1001.0020.txt

arXiv:1001.0020v2  [nlin.SI]  3 Mar 2010Classification of integrable hydrodynamic chains
A.V. Odesskii1,2, V.V. Sokolov1
1L.D. Landau Institute for Theoretical Physics (Russia)
2Brock University (Canada)
Abstract Using the method of hydrodynamic reductions, we find all inte-
grable infinite (1+1)-dimensional hydrodynamic-type chains of shif t one. A
class of integrable infinite (2+1)-dimensional hydrodynamic-type c hains is
constructed.
MSC numbers: 17B80, 17B63, 32L81, 14H70
Address : L.D. Landau Institute for Theoretical Physics of Russian Academ y of Sciences,
Kosygina 2, 119334, Moscow, Russia
E-mail: aodesski@brocku.ca, sokolov@itp.ac.ru
1Contents
1 Introduction 3
2 Integrable chains and hydrodynamic reductions 4
3 GT-systems 5
4 Canonical forms of GT-systems associated
with integrable chains 7
5 Generic case 12
6 Trivial GT-system and 2+1-dimensional integrable hydrodynamic chains 14
7 Infinitesimal symmetries of triangular GT-systems 17
21 Introduction
We consider integrable infinite quasilinear chains of the form
uα,t=φα,1u1,x+···+φα,α+1uα+1,x, α= 1,2,..., φ α,α+1/negationslash= 0, (1.1)
whereφα,j=φα,j(u1,...,uα+1).Two chains are called equivalent if they are related by a trans-
formation of the form
uα→Ψα(u1,...,uα),∂Ψα
∂uα/negationslash= 0, α= 1,2,... (1.2)
By integrability we mean the existence of an infinite set of hydrodyna mic reductions [1, 2,
3, 4, 5, 6].
Example 1. The Benney equations [7, 8, 9]
u1,t=u2,x, u 2,t=u1u1,x+u3,x,... u αt= (α−1)uα−1u1,x+uα+1,x,... (1.3)
provide the most known example of integrable chain (1.1). The hydro dynamic reductions for
the Benney chain were investigated in [10]. /square
In [4, 5, 6] integrable divergent chains of the form
u1t=F1(u1,u2)x, u2t=F2(u1,u2,u3)x,···, uit=Fi(u1,u2,...,ui+1)x,··· (1.4)
were considered. In [6] some necessary integrability conditions we re obtained. Namely, a non-
linear overdetermined system of PDEs for functions F1,F2was presented. The general solution
of the system was not found. Another open problem was to prove t hat the conditions are
sufficient. In other words, for any solution F1,F2of the system one should find functions
Fi,i>2 such that the resulting chain is integrable.
Probably any integrable chain (1.1) is equivalent to a divergent chain. However, the diver-
gent coordinates are not suitable for explicit formulas. Our main obs ervation is that a conve-
nient coordinates are those, in which the so-called Gibbons-Tsarev type system (GT-system)
related to integrable chain is in a canonical form.
Using our version (see [11, 12]) of the hydrodynamic reduction meth od, we describe all
integrable chains (1.1). We establish an one-to-one corresponden ce between integrable chains
(1.1) and infinite triangular GT-systems of the form
∂ipj=P(pi,pj)
pi−pj∂iu1, i/negationslash=j, (1.5)
∂i∂ju1=Q(pi,pj)
(pi−pj)2∂iu1∂ju1, i/negationslash=j, (1.6)
∂ium= (gm,0+gm,1pi+···+gm,m−1pm−1
i)∂iu1, g m,j=gm,j(u1,...,um), gm,m−1/negationslash= 0,
3wherem= 2,3,...andi,j= 1,2,3.The functions P,Qare polynomials quadratic in each of
variablespiandpj,with coefficients being functions of u1,u2.The functions p1,p2,p3,u1,u2,...
in (3.11) depend on r1,r2,r3,and∂i=∂
∂ri.
Example 1-1 (continuation of Example 1.) The system (1.5),(1.6) corresponding t o the
Benney chain has the following form
∂ipj=∂iu1
pi−pj, ∂ i∂ju1=2∂iu1∂ju1
(pi−pj)2, (1.7)
∂ium= (−(m−2)um−2−···−2u2pm−2
i−u1pm−3
i+pm−1
i)∂iu1. (1.8)
Equations (1.7) were firstly obtained in [10]. /square
Given GT-system (1.5), (1.6) the coefficients of (1.1) are uniquely de fined by the following
relations
pi∂ium=φm,1∂iu1+···+φm,m+1∂ium+1, m= 2,3,... (1.9)
Namely, equating the coefficients at different powers of piin (1.9), we get a triangular system
of linear algebraic equations for φi,j. Thus, the classification problem for chains (1.1) is reduced
to a description of all GT-systems (1.5), (1.6) . The latter problem is solved in Section 4-6.
The paper is organized as follows. Following [11, 12], we recall main defin itions in Section
2 (see [1, 2, 3, 11] for details). We consider only 3-component hyd rodynamic reductions since
the existence of reductions with N >3 gives nothing new [1]. In Section 3 we formulate
our previous results that are needed in the paper. Section 4 is devo ted to a classification of
admissible polynomials PandQin (1.5), (1.6). In Sections 5,6 we construct integrable chains
for the generic case and for some degenerations. Section 6 also co ntains examples of (2+1)-
dimensional infinite hydrodynamic-type chains integrable from the v iewpoint of the method
of hydrodynamic reductions. Infinitesimal symmetries of GT-syst ems are studied in Section 7.
These symmetries seem to be important basic objects in the hydrod ynamic reduction approach.
Acknowledgments. Authors thank M.V. Pavlov for fruitful discussions. V.S. is gratefu l to
Brock University for hospitality. He was partially supported by the R FBR grants 08-01-464,
09-01-22442-KE, and NS 3472.2008.2.
2 Integrable chains and hydrodynamic reductions
According to [1, 2, 3, 4, 5, 6] a chain (1.1) is called integrable if it admits sufficiently many
so-called hydrodynamic reductions.
Definition. A hydrodynamic (1+1)-dimensional N-component reduction of a chain (1.1)
is a semi-Hamiltonian (see formula (3.18) ) system of the form
ri
t=pi(r1,...,rN)ri
x, i= 1,..,N (2.10)
4and functions uj(r1,...,rN), j= 1,2,...such that for each solution of (2.10) functions uj=
uj(r1,...,rN), i= 1,...satisfy (1.1).
Substituting ui=ui(r1,...,rN), i= 1,...into (1.1), calculating tandx-derivatives by virtue
of (2.10) and equating coefficients at rs
xto zero, we obtain
∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
It is clear from this system that
∂suk=gk(ps,u1,...,uk)∂su1, k= 2,3,...
wheregk(p,u1,...,uk) is a polynomial of degree k−1 inpfor eachk= 2,3,...Compatibility
conditions∂i∂juk=∂j∂iukgive us a system of linear equations for ∂ipj, ∂jpi, ∂i∂ju1, i/negationslash=j.
This system should have a solution (otherwise we would not have suffic iently many reductions).
Moreover, expressions for ∂suk, k= 2,3,..., ∂jpi, ∂i∂ju1, i/negationslash=jshould be compatible and form
a so-called GT-system.
Remark. In the sequel we assume N= 3 because the case N >3 gives nothing new [1].
3 GT-systems
Definition. A compatible system of PDEs of the form
∂ipj=f(pi,pj,u1,...,un), ∂iu1j/negationslash=i,
∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, j/negationslash=i, (3.11)
∂iuk=gk(pi,u1,...,un)∂iu1, k= 1,...,n−1,
wherei,j= 1,2,3 is called n-fields GT-system . Herep1,p2,p3,u1,...,unare functions of
r1,r2,r3and∂i=∂
∂ri.
Definition. Two GT-systems are called equivalent if they are related by a transformation
of the form
pi→λ(pi,u1,...,un), (3.12)
uk→µk(u1,...,un), k= 1,...,n. (3.13)
Example 2 [13]. Leta0,a1,a2be arbitrary constants, R(x) =a2x2+a1x+a0. Then the
system
∂ipj=a2p2
j+a1pj+a0
pi−pj∂iu1, ∂ i∂ju1=2a2pipj+a1(pi+pj)+2a0
(pi−pj)2∂iu1∂ju1(3.14)
is an one-field GT-system. The original Gibbons-Tsarev system (1.7 ) corresponds to a2=a1=
0,a0= 1.The polynomial R(x) can be reduced to one of the following canonical forms: R= 1,
5R=x,R=x2, orR=x(x−1) by a linear transformation (3.12). A wide class of integrable
3D-systems of hydrodynamic type related to (3.14) is described in [1 3]. An elliptic version of
this GT-system and the corresponding integrable 3D-systems wer e constructed in [15]. /square
Definition. An additional system
∂iuk=gk(pi,u1,...,un+m)∂iun, k=n+1,...,n+m (3.15)
suchthat(3.11)and(3.15)arecompatibleiscalled an extension of(3.11)byfields un+1,...,un+m.
It turns our that
∂iun+1=f(pi,un+1,u1,...,un)∂iu1
is an extension for GT-system (3.11). Stress that here fis the same function as in (3.11). We
call this extension the regular extension byun+1.
Example 2-1. The generic case of Example 2 corresponds to R=x(x−1). The regular
extension by u2is given by
∂iu2=u2(u2−1)
pi−u2∂iu1.
If we express u1from this formula and substitute it to (3.14), we get the following one -field
GT-system
∂ipj=pj(pj−1)(pi−u1)
u1(u1−1)(pi−pj)∂iu1,
∂i∂ju1=pipj(pi+pj)−p2
i−p2
j+(p2
i+p2
j−4pipj+pi+pj)u1
u1(u1−1)(pi−pj)2∂iu1∂ju1./square(3.16)
The second basic notion of the hydrodynamic reduction method is so -called GT-family of
(1+1)-dimensional hydrodynamic-type systems.
Definition. An (1+1)-dimensional 3-component hydrodynamic-type system o f the form
ri
t=vi(r1,...,rN)ri
x, i= 1,2,3, (3.17)
is called semi-Hamiltonian if the following relation holds
∂j∂ivk
vi−vk=∂i∂jvk
vj−vk, i/negationslash=j/negationslash=k. (3.18)
Definition. A Gibbons-Tsarev family associated with the Gibbons-Tsarev type s ystem
(4.25) is a (1+1)-dimensional hydrodynamic-type system of the fo rm
ri
t=F(pi,u1,...,um)ri
x, i= 1,2,3, (3.19)
semi-Hamiltonian by virtue of (3.11).
6Example 2-2 [13]. Applying the regular extension to the generic GT-system (3.14) two
times, we get the following GT-system:
∂ipj=pj(pj−1)
pi−pj∂iw, ∂ ijw=2pipj−pi−pj
(pi−pj)2∂iw∂jw, i/negationslash=j, (3.20)
∂iuj=uj(uj−1)∂iw
pi−uj, j= 1,2. (3.21)
Consider the generalized hypergeometric [14] linear system of the f orm
∂2h
∂uj∂uk=sj
uj−uk·∂h
∂uk+sk
uk−uj·∂h
∂uj, j/negationslash=k, (3.22)
∂2h
∂uj∂uj=−/parenleftBigg
1+n+2/summationdisplay
k=1sk/parenrightBigg
sj
uj(uj−1)·h+sj
uj(uj−1)n/summationdisplay
k/negationslash=juk(uk−1)
uk−uj·∂h
∂uk+
/parenleftBiggn/summationdisplay
k/negationslash=jsk
uj−uk+sj+sn+1
uj+sj+sn+2
uj−1/parenrightBigg
·∂h
∂uj.(3.23)
Herei,j= 1,2 ands1,...,s4are arbitrary parameters. It easy to verify that this system is in
involution and therefore the solution space is 3-dimensional. Let h1,h2,h3be a basis of this
space. For any hwe put
S(p,h) =u1(u1−1)(p−u2)hh1,u1−hu1h1
h1+u2(u2−1)(p−u1)hh1,u2−hu2h1
h1.
Then the formula
F=S(p,h3)
S(p,h2)(3.24)
defines the generic linear fractional GT-family for (3.20). /square
4 Canonical forms of GT-systems associated
with integrable chains
For integrable chains the corresponding GT-systems involve infinite number of fields ui, i=
1,2,...(see Example 1-1). In this Section we show that these GT-systems are equivalent to
infinite triangular extensions of one-field GT-systems from Example s 2,3.
A compatible system of PDEs of the form
∂ipj=f(pi,pj,u1,...,un)∂iu1, i/negationslash=j,
∂iuk=gk(pi,u1,...,uk)∂iu1, k= 1,2,...,, (4.25)
7∂i∂ju1=h(pi,pj,u1,...,un)∂iu1∂ju1, i/negationslash=j,
wherei,j= 1,2,3 is called triangular GT-system . Herep1,p2,p3,u1,u2,...are functions of
r1,r2,r3,and∂i=∂
∂ri.
Definition. A chain (1.1) is called integrable if there exists a Gibbons-Tsarev type system
of the form (4.25) and a Gibbons-Tsarev family
ri
t=F(pi,u1,...,um)ri
x, i= 1,2,3, (4.26)
such that (1.1) holds by virtue of (4.25), (4.26).
Due to the equivalence transformations (3.12) we can assume witho ut loss of generality that
F(p,u1,...,um) =p. (4.27)
Under this assumption we have
uj,t=/summationdisplay
s∂sujrs
t=/summationdisplay
s∂sujpsrs
x.
and similar
uj,x=/summationdisplay
s∂sujrs
x.
Substituting these expressions into (1.1) and equating coefficients atrs
xto zero, we obtain
∂suαps=φα,1∂su1+···+φα,α+1∂suα+1, α= 1,2,...
Using (4.25) and replacing psbyp, we get
p=φ1,1+φ1,2g2, pg2=φ2,1+φ2,2g2+φ2,3g3, pg3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...
Solving this system with respect to g2, g3,..., we obtain
gi(p) =ψi,0+ψi,1p+...+ψi,i−1pi−1.
Hereψi,jare functions of u1,...,ui. For example,
g2=−p
φ1,2−φ1,1
φ1,2. (4.28)
Remark. Since we assume that φi,i−1/negationslash= 0,we haveψi,i−1/negationslash= 0 for all i. Therefore g1=
1,g2,...is a basis in the linear space of all polynomials in p. The coefficients φi,jof our chain
are just entries of the matrix of multiplication by pin this basis. More generally, if we don’t
normalizeF=p, then the coefficients φi,jcan be found from the equations
F(p) =φ1,1+φ1,2g2, F(p)g2=φ2,1+φ2,2g2+φ2,3g3,
F(p)g3=φ3,1+φ3,2g2+φ3,3g3+φ3,4g4,...(4.29)
8Compatibility conditions ∂i∂juα=∂j∂iuα, α= 2,3,4 give a system of linear equations for
∂ipj, ∂jpi, ∂i∂ju1. Solving this system, we obtain formulas (1.5),(1.6), where in principa lP, Q
coulddependon u1,u2,u3,u4. However, itfollowsfromcompatibility conditions ∂i∂jpk=∂j∂ipk
thatP, Qdepend onu1, u2only.
Written (1.5) in the form
∂ipj=/parenleftbiggR(pj)
pi−pj+(z4p2
j+z5pj+z6)pi+z4p3
j+z3p2
j+z7pj+z8/parenrightbigg
∂iu1, (4.30)
whereR(x) =z4x4+z3x3+z2x2+z1x+z0,one can derive from the compatibility conditions
∂i∂jpk=∂j∂ipk,∂i∂ju1=∂j∂iu1that the equation (1.6) has the following form
∂i∂ju1=/parenleftbigg2z4p2
ip2
j+z3pipj(pi+pj)+z2(p2
i+p2
j)+z1(pi+pj)+2z0
(pi−pj)2+z9/parenrightbigg
∂iu1∂ju1.(4.31)
It is easy to verify that we can normalize z9=z6−z7, g2=pby a transformation (1.2).
Then the coefficients zi(x,y),i= 0,...,8 satisfy the following pair of compatible dynamical
systems with respect to yandx:
z0,y= 2z0z5−z1z6, z 1,y= 4z0z4+z1z5−2z2z6, z 2,y= 3z1z4−3z3z6,
z3,y= 2z2z4−z3z5−4z4z6, z 4,y=z3z4−2z4z5, z 5,y=z4z7−z4z6−z2
5,
z6,y=z4z8−z5z6, z 7,y= 2z1z4−2z3z6−z5z6+z4z8, z 8,y= 2z0z4−z2
6−z6z7+z5z8,
and
z0,x=−z0z2−z0z6+3z0z7−z1z8, z 1,x=−z1z2+3z0z3−z1z6+2z1z7−2z2z8,
z2,x=−z2
2+2z1z3+4z0z4−z2z6+z2z7−3z3z8, z 3,x= 3z1z4−z3z6−4z4z8,
z4,x=z2z4−z4z6−z4z7, z 5,x=z1z4−z5z6−z4z8, z 6,x=z0z4−z2
6,
z7,x=z1z3+3z0z4+z1z5−z2z6−z2z7+z2
7−z3z8−2z5z8,
z8,x=z0z3+z0z5−z2z8−2z6z8+z7z8.
These is a complete description of the GT-systems related to integr able chains (1.1).
To solve the dynamical systems we bring the polynomial Rto a canonical form sacrificing
to the normalization (4.27).
It is obvious that linear transformations pi→api+b, wherea,bare functions of u1,u2,
preserve the form of GT-system (4.30),(4.31). Moreover, there exist transformations of the
form
pi=a¯pi+b
¯pi−ψ, i= 1,2,3 (4.32)
9preserving the form of GT-system (4.30),(4.31). Such admissible tr ansformations are described
by the following conditions:
au2=z4(b+aψ), b u2=z4bψ+z5b−z6a, ψ u2=z4ψ2+z5ψ+z6.
Under transformations (4.32) the polynomial Ris transformed by the following simple way:
R(pi)→(pi−ψ)4R/parenleftBigapi+b
pi−ψ/parenrightBig
.
Suppose that Rhas distinct roots. It is possible to verify that by an admissible trans formation
(4.32) we can move three of the four roots to 0 ,1 and∞. It follows from compatibility
conditionsfortheGT-system thatthenthefourthroot λ(u1,u2)doesnotdependon u2. Making
transformation of the form u1→q(u1) we arrive at the canonical forms λ=u1orλ=const. It
is straightforwardly verified that in the first case equations (4.30) , (4.31) coincides with (3.16).
In the second case the GT-system does not exist.
In the case of multiple roots the polynomial R(x) can be reduced to one of the following
forms:R= 0,R= 1,R=x,R=x2, orR=x(x−1).In all these cases equations (4.30),
(4.31) coincides with the corresponding equations from Example 2.
Thus, the following statement is valid:
Proposition 1. There are 6 non-equivalent cases of GT-systems (4.30), (4.31). T he canon-
ical forms are:
Case 1: (3.16) (generic case);
Case 2: (3.14) with R(x) =x(x−1);
Case 3: (3.14) with R(x) =x2;
Case 4: (3.14) with R(x) =x;
Case 5: (3.14) with R(x) = 1.
Case 6: (3.14) with R(x) = 0./square
Remark. Cases 2-6 can be obtained from Case 1 by appropriate limit procedur es. For
example, Case 2 corresponds to the limit u1→u1
ε, ε→0.
It follows from (4.27), (4.28) that for any canonical form the func tionsFandg2have the
following structure:
g2(pi) =k1pi+k2
k3pi+k4, F(pi) =f1pi+f2
k3pi+k4, (4.33)
where the coefficients are functions of u1,u2.
Lemma 1. For theCase 1 any function g2can bereduced by anappropriatetransformation
10¯u2=σ(u1,u2) to one of the following canonical forms:
a1:g2(p) =u2(u2−1)(p−u1)
u1(u1−1)(p−u2)(regular extension);
b1:g2(p) =1
p−u1;
c1:g2(p) =u−λ
1(u1−1)λ−1
p−λλ= 1,0;
d1:g2(p) =u1−u2
u1(u1−1)p+u2−1
u1−1./square
The GT-system from the Case 1 possesses a discrete automorphis m groupS4interchanging
the points 0 ,1,∞,u1. The group is defined by generators
σ1:u1→1−u1, pi→1−pi, σ 2:u1→u1
u1−1, pi→pi
pi−1,
and
σ3:u1→1−u1, pi→(1−u1)pi
pi−u1.
Up to this group the cases b1,c1,d1are equivalent and one can take say the case d1for further
consideration. The case a1is invariant with respect to the group.
Remark. The casesb1, c1, d1are degenerations of the case a1. Namely, they can be
obtained as appropriate limit u2→u1,u2→λ, u2→ ∞correspondingly.
All possible functions g2for Cases 2-5 are described in the following
Lemma 2. For the GT-system (3.14) (excluding Case 6) any function g2can be reduced
by an appropriate transformation ¯ u2=σ(u1,u2) to one of the following canonical forms:
a2:g2(p) =R(u2)
p−u2(regular extension);
b2:g2(p) =1
p−λ,whereR(λ) = 0;
c2:g2(p) =p−a2u2.
The discrete automorphism of the GT-system interchanges the ro ots ofRin the case b2./square
Lemma 3. For the GT-system (3.14) with R(x) = 0 (Case 6) any function g2can be
reduced to g2(p) =pby an appropriate transformation ¯ u2=σ(u1,u2). Furthermore, the
corresponding triangular GT-system has the form
∂ipj= 0, ∂ i∂ju1= 0, ∂ iuk=pk−1
iu1, k= 2,3,.../square (4.34)
115 Generic case
The next step in the classification is to find all functions Fof the form (4.28) for each pair
consisting of a GT-system from Proposition 1 and the correspondin gg2from Lemmas 1-3.
The semi-Hamiltonian condition (3.18) yields a non-linear system of PDE s for the functions
f1(u1,u2),f2(u1,u2).For each case this system can be reduced to the linear generalized h yper-
geometric system (3.22), (3.23) with a special set of parameters s1,s2,s3,s4or to a degeneration
of this system.
The general linear fractional GT-family for the generic case 1, a1is given by (3.24). Ac-
cording to (4.33), the additional restriction is that the root of the denominator has to be equal
u2.It is easy to verify that this is equivalent to s2= 0,h1,u2=h2,u2= 0. The latter means that
h1(u1),h2(u1) are linear independent solutions of the standard hypergeometric equation
u(u−1)h(u)′′+[s1+s3−(s3+s4+2s1)u]h(u)′+s1(s1+s3+s4+1)h(u) = 0.(5.35)
The function h3(u1,u2) is arbitrary solution of (3.22), (3.23) with s2= 0 linearly independent
ofh1(u1),h2(u1). Without loss of generality we can choose
h3(u1,u2) =/integraldisplayu2
0(t−u1)s1ts3(t−1)s4dt.
Formula (3.24) gives
F(p,u1,u2) =f1(u1,u2)p−f2(u1,u2)
p−u2, (5.36)
where
f1=u2(u2−1)h1h3,u2+u1(u1−1)(h1h3,u1−h3h′
1)
u1(u1−1)(h1h′
2−h2h′
1),
f2=u1u2(u2−1)h1h3,u2+u2u1(u1−1)(h1h3,u1−h3h′
1)
u1(u1−1)(h1h′
2−h2h′
1).
Notice that h1h′
2−h2h′
1=const(u1−1)s1+s4us1+s3
1.
For integer values of s1,s3,s4the hypergeometric system can be solved explicitly. For
example, if s1=s3=s4= 0, the above formulas give rise to F=g2.Ifs4=−2−s1−s3then
F=(u2−u1)s1+1us3+1
2(u2−1)−1−s1−s3
p−u2;
ifs4= 0,then
F=(p−1)(u2−u1)s1+1us3+1
2(u1−1)−1−s1
p−u2.
Nowwearetofindthefunctions g3,g4,...in(4.25). Thesefunctionsaredefineuptoarbitrary
transformation (1.2), where α= 3,4,.... In practice, one can look for functions g3,g4,...linear
inui,i>2 (cf. (1.8)). An extension linear in ui,i>2 is given by
g3(p) =−(u1−u2)(u2−1)p
u1(u1−1)(p−u2)2,
12gi(p) =(i−3)(u1−u2)(u2−1)pui
u1(u1−1)(p−u2)2−(u1−u2)i−3(u2−1)2p(p−u1)(p−1)i−4
u1(u1−1)i−2(p−u2)i−1−
i−4/summationdisplay
s=1(i−s−2)(u1−u2)s(u2−1)2p(p−u1)(p−1)s−1ui−s
u1(u1−1)s+1(p−u2)s+2.
The coefficients of the chain (1.1) corresponding to Case 1, a1are determined from (4.29),
whereFis given by (5.36). Relations (4.29) are equivalent to a triangular syst em of linear
algebraic equations. Solving this system, we find that for i>4 coefficients of the chain read:
φi,i+1=(u1−1)(f1u2−f2)
(u2−1)(u1−u2)def=Q1, φ i,i=f2−f1
u2−1def=Q2,
φi,4=−uiQ1, φ i,3=−/parenleftBig
(u4+i−3)ui+(2−i)ui+1/parenrightBig
Q1def=Ai,
andφi,j= 0 for all remaining i,j.Fori≤4 we have
φ1,1=f1u1−f2
u1−u2, φ 1,2=−u1
u2Q1,
φ2,1=(u2−1)(f1u2−f2)
(u1−1)(u1−u2), φ 2,2=f2u1−f1u2
2
u2(u1−u2), φ 2,3=f1u2−f2,
φ3,1=φ3,2= 0, φ 3,3=Q2−(u4−1)Q1, φ 3,4=−Q1,
φ4,1=φ4,2= 0, φ 4,3=A4, φ 4,4=Q2−u4Q1, φ 4,5=Q1.(5.37)
The explicit formulas for other cases of Proposition 1 can be obtaine d by limits from the
above formulas. We outline the limit procedures for the case 1, d1. In this case the limit is
given byu2→u1+εu2, ε→0.It is easy to check that under this limit the extension a1
turns tod1. The limit of the system (3.22), (3.23) with s2= 0 can be easily found. The general
solution of the system thus obtained is given by h=c1(u2−u1)1+s1+s3+s4+h1,whereh1is the
general solution of (5.35). Let h1,h2be solutions of (5.35), and h3= (u2−u1)1+s1+s3+s4. Then
the limit procedure in (5.36) gives rise to
F(p,u1,u2) =Q×/parenleftBig
(1+s1+s3+s4)h1(p−u1)+u1(u1−1)h′
1/parenrightBig
,
where
Q= (u2−u1)1+s1+s3+s4(u1−1)−1−s1−s4u−1−s1−s3
1.
As usual, the most degenerate cases in classification of integrable P DEs could be interesting
for applications. In our classification they are Case 5, c2and Case 6. The Benney chain
(see Examples 1 and 1-1) belongs to Case 5, case c2(i.eg2=p). Any GT-family has the form
F=f1(u1,u2)p+f2(u1,u2). Iff1= 1 thenF=p+k2u2+k1u1.The Benney case corresponds to
13k1=k2= 0. For arbitrary kiwe get the Kupershmidt chain [16]. In the case f1=A(u1),A′/negationslash= 0
we obtain:
f1=k2exp(λu1)+k1, f 2=k2k3exp(λu1)+λk1(k3u1−u2).
In the generic case
F= exp(λu2)(S1(u1)p+S2(u1)),
where the functions Sican be expressed in terms of the Airy functions.
6 Trivial GT-system and 2+1-dimensional integrable hy-
drodynamic chains
It was observed in [11] that (2+1)-dimensional systems of hydro dynamic type with the trivial
GT-system usually admit some integrable multi-dimensional generaliza tions. For the chains
such GT-system is defined by (4.34). That is why the Case 6 is of a gre at importance in our
classification. The automorphisms of (4.34) are given by
pj→pj, j= 1,...,N, u i→νui+γi, i= 1,2,...; (6.38)
pj→apj+b, j= 1,...,N, u i→ai−1ui+(i−1)ai−2bui−2+...+bi−1u1, i= 1,2,...
The corresponding GT-families are of the form F(p) =A(u1,u2)p+B(u1,u2), where
A(x,y),B(x,y) satisfies the following system of PDEs:
AByy=AyBy, AB xy=AyBx, AB xx=AxBx,
AAyy=A2
y, AA xy=AxAy, AA xx=A2
x+AxBy−AyBx.(6.39)
This system can be easily solved in elementary functions. For each so lution formula (4.29)
defines the corresponding integrable chain (1.1).
It follows from (6.39) that there are two types of u2-dependence:
1(generic case). F(p) = exp(λu2)/parenleftBig
a(u1)p+b(u1)/parenrightBig
,
2. F(p) =a(u1)p+λu2+b(u1).
In the first case there are two subcases: b′/negationslash= 0 andb′= 0.The first subcase gives rise to
a=σ′, b=k1σ σ(x) =c1exp(µ1x)+c2exp(µ2x),wherec1c2(λk1−µ1µ2) = 0.
The second subcase leads to
b=c1, a(x) =c2exp(µx)+c3,wherec2(c1λ−c3µ) = 0.
The same subcases for the case 2 yield
a=σ′, b=k1σ σ(x) =c1+c2x+c3exp(µx),wherec3(λ−c2µ) = 0,
14and
b=c1, a(x) =c2exp(µx)+c3,wherec2(λ−c3µ) = 0.
It is easy to verify that in the generic case the function Fcan be reduced by (6.38) to the
form
F(p) =eu2+u1(p−1)+eu2−u1(p+1).
In this case the corresponding chain reads as
uk,t= (eu2+u1+eu2−u1)uk+1,x+(eu2−u1−eu2+u1)uk,x, k= 1,2,3,... (6.40)
Asusual, thischainisthefirstmember ofaninfinitehierarchy. These condflowofthishierarchy
is given by
uk,τ= (eu2+u1+eu2−u1)uk+2,x+(u3−u1)(eu2+u1+eu2−u1)uk+1,x+
(eu2+u1(u1−u3−1)+eu2−u1(u3−u1−1))uk,x, k= 1,2,3,...
In the case 2 with c3=λ= 0,k1= 1 we get the chain
uk,t=uk+1,x+u1uk,x, k= 1,2,3,... (6.41)
This chain is equivalent to the chain of the so-called universal hierarc hy [17]. The chain (6.41)
is a degeneration of the chain
uk,t=uk+1,x+u2uk,x, k= 1,2,3,... (6.42)
Following the line of [3, 11] it is not difficult to find (2+1)-dimensional inte grable generaliza-
tions for all (1+1)-dimensional integrable chains constructed abo ve. Some families of functions
Fdescribed above linearly depend on two parameters. Denote these parameters by γ1,γ2.The
corresponding integrable chain
uk,t=γ1(φk,1u1,x+···+φk,k+1uk+1,x)+γ2(ψk,1u1,x+···+ψk,k+1uk+1,x)
is also linear in γ1,γ2.We claim that the following (2+1)-dimensional chain
uk,t= (φk,1u1,x+···+φk,k+1uk+1,x)+(ψk,1u1,y+···+ψk,k+1uk+1,y) (6.43)
is integrable from the viewpoint of the method of hydrodynamic redu ctions. For each case the
reductions can be easily described.
For example, in the generic case
F(p) =γ1eu2+u1(p−1)+γ2eu2−u1(p+1)
formula (6.43) yields (2+1)-dimensional chain
uk,t=eu2+u1(uk+1,x−uk,x)+eu2−u1(uk+1,y+uk,y), k= 1,2,3,... (6.44)
15After a change of variables of the form
x→ −1
2x, y→1
2y, u 1→1
2u0, u2→u1+1
2u0, u3→ −2u2+1
2u0,...
(6.44) can be written as
u0,t=eu1u0,y+eu1(u1,y−eu0u1,x), u i,t=eu0+u1ui,x+eu1(eu0ui+1,x−ui+1,y),(6.45)
wherei= 1,2,.... Probably (6.45) is a first example of a (2+1)-dimensional chain integ rable
from the viewpoint of the hydrodynamic reduction approach.
TriangularGT-systemsrelatedtointegrable(2+1)-dimensionalch ainswithfields u0,u1,u2,...
have the form
∂ipj=f1(pi,qi,pj,qj,u0,...,un)∂iu0, ∂ iqj=f2(pi,qi,pj,qj,u0,...,un)∂iu0,
∂i∂ju0=h(pi,qi,pj,qj,u0,...,un)∂iu0∂ju0, (6.46)
∂iuk=gk(pi,qi,u0,...,uk+1)∂iu0, k= 0,1,2,...
Herei/negationslash=j, i,j= 1,...,3,p1,...,p3, q1,...,q3,u0,u1,u2,...,arefunctionsof r1,r2,r3.Inparticular,
the GT-system associated with (6.45) has the form:
∂ipj=∂i∂ju0= 0, ∂ iqj=/parenleftBigpiqi−pjqj
pi−pj−qiqj/parenrightBig
∂iu0, ∂ iuk=−pi
(pi−1)k∂iu0.
Thehydrodynamicreductionsof(6.45)isgivenbythepairofsemi-ha miltonian(1+1)-dimensional
systems
ri
y=eu0/parenleftBig
1−1
qi/parenrightBig
ri
x, ri
t=eu0+u1/parenleftBig1
(pi−1)qi+1/parenrightBig
ri
x.
Chain (6.45) is the first member of an infinite hierarchy of pairwise com muting flows where
the corresponding ”times” are t1=t, t2, t3,.... These flows and their hydrodynamic reductions
can be described in terms of the generating function U(z) =u1+u2z+u3z2+...The hierarchy
is given by
D(z)u0=eU(z)/parenleftBig
u0,y+U(z)y−eu0U(z)x/parenrightBig
,
D(z1)U(z2) =eu0+U(z1)U(z2)x+(1+z1)eU(z1)/parenleftBig
eu0U(z1)x−U(z2)x
z1−z2−U(z1)y−U(z2)y
z1−z2/parenrightBig
,
whereD(z) =∂
∂t1+z∂
∂t2+z2∂
∂t3+...The reductions can be written as
D(z)ri=eu0+U(z)/parenleftBig
1+1+z
(pi−1−z)qi/parenrightBig
ri
x.
Other (2+1)-dimensional integrable chains related to 2-dimensiona l vector spaces of solu-
tions for system (6.39) are degenarations of (6.45). In particular F=γ1eu1p+γ2(p+u2) leads
to the following (2+1)-dimensional integrable generalization of (6.44 ):
uk,t=eu1uk+1,x+uk+1,y+u2uk,y, k= 1,2,3,....
16Conjecture. Any chain of the form (6.43) integrable by the hydrodynamic reduct ion
method is a degeneration of (6.45).
We are planning to consider the problem of classification of integrable chains (6.43) in a
separate paper.
7 Infinitesimal symmetries of triangular GT-systems
A scientific way to construct the functions g3,g4,...for different cases from Proposition 1 is
related to infinitesimal symmetries of the corresponding GT-syste m1. The whole Lie algebra
of symmetries is one the most important algebraic structures relat ed to any triangular GT-
system (4.25). In particular, this algebra acts on the hierarchy of the commuting flows for the
corresponding chain (1.1).
A vector field
S=N/summationdisplay
j=1X(pj,u1,...,us)∂
∂pj+∞/summationdisplay
m=1Ym(u1,...,ukm)∂
∂um,∂Ym
∂ukm/negationslash= 0 (7.47)
is called a symmetry of the triangular GT-system (4.25) if it commutes with all ∂i.Notice that
it follows from the definition that
S(∂iu1) =∂i(Y1).
We call (7.47) a symmetry of shift difkm=m+dform>>0.LetMbe the minimal integer
such thatkm=m+d,m>M. If the functions gi,i= 1,...,M+dfrom (4.25) are known, then
the functions X,Y1,...YMcan be found from the compatibility conditions
S(∂ipj) =∂iS(pj), S(∂iuk) =∂iS(uk), k= 1,...,M.
The functions YM+1,YM+2,...can be chosen arbitrarily. After that gM+d+1,gM+d+2,...are
uniquely defined by the remaining compatibility conditions.
The generic case 1, a 1. Looking for symmetries of shift one, we find X=Y1= 0 and
M= 1. Hence without loss of generality we can take
S=∞/summationdisplay
m=2um+1∂
∂um
for the symmetry. This fact gives us a way to construct all functio nsgi,i >3 in the infinite
triangular extension for the case 1, a1.Indeed, it follows from the commutativity conditions
S(∂iuk) =∂iS(uk) thatgk+1=S(gk),wherek= 2,3,.... In particular,
g3=(pj−u1)(2pju2−pj−u2
2)u3
u1(u1−1)(pj−u2)2.
1Note that these functions are not unique because of the triangula r group of symmetries (1.2) acting on the
fieldsu3,u4,...
17The functions githus constructed are not linear in u3.The corresponding chain (1.1) is equiv-
alent to the chain constructed in Section 5 but not so simple.
It would be interesting to describe the Lie algebra of all symmetries in this case. Here we
present the essential part for symmetry of shift 2:
X=pj(pj−1)u2
3
(pj−u2)u2(u2−1), Y 1=u1(u1−1)u2
3
(u1−u2)u2(u2−1),
Y2=−3
2u4+(2u1−1)u2
3
u2(u2−1)+u3./square
The case 1, d 1. One can add fields u3,...in such a way that the whole triangular GT-
system admits the following symmetry of shift 1:
S=u2
u1(u1−1)N/summationdisplay
i=1pi(pi−1)∂
∂pi+∞/summationdisplay
i=1ui+1∂
∂ui.
As in the previous example, one can easily recover the whole GT-syst em. For example,
∂iu3=/parenleftbiggu3(pi+u1−1)
u1(u1−1)+2u2
2pi(pi−1)
u2
1(u1−1)2/parenrightbigg
∂iu1./square
Below we describe the symmetry algebra for the case 5, c2(in particular, for the Benney
chain).
The case 5, c 2. For the triangular GT-system (1.7), (1.8) there exists an infinite L ie
algebra of symmetries Si,i∈Z,whereSiis a symmetry of shift i. The simplest symmetries
are the following:
S−2=∂
∂u1+∞/summationdisplay
i=3/parenleftBig
−ui−2+/summationdisplay
k+m=i−3ukum−/summationdisplay
k+m+l=i−4ukumul+···/parenrightBig∂
∂ui,
S−1=N/summationdisplay
j=1∂
∂pj+∞/summationdisplay
i=1(i−1)ui−1∂
∂ui,
S0=N/summationdisplay
j=1pj∂
∂pj+∞/summationdisplay
i=1(i+1)ui∂
∂ui,
S1=N/summationdisplay
j=1(p2
j+3u1)∂
∂pj+∞/summationdisplay
i=1(i+3)ui+1∂
∂ui+∞/summationdisplay
i=2/summationdisplay
k+m=iukum∂
∂ui+∞/summationdisplay
i=23(i−1)u1ui−1∂
∂ui,
S2=N/summationdisplay
j=1(p3
j+4u1pj+5u2)∂
∂pj+∞/summationdisplay
i=1(i+5)ui+2∂
∂ui+∞/summationdisplay
i=14iu1ui∂
∂ui+∞/summationdisplay
i=25(i−1)u2ui−1∂
∂ui+
18∞/summationdisplay
i=1/summationdisplay
k+m=i+13ukum∂
∂ui+∞/summationdisplay
i=3/summationdisplay
k+m+l=iukumul∂
∂ui.
The whole algebra is generated by S1,S2,S−1,S−2.It is isomorphic to the Virasoro algebra with
zero central charge.
LetDtibe the vector fields corresponding to commuting flows for the Benn ey chain. Here
Dt1=Dx, Dt2=Dt. Then the commutator relations
[S1,Dti] = (i+1)Dti+1
hold. Thus the vector field S1plays the role of a master-symmetry for the Benney hierarchy.
/square
The case 6 . In this case there exist infinitesimal symmetries of form
Ti=ui+1∂
∂u1+ui+2∂
∂u2+..., i= 0,1,2,...
Si=N/summationdisplay
j=1pi+1
j∂
∂pj+ui+2∂
∂u2+2ui+3∂
∂u3+3ui+4∂
∂u4+..., i=−1,0,1,2,...
Note that [Si,Sj] = (j−i)Si+j,[Ti,Tj] = 0,[Si,Tj] =jTi+j./square
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