n兲, 2,共n兲 = V22共n兲, 4,共n兲 = V44共n兲, r = a, ur共n兲 = − i au共n兲 + g共兲, 4= eu共n兲+兰g共兲d2,
where the action of operator U2is given by共35兲, V2is given by共36兲, U4is given by共39兲, and V4
is given by共40兲.
Remark 2: The above boundary value problems admit infinitely many solutions. To have a
unique solution, one can put additional conditions u共x,t,0兲= f共x,t兲 and u共x,t,1兲=g共x,t兲, where
f共x,t兲 is a smooth function compatible with the boundary condition when n=0 and g共x,t兲 is a
smooth function compatible with the boundary condition when n = 1.
IV. REDUCTIONS OF TWO-DIMENSIONAL TODA LATTICE EQUATION
In this section, we obtain integrable boundary conditions for the sinh-Gordon and Tzitzeica equations as reductions of integrable boundary conditions of the two-dimensional Toda lattice equation.
To reduce the two-dimensional Toda lattice equation to the sinh-Gordon equation, we put periodicity condition u共n兲=u共n+2兲 for all n, where u satisfies共29兲. Then, for p = u共0兲−u共1兲, we have prr+ 1 rpr+ 1 r2p= 4 sinh p, 共75兲
the sinh-Gordon equation in the polar coordinates. Only the boundary condition of the problem
pr+
i
ap= 0, 共76兲
on the boundary r = a. Evidently by changing p = iv, we get vrr+共1/r兲vr+共1/r2兲v= 4 sinv and
vr+共i/a兲v= 0.
To reduce the two-dimensional Toda lattice equation to the Tzitzeica equation, we put u共n兲 = u共n+3兲 and u共n兲=−u共2−n兲. Then, for q=u共0兲, we have
qrr+ 1 rqr+ 1 r2q= e 2q− e−q, 共77兲
the Tzitzeica equation in polar coordinates. Again, only the boundary condition of the problem
ur共n兲=−共i/a兲u共n兲+g共兲 is consistent with periodicity constraints u共n兲=u共n+3兲 and u共n兲=−u共2
− n兲. It gives
qr+
i
aq= 0, 共78兲
on the boundary r = a.
V. SOME SOLUTIONS OF THE BOUNDARY VALUE PROBLEM
In this section, we give an example of solutions for the special case of the boundary value problem urr+ 1 r2u+ 1 rur=共n − 1兲 −共n兲, 兩ur共n兲兩r=a= 2n a + g共兲, 共79兲
where共n兲=exp共u共n兲−u共n+1兲兲. We assume that g共兲=0 and look for a spherically symmetric
solution. That is, u is a function of r only. The boundary value problem共79兲reduces to
urr+
1
rur=共n − 1兲 −共n兲, 兩ur共n兲兩r=a=
2n
a . 共80兲
Let us introduce new variables t = ln共r/a兲 and v共n,t兲=u共n,r兲−2n ln r. Then, the boundary value
problem共80兲becomes
vtt=¯共n − 1兲 −¯共n兲, 兩vt共n兲兩t=0= 0, 共81兲
where¯共n兲=exp兵v共n兲−v共n+1兲其. As solutions of the above boundary value problem, we can take
even solitons of the Toda lattice equation in one dimension. Following Ref.19共see pp. 494–498兲,
the general N-soliton solution is given in terms of the data兵c,zj,␥j其 such that
共I兲 the quantities zjlie in the interval −1⬍zj⬍1 and are pairwise disjoint;
共II兲 e−c=兿Nj=1zj
2
;
共III兲 the quantities mj共0兲=␥j/ a˙共zj兲, where a共z兲=兿j=1 N
sgn zj共z−zj兲/共zzj− 1兲 and dot means
deriva-tive with respect to z, are posideriva-tive. The N-soliton solution is given by
v共n,t兲 = c + ln det M共n,t兲
det M共n − 1,t兲, 共82兲
where M共n,t兲 is a matrix with entries Mij共n,t兲=␦ij+
冑
mi共t兲mj共t兲共zizj兲n+1/共1−zizj兲 and mj共t兲= e−共zj−zj
−1
兲t␥
The even solitons are described by the following lemma.
Lemma 8: Let N = 2k and the data zj, ␥j, j = 1 , . . . , N satisfy zi= −zN−i+1, ␥i= −␥N−i+1, i
= 1 , . . . , k. Then, the N-soliton solution(82)is an even function of t.
Proof: With our choice of the initial data, the elements of matrix M共n,t兲, which are symmetric
with respect to the “center” of the matrix, are equal. If t is changed to −t, then every element of
M共n,t兲 is replaced by the element symmetric to it with respect the center of the matrix. Hence, the
determinant of M共n,t兲 is equal to the determinant of M共n,−t兲 and v共n,t兲=v共n,−t兲. From v共n,t兲
=v共n,−t兲, it follows that 兩v
⬘
共t兲兩t=0= 0. 䊏We give an example of the solutions described in the above lemma. For N = 2, we put z1
= z0, z2= −z0, c = −4 ln z0, and␥1= −␥0,␥2=␥0, where 0⬍z0⬍1 and␥0⬎0. Then, the data satisfy
conditions 共I兲, 共II兲, and 共III兲 and the conditions of the lemma. With such data, one has the following solution of共81兲: v共n,t兲 = c + ln1 +␥0共1 + z0 2兲z 0 2n+1cosh关共z 0− z0−1兲t兴 +␥02z04n+4 1 +␥0共1 + z02兲z02n−1cosh关共z0− z0−1兲t兴 +␥02z04n . 共83兲
Hence, the boundary value problem共80兲has the following solution:
u共n,r兲 = c + ln1 + 1 2␥0共1 + z0 2兲z 0 2n+1共r共z0−z0 −1兲 + r−共z0−z0 −1兲 兲 +␥0 2 z04n+4 1 +12␥0共1 + z02兲z02n−1共r共z0−z0 −1兲 + r−共z0−z0 −1兲 兲 +␥02z04n . 共84兲 VI. CONCLUSION
In the present paper, we apply the method of involutions to boundary value problems for soliton equations on bounded regions. As illustrative models, we consider the Neumann-type boundary value problem on a circle for the Liouville equation and initial boundary value problem for the two-dimensional Toda lattice equation. The Lax representations for the boundary value problems are represented. In the case of Liouville equation in a disk, we failed to find any effective approach to look for a regular solution satisfying the corresponding integrable boundary condition. For the Toda lattice in a cylinder, we have actually a mixed-type problem共time evolu-tion is given by the variable n兲. In our opinion, this mixed problem is well posed and can be solved by applying the inverse scattering transformation method. As it was shown in Ref. 18 for the Kadomtsev-Petviashvili共KP兲 equation on a strip, the Marchenko kernels of two equations con-nected by an involution are concon-nected by a very simple transformation. In this article, we consid-ered some reductions of the integrable boundary value problems in the case of the two-dimensional Toda lattice equation. We also constructed a class of solutions satisfying one of the found boundary conditions.
ACKNOWLEDGMENTS
The authors thank Scientific and Technological Research Council of Turkey and Turkish Academy of Science for Partial financial support.
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