Hydrodynamic type integrable equations on a segment
and a half-line
Metin Gürses,1,a兲Ismagil Habibullin,1,b兲 and Kostyantyn Zheltukhin2,c兲 1
Department of Mathematics, Faculty of Sciences, Bilkent University, 06800 Ankara, Turkey
2
Department of Mathematics, Faculty of Sciences, Middle East Technical University, 06531 Ankara, Turkey
共Received 4 June 2008; accepted 10 September 2008; published online 20 October 2008兲
The concept of integrable boundary conditions is applied to hydrodynamic type systems. Examples of such boundary conditions for dispersionless Toda systems are obtained. The close relation of integrable boundary conditions with integrable re-ductions in multifield systems is observed. The problem of consistency of boundary conditions with the Hamiltonian formulation is discussed. Examples of Hamil-tonian integrable hydrodynamic type systems on a segment and a semiline are presented. © 2008 American Institute of Physics.关DOI:10.1063/1.2993008兴
I. INTRODUCTION
The theory of the integrable hydrodynamic type of systems ut i =vj i共u兲u x j , i, j = 1,2, . . . N 共1兲
was initiated by Dubrovin and Novikov1and Tsarev.2Here in Eq.共1兲summation over the repeated indices is assumed and u is an N-component column vector of the form u =共u1, u2, . . . , uN兲t
. Such systems have a variety of applications in gas dynamics, fluid mechanics,3–6 chemical kinetics, Whitham averaging procedure,7–10differential geometry, and topological field theory. We refer to Refs.11and12for further discussions and references.
In the present article, a problem of finding boundary conditions for hydrodynamic type equa-tions consistent with the integrability property is studied for a special case of the system共1兲called dispersionless Toda lattices.13–15Actually we assume that Eq.共1兲admits a Lax representation on the algebra
A =
再
兺
−⬁ ⬁
ui共x兲pi:ui decay sufficiently rapidly as x→ ⫾ ⬁
冎
共2兲 with the following Poisson bracket:兵f,g其 = p
冉
f p g x− f x g p冊
, f,g苸 A.Such equations, for example, appear in the fluid mechanics as reductions in Benney moment equations.3–12
Our definition of consistency of boundary conditions with the integrability共see Refs.16–18兲
is based on the notion of symmetries. A constraint of the form
a兲Electronic mail: gurses@fen.bilkent.edu.tr.
b兲Electronic addresses: habibullin_i@mail.rb.ru and habib@fen.bilkent.edu.tr. On leave from Ufa Institute of Mathematics, Russian Academy of Science, Chernyshevskii Str. 112, Ufa 450077, Russia.
c兲Electronic mail: zheltukh@metu.edu.tr.
49, 102704-1
f共t,u,u关1兴, . . . ,u关k兴兲兩x=x0= 0, 共3兲 where u关j兴=j/xju imposed at some point x0is called a boundary condition at this point.
Bound-ary value problems共1兲 and共3兲关or simply boundary condition共3兲兴 are called consistent with the symmetry
ui=i共t,x,u,u关1兴, . . . ,u关m兴兲 共4兲 if 共1兲 and 共4兲 are compatible under the constraint 共3兲. More precisely, we mean the following: differentiation of共3兲with respect toyields
兺
fu关n兴i 共u关n兴 i 兲
= 0, 共5兲
where-derivatives are replaced by means of Eq.共4兲.
Definition 1: Boundary value problems(1)and(3)are consistent with the symmetry(4)if(5) holds identical by means of(3) and its differential consequences obtained by differentiating with respect to t.
Note that since constraint共3兲is valid only for x = x0, it cannot be differentiated with respect to x. For this reason, it is convenient to exclude the x-derivatives of dependent variable u from our scheme. By solving Eq. 共1兲 for uxi one gets ux=v−1ut, where v−1 is the matrix inverse to vj
i共u兲. Similarly, uxx=共v−1ut兲x=共v−1兲xut+v−1共v−1兲tut+v−2uttis expressed through u , ut, uttand so on. As a result one can rewrite boundary condition共3兲and symmetry 共4兲taken at point x0 as
f1共t,u,ut, . . .兲 = 0 共6兲
and
u=1共t,x0,u,ut, . . .兲. 共7兲 Now the consistency requirement can be reformulated as follows. Boundary condition 共3兲 is consistent with共4兲 if differential constraint 共6兲 is consistent with the associated-dynamics共7兲. We call the boundary condition consistent with integrability if it is consistent with an infinite dimensional subspace of symmetries. Hydrodynamic type system given in 共1兲 defines an N-dimensional dynamical system and the boundary condition 共6兲 defines a hypersurface in N-dimensional space of functions. Thus, due to the remark above, an integrable boundary condi-tion is closely connected to reduccondi-tions in the associated system共7兲compatible with integrability.15 Below we use this important observation in order to find symmetry consistent boundary condi-tions.
Boundary conditions that passed the symmetry test are then tested for consistency with the conserved quantities, Hamiltonian structures, and the complete integrability property of system
共1兲.19–22
It is remarkable that some of the boundary conditions also satisfy these additional require-ments and thus allow one to reduce 共1兲 to a completely integrable Hamiltonian system on a segment and a half-line. For more information see Ref.23and references therein.
This paper is organized as follows. In Sec. II some integrable boundary conditions for Toda system are derived and it is shown that these boundary conditions are compatible with infinite number of symmetries. The relation between the integrable reductions in N-system and the inte-grable boundary conditions is considered in Sec. III. It is observed that some inteinte-grable boundary conditions lead to integrable reductions. In Sec. IV we discuss the compatibility of the integrable boundary conditions found in the previous sections with the Hamiltonian formulation. We show that some boundary conditions are indeed compatible with the Hamiltonian formulation and also with an infinite class of symmetries. In all sections up to Sec. IV only N = 2 systems are consid-ered. In Sec. V we study N = 3 systems which give other examples of hydrodynamic type
equa-tions. For this case, some integrable boundary conditions compatible with infinite number of symmetries and boundary conditions compatible with the Hamiltonian formulation are found. II. INTEGRABLE BOUNDARY CONDITIONS FOR THE TODA SYSTEM
In this section we study the well known example of integrable model24 St= Px,
共8兲 Pt= PSx
called Toda system, admitting the Lax representation on the algebra of Lourent series共2兲
Lt=兵共L兲ⱖ0,L其, 共9兲
where
L = p + S + Pp−1. 共10兲
The corresponding hierarchy of symmetries of the Toda system共8兲 is
Ltn=兵L,共Ln兲ⱖ0其. 共11兲
Recursion operator corresponding to the above hierarchy is共for calculation of recursion operator see Refs.15and25兲
R=
冉
S 2 + PxDx−1· P−1
2P S + SxPDx
−1
· P−1
冊
. 共12兲In some cases, it is convenient to consider the Toda system共8兲in other variables. We write the Lax function共10兲as L = p−1共p+u兲共p+v兲, that is,
S = u +v, P = uv. 共13兲
Then the Toda system共9兲gives
ut= uvx,
共14兲 vt=vux.
Let us find boundary conditions compatible with an infinite number of symmetries from the hierarchy共11兲. As a boundary, we take x = 0. First we find boundary conditions compatible with the first symmetry of the hierarchy共11兲. Assume that the boundary condition depends on P and S and can be solved with respect to S. So the boundary condition can be written as
S = F共P兲, x = 0. 共15兲
Lemma 1: On the boundary x = 0, the boundary condition of the form (15)compatible with the first symmetry of the hierarchy(11)
St1= 2SPx+ 2PSx, 共16兲 Pt1= 2PPx+ 2SPSx is given by P =共S + c兲 2 4 , x = 0. 共17兲
Proof: The boundary condition共15兲is compatible with the symmetry共16兲if on the boundary x = 0
St1= F
⬘
共P兲Pt1 共18兲for all solutions of Eq.共8兲. Let us find functions F for which the above equality holds. We rewrite the symmetry共16兲in terms of variables S, P, and their t derivatives using Eq.共8兲
St1= 2SSt+ 2Pt,
共19兲 Pt1= 2PSt+ 2SPt.
Then we substitute St1 and Pt1into共18兲, so
2SSt+ 2Pt= F
⬘
共P兲共2PSt+ 2SPt兲. 共20兲 From共15兲it follows that St= F⬘
共P兲Pt, so2SF
⬘
共P兲Pt+ 2Pt= F⬘
共P兲共2PF⬘
共P兲Pt+ 2SPt兲. 共21兲 Hence,F
⬘
2共P兲 = 1P. 共22兲
The above equation has a solution共17兲. 䊐
It is convenient to write the boundary condition 共17兲as
P =S
2
4, x = 0. 共23兲
By shifting S, the Toda system共8兲is invariant with respect to such shift.
Lemma 2: All the symmetries of the hierarchy(11)are compatible with the boundary condi-tion(23).
Proof: The boundary condition 共23兲is compatible with an evolution symmetry
冉
S P冊
=冉
冊
共24兲if=12Sfor P =14S2. That is, under the constraint共23兲the symmetry共24兲should take the form
冉
S 1 2SS冊
=冉
1 2S冊
. 共25兲Evidently, the first symmetry of the hierarchy共11兲has such a form. Let us show that the recursion operator共12兲preserves the property共25兲. On the boundary x = 0, we rewrite the recursion operator
共12兲in terms of t derivatives using the Toda system共8兲as follows:
R=
冉
S + StDt−1
2 2P + PtDt−1 S
冊
. 共26兲
Applying the recursion operator共26兲into a symmetry共25兲, we obtained a symmetry
冉
S P冊
˜=冉
˜ 1 2S˜冊
. 共27兲 䊐We also have the following boundary condition compatible with the hierarchy 共11兲. Lemma 3: On the boundary x = 0, the boundary condition
P = 0 共28兲
is compatible with all symmetries of the hierarchy(11). The above lemma is proved in the same way as Lemma 2.
Another boundary condition comes from consideration of odd and even solutions of the Toda system共8兲. This boundary condition is not compatible with all symmetries of the hierarchy共11兲 but only with even ones.
Lemma 4: On the boundary x = 0, the boundary condition
S = 0 共29兲
is compatible with all even numbered symmetries of the hierarchy(11).
The above lemma is proved in the same way as Lemma 2 using the square of the recursion operator共26兲.
III. INTEGRABLE REDUCTIONS
Let us consider other equations admitting a Lax representation on the algebra共2兲. For a Lax function L = p−1共p−u
N兲共p−uN−1兲¯共p−u1兲, where N⬎2, we consider the Lax equation,
Lt=兵L,共L兲ⱖ0其 共30兲
and an infinite hierarchy of symmetries
Ltn=兵L,共Ln兲ⱖ0其, n = 1,2, ... . 共31兲 For such equations we cannot directly find boundary conditions compatible with symmetries共see Sec. V兲. So we use integrable reductions.15
Definition 2: A reduction in an integrable equation is called integrable if reduced equation is also integrable. That is, the reduced equation admits an infinite hierarchy of symmetries.
In Ref.15it was shown that the following reductions uN= uN−1= ¯ = ui= 0, iⱖ 2,
uN= uN−1= ¯ = uj, jⱖ 1 共32兲
of the above equations are integrable. We note that for these reductions the symmetries of the reduced equation are obtained by the reduction in the symmetries of the original system.
If we have an integrable reduction such that symmetries of the reduced system are obtained by the reduction in the symmetries of the original system, then the reduction can be taken as inte-grable boundary conditions. Indeed, the original system is invariant under the hierarchy of sym-metries and the reduced system is invariant under the symsym-metries. Since reduction can be recov-ered from the original system and the reduced system, it is also invariant under the symmetries. So, taking the reductions 共32兲 as boundary conditions we obtain symmetry invariant boundary conditions.
Theorem 1: For a system (30) the boundary condition 共uN= uN−1=¯ =ui兲兩x=a= 0 or 共uN = uN−1=¯ =uj兲兩x=a (taking x = a as the boundary) is integrable.
Let us take boundary conditions obtained in Sec. II. The condition 共P=S2/4兲兩
x=0 in u ,v variables共13兲is共u−v兲兩x=0= 0. It corresponds to a reduction u =v. The condition P兩x=0= 0 in u ,v variables is共uv兲兩x=0= 0. It corresponds to a reduction u = 0共or v=0兲. The condition S兩x=0= 0 in u ,v variables is共u+v兲兩x=0= 0. It does not correspond to reductions considered above.
Remark: If we take a reduction mentioned above as a boundary condition, then we can consider the corresponding reduced system. Solutions of the reduced system obviously satisfy the
main system equations and the boundary condition. For Toda system, the reduction P = 0 leads to the equation
St= 0. 共33兲
Its solution S = f共x兲, for any differentiable function f, gives the solution of Toda system 共8兲 satisfying the corresponding boundary condition 共28兲. The reduction P = S2/4 leads to the Hopf
equation,
St=
1
2SSx. 共34兲
Its solution S = h共2x+tS兲 gives the solution of Toda system共8兲satisfying the corresponding bound-ary condition. Here h is any differentiable function of x and t. To find a solution of N-system satisfying the integrable boundary condition, the method described above is very effective. We take the corresponding reduction and the corresponding reduced共N−1兲 system. Solving the re-duced system gives automatically the solution of the N-system, satisfying the integrable boundary condition.
IV. HAMILTONIAN REPRESENTATION OF THE INTEGRABLE BOUNDARY VALUE PROBLEMS
To obtain the Hamiltonian formulation of the Toda system 共8兲, we use its Lax representation on the algebra共2兲.
We define, for the algebra of Lourent series 共2兲, a trace functional
tr f =
冕
−⬁ ⬁ u0dx, f苸 A, f =兺
−⬁ ⬁ ui共x兲pi 共35兲and a nondegenerate ad-invariant pairing
共f,g兲 = tr共f · g兲, f,g 苸 A. 共36兲
Thus we have a Poisson algebra with a commutative multiplication and unity, the multiplication satisfies the derivation property with respect to the Poisson bracket, and the algebra is equipped with a nondegenerate ad-invariant pairing. Following Ref.14we can define an infinite family of Poisson structures for smooth functions on the algebra A. A function F on A is smooth if there is a map dF : A→A such that
F
⬘
兩t=0共f + tg兲 = 共dF共f兲,g兲, f,g 苸 A. The following theorem共Ref. 14see also Refs.26and27兲 holds.Theorem 2: Let A be a Poisson algebra with unity, commutative multiplication, Poisson bracket {.,.}, and nondegenerate, ad-invariant pairing (.,.). Assume that the multiplication satisfies the derivation property with respect to the Poisson bracket and is symmetric with respect to the pairing共fg,h兲=共f ,gh兲. If R:A→A is a classical r-matrix, then for smooth functions F and G on A:
共a兲
兵F,G其共n兲共f兲 = 共f,兵R共fn+1dF共f兲兲,dG共f兲其兲 + 共f,兵dF共f兲,R共fn+1dG共f兲兲其兲 共37兲
defines a Poisson structure for each integer nⱖ−1.
共b兲 The structures 兵.,.其共n兲 are compatible with each other (their sum is again a Poisson
structure).
关f,g兴 =1
2共兵Rf,g其 + 兵f,Rg其兲
is a Lie bracket.
To apply the above theorem, we take an r-matrix
R =12共Pⱖ0− Pⱕ−1兲, 共38兲
where Pⱖ0 and Pⱕ−1 are projectors on Poisson subalgebras
Aⱖ0=
再
u =兺
0 ⬁ uipi:u苸 A冎
and Aⱕ0=再
u =兺
−⬁ −1 uipi:u苸 A冎
, respectively. Note that the Lax equation共9兲isLt=兵R共L兲,L其, 共39兲
where L = p + S + Pp−1.
Using the Poisson structures given by Theorem 2 we obtain bi-Hamiltonian formulation of the Toda lattice.
The submanifold M =兵L苸A:L=p+S+ Pp−1其 is a Poisson submanifold for the Poisson
struc-ture共37兲with n = −1. Restricting this structure on M we obtain the following Hamiltonian opera-tor:
D−1=
冉
0 PDx+ PxPDx 0
冊
. 共40兲
We have first Hamiltonian formulation for共8兲
冉
S P冊
t = D−1冉
␦H−1/␦S ␦H−1/␦P冊
, 共41兲 where H−1= 1 2tr L 2, that is, H −1= 1 2冕
−⬁ ⬁ 共S2+ 2P兲dx. 共42兲The second Hamiltonian operator can be obtained by restricting the Poisson structure 共37兲with n = 0 on the submanifold M or by application of the recursion operator共12兲 to the Hamiltonian operator共40兲. The second Hamiltonian operator is
D0=
冉
2PDx+ Px SPDx+ SPx SPDx+ SxP P2Dx+ PPx冊
. 共43兲
The corresponding Hamiltonian functional is
H0= tr L, that is, H0=
冕
−⬁⬁
Sdx. 共44兲
Since Hamiltonian operators D−1and D0are compatible, we have a bi-Hamiltonian representation of Eq.共8兲.
In u ,v variables共13兲the Hamiltonian operators and functionals take form
B−1= uv 共u − v兲2
冉
− 2u u +v
+ 1 共u − v兲3
冉
2uv2ux− u3vx− u2vvx u2vvx+ u3vx− 2uv2ux 2uv2vx− uv2ux−v3ux v3ux+ uv2ux− 2u2vvx冊
共46兲 and G−1=冕
−⬁ ⬁ 共u2+v2+ 4uv兲dx, 共47兲 B0=冉
0 uvx+ uvDx vux+ uvDx 0冊
, 共48兲 and G0=冕
−⬁ ⬁ 共u + v兲dx. 共49兲A different approach was used in Ref.13 to obtain the Hamiltonian operator B0共see also
refer-ences in Ref.13兲. The explicit expressions of an infinite number of conservation laws for the Toda
system共14兲was given in Ref. 13,
Qn,t= Fn,x, n = 1,2 . . . , 共50兲 where Qn=
兺
j=0 n冉
n j冊
2 ujvn−j, n = 1,2,3. . . 共51兲 and Fn=兺
j=0 n n − j j + 1冉
n j冊
2 uj+1vn−j, n = 1,2,3 . . . . 共52兲The conserved quantitiesQn=兰−⬁⬁ Qndx are in involution with respect to the Hamiltonian operators B−1 and B0. One can easily check if the boundary conditions preserve the conserved quantities.
Lemma 5: For the Toda system (14)with the boundary condition
共a兲 共u−v兲兩x=0= 0共共P=S2/4兲兩x=0兲 the above conservation laws are not preserved; 共b兲 uv兩x=0= 0共P兩x=0= 0兲 the quantities
冕
0 ⬁Qndx, n = 1,2,3. . . 共53兲
are conserved; and
共c兲 共u+v兲兩x=0= 0共S兩x=0= 0兲 the quantities
冕
0 ⬁Qndx, n = 2,4,6. . . 共54兲
are conserved.
We can use the above Hamiltonian operators to obtain the Hamiltonian representation of some of the boundary value problems.
uv兩x=0= 0 and uv兩x=1= 0 共55兲 admits the bi-Hamiltonian representation with Hamiltonian operators B共n兲, n = −1 , 0, and Hamil-tonians G−1=
冕
0 1 共u2+v2+ 4uv兲dx =冕
−⬁ ⬁ 共u2+v2+ 4uv兲共x兲共1 − x兲dx 共56兲 and G0=冕
0 1 共u + v兲dx =冕
−⬁ ⬁ 共u + v兲共x兲共1 − x兲dx, 共57兲respectively, where共x兲 is the Heaviside step function. Proof: The Hamiltonian equations
冉
u v冊
t = Bn冉
␦Gn/␦u ␦Gn/␦v冊
, n = − 1,0 共58兲 are for n = −1 ut= uvx− uv u −v共␦共x兲 −␦共1 − x兲兲, 共59兲 vt=vux+ uv u −v共␦共x兲 −␦共1 − x兲兲 and for n = 0 ut= uvx+ uv共␦共x兲 −␦共1 − x兲兲, 共60兲 vt=vux+ uv共␦共x兲 −␦共1 − x兲兲,where x苸关0,1兴. Under the boundary conditions uv兩x=0= 0 and uv兩x=1= 0 we have the Toda system
共14兲on关0,1兴. Note that the Poisson brackets are given by
兵K,N其 =
冕
−⬁ ⬁冉
␦K/␦u ␦K/␦v冊
B共n兲冉
␦N/␦u ␦N/␦v冊
, 共61兲 where n = −1 , 0. 䊐V. INTEGRABLE BOUNDARY CONDITIONS FOR THE THREE FIELD SYSTEMS
Let us consider a three field hydrodynamic type system on the algebra 共2兲. We take a Lax function
L = p2+ Sp + P + Qp−1. 共62兲
We can construct two integrable hierarchies with this Lax function. The first hierarchy is given by
Lt=兵共Ln+1/2兲ⱖ0,L其, n = 0,1,2, ... , 共63兲 the first equation of the hierarchy is
St= Px−
1 2SSx,
Pt= Qx, 共64兲
Qt=
1 2QSx. The second hierarchy is given by
Lt=兵共Ln兲ⱖ0,L其, n = 1,2,3 ... , 共65兲 the first equation of the hierarchy is
St= 2Qx,
Pt= SQx+ QSx, 共66兲
Qt= QPx.
We also have a recursion operator15 of the hierarchies共63兲and共65兲,
冢
P −14S2+共
12Px− 1 4SSx兲
Dx−1 1 2S 3 + 2QxDx−1Q−1 3 2Q + 1 2QxDx −1 P 2S +共SQ兲xDx −1 Q−1 1 4SQ + 1 4SxQDx−1 3 2Q P + QPxDx −1 Q−1冣
. 共67兲The bi-Hamiltonian representation of Eqs.共64兲and共66兲is obtained by restricting the Poisson structure 共37兲with n = −1 and n = 0 on the submanifold M =兵L苸A:L=p2+ Sp + P + Qp−1其. So we
have Hamiltonian operators
C−1=
冢
2Dx 0 0 0 0 QDx+ Qx 0 QDx 0冣
共68兲 and C0=冢
共
2P −12S2兲
Dx+ Px− 1 2SSx 3QDx+ 2Qx 1 2QDx+ 1 2SQx QDx+ Qx 2SQDx+ SQx+ QSx PQDx+ PDx 1 2SQDx+ 1 2QSx PDx+ PxQ 3 2Q 2D x+ 3 2QQx冣
. 共69兲Equation共64兲can be written as
冢
S P Q冣
t = C−1冢
␦H−1/␦S ␦H−1/␦P ␦H−1/␦Q冣
= C0冢
␦H0/␦S ␦H0/␦P ␦H0/␦Q冣
, 共70兲 where H−1= 2 3tr L 3/2, that is, H −1=冕
−⬁ ⬁冉
Q +1 2SP − 1 24S 3冊
dx 共71兲 and H0= 2 tr L1/2, that is, H0=冕
−⬁ ⬁ Sdx. 共72兲冢
S P Q冣
t = C−1冢
␦H˜ −1/␦S ␦H˜ −1/␦P ␦H˜ −1/␦Q冣
= C0冢
␦H˜ 0/␦S ␦H˜ 0/␦P ␦H˜ 0/␦Q冣
, 共73兲 where H˜ −1= 1 2tr L 2, that is, H˜ −1=冕
−⬁ ⬁冉
SQ +1 2P 2冊
dx 共74兲 and H˜ 0= tr L, that is, H˜0=冕
−⬁ ⬁ Pdx. 共75兲We can give both hierarchies in modified variables, writing the Lax function 共62兲 as L = p−1共p−u兲共p−v兲共p−w兲, that is,
S = u +v + w,
P = uv + uw + vw, 共76兲
Q = uvw.
It is quite difficult to find integrable boundary condition directly for three field systems. For example, consider hierarchy共63兲. In the following lemmas, we use P , Q , R variables since sym-metries and recursion operator have a simple form in these variables.
Lemma 6: Let x = 0 be the boundary. The boundary conditions of the form P = F共S兲 and Q = G共S兲 are compatible with the first symmetry of the hierarchy(63)if the functions F and G satisfy the following differential equations:
3 2S共F
⬘
兲 2+ 3F⬘
G⬘
−3 4F⬘
S 2− 3G⬘
S −3 2G = 0, 共77兲 3 2SF⬘
G⬘
+ 3共G⬘
兲 2−3 2F⬘
G − 3 4G⬘
S 2−3 4SG = 0. 共78兲Proof: The first symmetry of the hierarchy共63兲is St1= 3 2
共
P − 1 4S2兲共
Px− 1 2SSx兲
+ 3 2SQx+ 3 2SxQ, Pt1= 3 2PQx+ 3 2PxQ 3 4QSSx+ 3 8S 2Q x, 共79兲 Qt1= 1 4SQ共
Px− 1 2SSx兲
+ 1 4Q共
P − 1 4S 2兲
+3 2QQx+ 1 2QPSx+ 1 2QSPx.Differentiating the boundary conditions P = F共S兲 and Q=G共S兲 with respect to the above symmetry and expressing all the x derivatives in terms of t derivatives using Eq.共64兲, we obtain Eqs.共77兲
and共78兲. 䊐
Lemma 7: Let x = 0 be the boundary. The boundary condition of the form S = F共P,Q兲 is compatible with the first symmetry of the hierarchy (63) if function F satisfies the following differential equations: 3 2
共
P − 1 4F 2兲
F P+ 3 2F = 3 2PFP+ 3 2QFP 2 +38QFP 2 +38F2FP+ 1 4QFFQ+ 3 2QFQ+ 1 2FFPFQ, 共80兲3 2
共
P − 1 4F 2兲
F Q+ 3 2= 3 2QFPFQ+ 3 2FFP+ 1 2共
P − 1 4S 2兲
F Q+ PFQ+ 1 2QFFQ2+ 1 2F 2F Q. 共81兲 Proof: We differentiate the boundary condition S = F共P,Q兲 with respect to the symmetry共63兲 and express all the x derivatives in terms of t derivatives using Eq. 共64兲. Then separating termscontaining Ptand Qt, we obtain Eqs.共80兲and共81兲. 䊐
The differential equations obtained in the above lemmas are nonlinear partial differential equations which are rather complicated. So, to obtain integrable boundary conditions it is easy to use integrable reductions discussed in Sec. III. Let x = 0 be a boundary.
共a兲 Integrable reduction u=v gives integrable boundary condition u兩x=0=v兩x=0 or 共S3Q − S2P2 + 4Q3− 18SPQ + 27Q2兲兩x=0= 0共condition on coefficients of cubic equation to have two equal roots兲 in S, P,Q variables.
共b兲 Integrable reduction u=v=w gives integrable boundary conditions u兩x=0=v兩x=0= w兩x=0 or P兩x=0=
1
3S2兩x=0, Q兩x=0=
1
27S3兩x=0共condition on coefficients of cubic equation to have all roots equal兲.
共c兲 Integrable reduction u=0 gives integrable boundary condition u兩x=0= 0 or Q兩x=0= 0. 共d兲 Integrable reduction u=0, v=0 gives integrable boundary conditions u兩x=0= 0, v兩x=0= 0 or
P兩x=0= 0, Q兩x=0= 0.
To obtain boundary value problems that admit bi-Hamiltonian representation we modify Hamiltonian functions, as in the case of Toda system. We use S , P , Q variables, the Hamiltonian operators have simpler form in this variables.
For Eq.共64兲we have the following.
Theorem 4: Equation (64)on a segment 关0,1兴 with boundary conditions
冉
P −1 4S 2冊
兩 x=0= 0, Q兩x=0= 0 and冉
P − 1 4S 2冊
兩 x=1= 0, Q兩x=1= 0 共82兲 admits the bi-Hamiltonian representation with Hamiltonian operators(68)and(69), and Hamil-tonians Hញ−1=冕
−⬁ ⬁冉
Q +1 2SP − 1 24S 3冊
共x兲共1 − x兲dx 共83兲 and Hញ0=冕
−⬁ ⬁ S共x兲共1 − x兲dx, 共84兲respectively, where共x兲 is the Heaviside step function. Proof: The Hamiltonian equations
冢
S P Q冣
t = Cn冢
␦Hញn/␦S ␦Hញn/␦P ␦Hញn/␦Q冣
, n = − 1,0 共85兲 are for n = −1St= Px− 1 2SSx+
共
P − 1 4S 2兲
共␦共x兲 −␦共1 − x兲兲, Pt= Qx+ 1 2SQ共␦共x兲 −␦共1 − x兲兲, 共86兲 Qt= 1 2QSx+ Q共␦共x兲 −␦共1 − x兲兲 and for n = 0 St= Px− 1 2SSx+共
2P − 1 2S 2兲
共␦共x兲 −␦共1 − x兲兲, Pt= Qx+ Q共␦共x兲 −␦共1 − x兲兲, 共87兲 Qt= 1 2QSx+ 1 2SQ共␦共x兲 −␦共1 − x兲兲,where x苸关0,1兴. Under the boundary conditions共82兲we have Eq.共64兲on关0,1兴. 䊐 The boundary conditions共82兲are symmetry integrable.
Lemma 8: All the symmetries of the hierarchy(63)are compatible with the boundary condi-tion(82).
Proof: The boundary condition 共82兲is compatible with an evolution symmetry
冢
S P Q冣
=冢
冣
共88兲if=21Sand= 0 for P =14S2and Q = 0 on the boundary x = 0. That is, under the conditions共82兲
the symmetry共88兲should take the form
冢
S 1 2SS 0冣
=冢
1 2S 0冣
. 共89兲One can check that the first symmetry of the hierarchy共63兲has such a form. Let us show that the recursion operator共67兲preserves the form共89兲. On the boundary x = 0, we rewrite the recursion operator共67兲in terms of t derivatives using Eq.共64兲as follows:
冢
P −14S2− 1 4StDt−1S 1 2S + 1 2StDt−1 3 + PtDt−1 3 2Q − 1 4PtDt −1 S P +12PtDt −1 2S +共
12SPt+ Qt兲
Dt −1 1 4SQ − 1 4QtDt −1 S 3 2Q + 1 2QtDt −1 P +12QPtDt −1冣
. 共90兲Applying the recursion operator共90兲to a symmetry共89兲we obtained a symmetry
冢
S˜ 1 2SS˜ 0冣
=冢
˜ 1 2S˜ 0冣
. 共91兲 䊐 For Eq.共66兲we have the following.Theorem 5: Equation (66)on a segment 关0,1兴 with boundary conditions
admits the bi-Hamiltonian representation with Hamiltonian operators(68)and(69), and Hamil-tonians H˜˜ −1=
冕
−⬁ ⬁冉
SQ +1 2P 2冊
共x兲共1 − x兲dx 共93兲 and H˜˜ 0=冕
−⬁ ⬁ P共x兲共1 − x兲dx, 共94兲respectively, where共x兲 is the Heaviside step function. Proof: The Hamiltonian equations
冢
S P Q冣
t = Cn冢
␦H˜˜ n/␦S ␦H˜˜ n/␦P ␦H˜˜ n/␦Q冣
, n = − 1,0 共95兲 are for n = −1 St= 2Qx+ 2Q共␦共x兲 −␦共1 − x兲兲, Pt= SQx+ QSx+ PQ共␦共x兲 −␦共1 − x兲兲, 共96兲 Qt= QPx+ SQ共␦共x兲 −␦共1 − x兲兲 and for n = 0 St= 2Qx+ Q共␦共x兲 −␦共1 − x兲兲, Pt= SQx+ QSx+ 2SQ共␦共x兲 −␦共1 − x兲兲, 共97兲 Qt= QPx+ PQ共␦共x兲 −␦共1 − x兲兲,where x苸关0,1兴. Under the boundary conditions共92兲we have Eq.共66兲on关0,1兴. 䊐 In the same way as in Lemma 8, one can show that the boundary condition共92兲is symmetry integrable. This case is similar to the case of Toda system 共the boundary condition Q兩⌫= 0 in modified variables is uvw兩⌫= 0兲.
VI. CONCLUSION
In this article we studied the problem of integrable boundary conditions for hydrodynamic type integrable systems. To our knowledge, the problem has never been discussed in literature before. Since the term integrability has various meanings, the notion of integrable boundary conditions has also several definitions. As basic ones we take three definitions, namely, consis-tency with infinite set of symmetries, consisconsis-tency with infinite set of conserved quantities, and consistency with the Hamiltonian integrability共or bi-Hamiltonian structure兲. Comparison of these three kinds of integrable boundary conditions shows that the consistency with the bi-Hamiltonian structure is a very severe restriction. Only very special kind of boundary conditions passes this test. The class of symmetry consistent boundary conditions seems to be relatively larger. As an example we studied the dispersionless Toda system. We found all symmetry compatible boundary conditions of this system and showed that only a subclass of these boundary conditions is
com-patible with the Hamiltonian formulation of the system. We pointed out that the integrable reduc-tions in the N-system of hydrodynamical type of equareduc-tions are directly related to the integrable boundary conditions of the same systems. Using this property, a method for constructing exact solutions satisfying the integrable boundary conditions is given. We considered also an N = 3 system. Integrable boundary conditions compatible with symmetries and compatible with the Hamiltonian formulation of this system were found.
ACKNOWLEDGMENTS
This work was partially supported by the Scientific and Technological Research Council of Turkey共TUBITAK兲 and Turkish Academy of Sciences 共TUBA兲. One of the authors 共I.H.兲 also thanks Russian Foundation for Basic Research under Grant No. 06-01-92051 KE-a.
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