On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems
The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A dispersionless Lax formulation is found.
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Інститут математики НАН України
2009
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| Назва видання: | Symmetry, Integrability and Geometry: Methods and Applications |
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| Цитувати: | On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems / M.V. Pavlov, Z. Popowicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 7 назв. — англ. |
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irk-123456789-1492422019-02-20T01:28:05Z On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems Pavlov, M.V. Popowicz, Z. The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A dispersionless Lax formulation is found. 2009 Article On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems / M.V. Pavlov, Z. Popowicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 7 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 35Q53 http://dspace.nbuv.gov.ua/handle/123456789/149242 en Symmetry, Integrability and Geometry: Methods and Applications Інститут математики НАН України |
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The particular case of the integrable two component (2+1)-dimensional hydrodynamical type systems, which generalises the so-called Hamiltonian subcase, is considered. The associated system in involution is integrated in a parametric form. A dispersionless Lax formulation is found. |
| format |
Article |
| author |
Pavlov, M.V. Popowicz, Z. |
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Pavlov, M.V. Popowicz, Z. On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems Symmetry, Integrability and Geometry: Methods and Applications |
| author_facet |
Pavlov, M.V. Popowicz, Z. |
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Pavlov, M.V. |
| title |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems |
| title_short |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems |
| title_full |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems |
| title_fullStr |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems |
| title_full_unstemmed |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems |
| title_sort |
on integrability of a special class of two-component (2+1)-dimensional hydrodynamic-type systems |
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Інститут математики НАН України |
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2009 |
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http://dspace.nbuv.gov.ua/handle/123456789/149242 |
| citation_txt |
On Integrability of a Special Class of Two-Component (2+1)-Dimensional Hydrodynamic-Type Systems / M.V. Pavlov, Z. Popowicz // Symmetry, Integrability and Geometry: Methods and Applications. — 2009. — Т. 5. — Бібліогр.: 7 назв. — англ. |
| series |
Symmetry, Integrability and Geometry: Methods and Applications |
| work_keys_str_mv |
AT pavlovmv onintegrabilityofaspecialclassoftwocomponent21dimensionalhydrodynamictypesystems AT popowiczz onintegrabilityofaspecialclassoftwocomponent21dimensionalhydrodynamictypesystems |
| first_indexed |
2025-07-12T21:39:57Z |
| last_indexed |
2025-07-12T21:39:57Z |
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1837478865281220608 |
| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 (2009), 011, 10 pages
On Integrability of a Special Class of Two-Component
(2+1)-Dimensional Hydrodynamic-Type Systems?
Maxim V. PAVLOV † and Ziemowit POPOWICZ ‡
† Department of Mathematical Physics, P.N. Lebedev Physical Institute of RAS,
53 Leninskii Ave., 119991 Moscow, Russia
E-mail: M.V.Pavlov@lboro.ac.uk
‡ Institute of Theoretical Physics, University of Wroc law,
pl. M. Borna 9, 50-204 Wroc law, Poland
E-mail: ziemek@ift.uni.wroc.pl
Received August 28, 2008, in final form January 20, 2009; Published online January 27, 2009
doi:10.3842/SIGMA.2009.011
Abstract. The particular case of the integrable two component (2+1)-dimensional hydro-
dynamical type systems, which generalises the so-called Hamiltonian subcase, is considered.
The associated system in involution is integrated in a parametric form. A dispersionless Lax
formulation is found.
Key words: hydrodynamic-type system; dispersionless Lax representation
2000 Mathematics Subject Classification: 37K10; 35Q53
1 Introduction
Quasilinear (2+1)-dimensional systems of the first order,
Ai
k(u)uk
t +Bi
k(u)uk
y + Ci
k(u)uk
x = 0,
play an important role in the description of variety of physical phenomena. The method of the
hydrodynamical reductions (see e.g. [1]) enables us to pick from this class the integrable systems
which possess sufficiently many hydrodynamic reductions, and thus infinitely many particular
solutions. Recently, a system in involution describing the integrable (2+1)-dimensional hydro-
dynamical type systems(
v
w
)
t
=
(
A11 A12
A21 A22
) (
v
w
)
y
+
(
B11 B12
B21 B22
) (
v
w
)
x
, (1)
where Aik and Bik are functions of v and w, was derived in [2] using the method of hydrodynamic
reductions. In a particular case(
v
w
)
t
=
(
α 0
0 β
) (
v
w
)
y
+
(
p q
r s
) (
v
w
)
x
, (2)
where α and β are constants, the corresponding system in involution for two functions r(v, w)
and q(v, w) simplifies to the form
qvv = (qr)w, qvw =
qvqw
q
+ q2r, qww =
q2w
q
+ 2qqv −
qwrw
r
,
?This paper is a contribution to the Proceedings of the XVIIth International Colloquium on Integrable Sys-
tems and Quantum Symmetries (June 19–22, 2008, Prague, Czech Republic). The full collection is available at
http://www.emis.de/journals/SIGMA/ISQS2008.html
mailto:M.V.Pavlov@lboro.ac.uk
mailto:ziemek@ift.uni.wroc.pl
http://dx.doi.org/10.3842/SIGMA.2009.011
http://www.emis.de/journals/SIGMA/ISQS2008.html
2 M.V. Pavlov and Z. Popowicz
rww = (qr)v, rvw =
rvrw
r
+ r2q, rvv =
r2v
r
+ 2rrw −
qvrv
q
, (3)
and a general solution therefore depends on 6 arbitrary constants. Two other functions p(v, w)
and s(v, w) can be found in quadratures
dp = rqdv + qvdw, ds = rwdv + rqdw. (4)
A general solution of system (3) was presented in [2] in a parametric form.
Under a simple linear transformation of independent variables (y, t) such that
∂t − α∂y → ∂t, ∂t − β∂y → ∂y,
system (2) reduces to the form
vt = p(v, w)vx + q(v, w)wx, wy = r(v, w)vx + s(v, w)wx. (5)
Our first result is that the equations (5) reduces to the most compact form
vt = ∂xHww, wy = ∂xHvv (6)
by introducing the potential function H(v, w) such that r = Hvvv, s = Hvvw, p = Hvww,
q = Hwww. Then the equations (3) can be written in the form
Hvvww = HvvvHwww,
Hvvvvw =
HvvvvHvvvw
Hvvv
+H2
vvvHwww,
Hvwwww =
HwwwwHvwww
Hwww
+H2
wwwHvvv,
Hvvvvv =
H2
vvvv
Hvvv
− HvvvvHvwww
Hwww
+ 2HvvvHvvvw,
Hwwwww =
H2
wwww
Hwww
− HwwwwHvvvw
Hvvv
+ 2HwwwHvwww. (7)
Our second result is that we present a general solution of system in involution (7) in a new
parametric form (33), which is more convenient for the investigation in the special cases. This
approach is universal, i.e. any other (2+1)-dimensional quasilinear equations can be investigated
effectively in the same fashion (see [6, 5]).
In the general case (1), a dispersionless Lax representation has the form (see [2])
ψt = a(ψx, v, w), ψy = c(ψx, v, w), (8)
where a(µ, v, w) and c(µ, v, w) are some functions. The function ψ is called a pseudopotential.
Equations (3) and (4) can be derived directly from the compatibility condition (ψt)y = (ψy)t.
Our third result is that the dispersionless Lax representation (8) for (2+1)-dimensional hyd-
rodynamical type system (5) reduces to the most compact form
ψt = a(ψx, w), ψy = c(ψx, v), (9)
i.e., each of the functions a and c depends on two arguments only.
Moreover, we were able to integrate the corresponding nonlinear system for the functions
a(µ,w) and c(µ, v), i.e. we found the dispersionless Lax pair (9) for the (2+1)-dimensional
quasilinear equations (6). In general, the compatibility condition (ψt)y = (ψy)t for the system
ψt = a, ψy = c,
On Integrability of a Special Class of Hydrodynamic-Type Systems 3
where a = a(µ, u1, u2, . . . , uN ) and c = c(µ, u1, u2, . . . , uN ) are some functions (cf. (9)), yields
(2+1)-dimensional quasilinear systems of the nonlinear equations of the first order. Recently,
a complete classification of pseudopotentials that satisfy the single equation ψt = a(ψx, w) was
given in [6].
Our fourth result is that we were able to extract the functions a(µ,w) and c(µ, v) such that (9)
yields a system of the form (6).
Let us mention that the particular case, when Hvvv = Hwww,
vt = ∂xhv, wy = ∂xhw. (10)
was considered in [1] and a complete classification of all admissible functions h(v, w) was pre-
sented in [4].
The paper is organized as follows. In Section 2, we prove that quasilinear system (6) admits
the dispersionless Lax representation (9), which is a special case of (8), and conversely, a disper-
sionless Lax representation (9) yields an integrable (2+1)-dimensional quasilinear system (6).
We also derive the associated system in involution (3) which in the case under study takes the
form (7). The problem of computation of a single function H(v, w) that solves (7) is reduced
to quadratures. In Section 3, we present an effective method for direct integration of the sys-
tem (3) and the corresponding equations are integrated in the parametric form. The last section
contains the conclusion.
2 System in involution
A classification of integrable (2+1)-dimensional two-component Hamiltonian hydrodynamic-
type systems (10) was obtained in [1] using the method of hydrodynamic reductions. In
this paper, we use dispersionless Lax representations following the original papers [2] and [7]
(see [5] for further details). It was proved in [2] that the pseudopotentials ψ(x, y, t) for (2+1)-
dimensional hydrodynamic-type systems (1) must satisfy the dispersionless Lax representation
of the form (8). Also, it was proved in [4] that there exist the dispersionless Lax representations
for (10) of the form (9). Moreover, in our case the dispersionless Lax representation (9) remains
valid for more general case (5). Indeed, the following assertion holds.
Lemma 1. The hydrodynamic-type system (5) can be obtained from the compatibility condition
(ψt)y = (ψy)t, where the pseudopotential ψ satisfies dispersionless Lax representation (9).
Proof. The compatibility condition for (8) implies
∂ya(ψx, v, w) = ∂tc(ψx, v, w).
Substituting (5) into this equation yields
aµ(cvvx + cwwx) + avvy + aw(rvx + swx)
= cµ(avvx + awwx) + cv(pvx + qwx) + cwwt,
where µ = ψx. Since we are interested in a general solution, the coefficients at the t- and
x-derivatives v and w on the left- and right-hand sides of the above equations must be identical,
that is,
aµcv + raw = cµav + pcv, aµcw + saw = cµaw + qcv,
and av = 0, cw = 0. Thus, a is independent of v, and c is independent of w. This means that (8)
reduces to (9). The lemma is proved. �
4 M.V. Pavlov and Z. Popowicz
Theorem 1. The dispersionless Lax representation (9) uniquely determines a hydrodynamic-
type system of the form (5).
Proof. The compatibility condition (ψt)y = (ψy)t implies
aµcvvx + awwy = cµawwx + cvvt, (11)
where the dispersionless Lax representation (9) is re-written in the form
µt = ∂xa(µ,w), µy = ∂xc(µ, v) (12)
Since aw 6= 0, differentiating the equation (11)
aµcv
aw
vx + wy = cµwx +
cv
aw
vt
with respect to µ yields(
aµcv
aw
)
µ
vx = cµµwx +
(
cv
aw
)
µ
vt.
Suppose that (cv/aw)µ 6= 0, then the equation
vt =
(
aµcv
aw
)
µ(
cv
aw
)
µ
vx −
cµµ(
cv
aw
)
µ
wx (13)
cannot depend on µ. This means that (see (5)) we have(
aµcv
aw
)
µ(
cv
aw
)
µ
= p(v, w),
cµµ(
cv
aw
)
µ
= −q(v, w)
for some functions p(v, w) and q(v, w) that do not depend on µ. Then the above equation (13)
reduces to the form (cf. (5))
vt = pvx + qwx.
Since cv 6= 0, two other relations of the above type,
aµµ(
aw
cv
)
µ
= −r(v, w),
(
cµaw
cv
)
µ(
aw
cv
)
µ
= s(v, w),
where r(v, w) and s(v, w) are some other functions independent of µ, can be derived in analogy
with the above, along with the second equation of the hydrodynamic-type system (5). Thus,
the hydrodynamic-type system (5) is indeed uniquely determined by the compatibility condition
(ψt)y = (ψy)t. The theorem is proved. �
Substituting (5) into (11) implies the relations
aw =
qcv
s− cµ
, rq = (s− cµ)(p− aµ). (14)
among the first derivatives of the functions a(µ,w) and c(µ, v).
On Integrability of a Special Class of Hydrodynamic-Type Systems 5
The compatibility conditions (aw)µ = (aµ)w, (aw)v = 0 and (aµ)v = 0 yield
cvv =
pv
rq
cµcv +
qrv − spv
rq
cv,
cµv =
pv
rq
c2µ +
(
qv
q
+
rv
r
− 2
spv
rq
)
cµ +
s2pv
rq
−
(
qv
q
+
rv
r
)
s+ sv, (15)
cµµ =
1
qcv
[pv
r
(cµ − s)3 +
(
pw + qv +
qrv
r
)
(cµ − s)2 + (rqw + qrw + qsv)(cµ − s) + rqsw
]
.
Similar formulas can be obtained from the other compatibility conditions, namely, (cv)µ = (cµ)v,
(cv)w = 0 and (cµ)w = 0. The compatibility conditions (cµµ)v = (cµv)µ, (cµv)v = (cvv)µ,
(aµµ)w = (aµw)µ, (aµw)w = (aww)µ yield the system
pvv =
pv
rq
(rpw + qrv), pvw =
pv
rq
(rqw + qrw), pww =
1
rq
(rpwqw + qpvsw),
sww =
sw
rq
(rqw + qsv), svw =
sw
rq
(rqv + qrv), svv =
1
rq
(qrvsv + rpvsw),
rqvv + qrvv =
1
rq
(r2pvqw + q2r2v) + 2svpv + rwpv − rvqv, (16)
rqww + qrww =
1
rq
(q2swrv + r2q2w) + 2pwsw + qvsw − qwrw,
rqvw + qrvw =
1
rq
(r2qvqw + q2rvrw) + 2pvsw.
The last equation can be replaced by the pair of equations
rvw =
1
rq
(rpvsw + qrvrw), qvw =
1
rq
(rqvqw + qpvsw),
which can be obtained from the compatibility conditions (pvw)v = (pvv)w and (svw)w = (sww)v.
The second and fifth equations of (16) can be integrated once to yield
pv = rqϕ1(v), sw = rqϕ2(w), (17)
where ϕ1(v) and ϕ2(w) are arbitrary functions. However, without loss of generality these
functions can be set equal to 1, because these functions can be eliminated from all of the
above equations upon using the scaling
∫
ϕ1(v)dv → v,
∫
ϕ2(w)dw → w, ϕ1(v)p/ϕ2(w) → q,
ϕ2(w)r/ϕ1(v)→ r. Thus, substituting pv = rq, sw = rq into (16), we finally obtain the system
in involution (3) together with (4) (cf. [2]).
Thus, integrable (2+1)-dimensional system (5) can be written in the most compact form (6),
and the reduction to the Hamiltonian case (10) is given by the symmetric constraint r = q. The
function H can be reconstructed via the complete differentials,
dHvv = rdv + sdw, dHvw = sdv + pdw, dHww = pdv + qdw,
dHv = Hvvdv +Hvwdw, dHw = Hvwdv +Hwwdw, dH = Hvdv +Hwdw.
Remark 1. The above choice (17) uniquely fixes system (5) in conservative form (6). This
means that the integrable (2+1)-dimensional hydrodynamic-type system (5) is reducible to (6)
under appropriate choice of functions ϕ1(v) and ϕ2(w) in (17).
6 M.V. Pavlov and Z. Popowicz
3 Dispersionless Lax representation
In this section, we obtain a general solution of the system in involution (7) and simultaneously
reconstruct the functions a(µ,w) and c(µ, v) (see (9)).
Rewrite system (15) for the function c(µ, v) in the form
cvv = A1cµcv +A2cv,
cµv = B1c
2
µ +B2cµ +B3, (18)
cvcµµ = D1c
3
µ +D2c
2
µ +D3cµ +D4,
where all the coefficients Ak, Bn, Dm depend on v alone and are to be determined. The
compatibility conditions (cvv)µ = (cµv)v, (cµµ)v = (cµv)µ give rise to the following system for the
coefficients Ak, Bn, Dm:
A1 = B1 = D1,
B′
1 = A2B1 − 2B1B2 +B1D2,
B′
2 = A2B2 −B1B3 +B1D3 −B2
2 ,
B′
3 = A2B3 +B1D4 −B2B3, (19)
D′
2 = A2D2 − 3B1B3 + 2B1D3 −B2D2,
D′
3 = A2D3 + 3B1D4 − 2B3D2,
D′
4 = A2D4 +B2D4 −B3D3,
where the prime denotes the derivative with respect to v.
The derivative cv can be expressed from the last equation of (18). Plugging this expression
into the l.h.s. of the second equation of (18) yields
cµµµ
c2µµ
=
(3D1 −B1)c2µ + (2D2 −B2)cµ +D3 −B3
D1c3µ +D2c2µ +D3cµ +D4
, (20)
which can be expanded into simple fractions (see [6] for the general case)
cµµµ
c2µµ
=
k1
cµ − b1
+
k2
cµ − b2
+
k3
cµ − b3
, (21)
where ki(v) are some functions such that k1 + k2 + k3 = 2, and the functions bk(v) are roots of
the cubic polynomial in the last equation of (18):
cvcµµ = D1(cµ − b1)(cµ − b2)(cµ − b3).
Conversely, upon comparing (20) with (21) the above coefficients Ak, Bn, Dm can be expressed
via the functions bk(v) in the symmetric form
−B2/D1 = k1b1 + k2b2 + k3b3, B3/D1 = (1− k1)b2b3 + (1− k2)b1b3 + (1− k3)b1b2,
−D2/D1 = b1 + b2 + b3, D3/D1 = b1b2 + b1b3 + b2b3, −D4/D1 = b1b2b3.
Lemma 2. ki are constants.
Proof. Integrating (21) yields the following equation:
cµµ = b(cµ − b1)k1(cµ − b2)k2(cµ − b3)k3 ,
where b(v) is a function of v alone. Then (see the last equation in (18)) we have
cv =
D1
b
(cµ − b1)1−k1(cµ − b2)1−k2(cµ − b3)1−k3 . (22)
The compatibility condition (cv)µµ = (cµµ)v is satisfied only if ki are constants. The lemma is
proved. �
On Integrability of a Special Class of Hydrodynamic-Type Systems 7
Theorem 2. Under the above substitutions, the system (19) reduces to the form
b′i = D1(1− ki)
∏
j 6=i
(bi − bj), i = 1, 2, 3, (23)
In this case we have
A2 = D′
1/D1 +D1(b1 + b2 + b3)− 2D1(k1b1 + k2b2 + k3b3). (24)
Proof. The coefficient A2 in the form (24) can be expressed from the second equation of (19).
The last three equations of (19) are linear with respect to the first derivatives b′k. Solving
this linear system with respect to bk immediately yields (23). Moreover, the third and fourth
equations of (19) are then automatically satisfied. The theorem is proved. �
Let us choose A1 = B1 = D1 = 1 in this formulation (see formulae (18), (19), (23), (24))
in agreement with the normalization (17). This means that a solution of the system (see (23),
where D1 = 1)
b′i = (1− ki)
∏
j 6=i
(bi − bj), i = 1, 2, 3 (25)
determines the coefficients p, q, r, s of (2+1)-dimensional integrable hydrodynamic-type sys-
tem (5) written in the form (6).
Introducing the “intermediate” independent variable V (v) such that
V ′ = ξV k1(1− V )k3 , (26)
where ξ is an arbitrary constant we obtain the following theorem:
Theorem 3. General solution of system (25) can be written in the form
b2 = b1 + ξV k1−1(1− V )k3 , b3 = b1 + ξV k1−1(1− V )k3−1,
b1 = (1− k1)ξ
∫
V k1−2(1− V )k3−1dV.
Proof. Introduce the auxiliary functions b12 = b2− b1 and b13 = b3− b1. Then the system (25)
reduces to the form
b′1 = (1− k1)b12b13, b′12 = b12[(1− k2)b12 − k3b13],
b′13 = b13[(1− k3)b13 − k2b12]. (27)
The ratio of the last two equations
d ln b12
d ln b13
=
(1− k2)b12 − k3b13
(1− k3)b13 − k2b12
is nothing but a first-order ODE. Substituting the intermediate function V = 1 − b12/b13 into
this ODE reduces to the following quadrature:
d ln b13 = (k3 − 1)d ln(1− V ) + (k1 − 1)d lnV.
Taking into account that b12 = (1− V )b13, one can obtain the equality b12 = ξV k1−1(1− V )k3 ,
where the above quadrature is integrated in the parametric form b13 = ξV k1−1(1 − V )k3−1.
Substituting these expressions for b12 and b13 into the first equation of (27), b′1 = (1− k1)b12b13,
yields the quadrature
db1 = (1− k1)ξV k1−2(1− V )k3−1dV.
The remaining equations in (27) yield (26). The theorem is proved. �
8 M.V. Pavlov and Z. Popowicz
In turn, comparing the expressions for cµµ, cµv, and cvv from (15) with their counterparts (18)
and equating the coefficients at the powers of cµ and cv gives rise the following quadratures:
dp = q[rdv + (D2 −B2 + s)dw], (28)
d ln q =
2s2 + (2D2 −B2)s+D3 −B3
r
dw + (B2 + 2s)dv − d ln r, (29)
where
r =
s3 +D2s
2 +D3s+D4
sw
, (30)
and the function s(v, w) satisfies the Riccati equation
sv = s2 +B2s+B3. (31)
This equation can be reduced to the linear ODE
fv = [k3(b3 − b1) + k2(b2 − b1)]f − 1
by the substitution s = b1 + 1/f(v, w) with the solution given by
f =
1− VW
b2 − b1
,
where W (w) is an integration “constant”. With this in mind, the function r can be found
from (30). In turn, the function W (w) cannot be found from the compatibility conditions
∂v(∂w ln q) = ∂w(∂v ln q) (see (29)) or ∂v(pw) = ∂w(pv) (see (28)). A substitution of q = sw/r
(see (4)) into (29) yields an equation
W ′ = ξ̄W k2 (1−W )k3 (32)
which is similar to (26). Here ξ̄ is an arbitrary constant. Then the two quadratures (28) and (29)
can be performed explicitly.
Thus, the functions p, q, r, s are given by the following expressions:
s = ξ
V k1−1(1− V )k3
1− VW
+ (1− k1)ξ
∫
V k1−2(1− V )k3−1dV,
r = −ξ
2
ξ̄
W 1−k2 (1−W )1−k3
1− VW
V 2k1−1(1− V )2k3−1,
q = − ξ̄
2
ξ
W 2k2−1 (1−W )2k3−1 V
1−k1(1− V )1−k3
1− VW
,
p = ξ̄
W k2−1 (1−W )k3
1− VW
+ (1− k2)ξ̄
∫
W k2−2(1−W )k3−1dW. (33)
This means that the function H(v, w) (see (7)) is determined via its third derivatives (see the
end of Section 2). Thus, we proved that a single function c(µ, v) completely determines a (2+1)-
dimensional quasilinear system (6), and the second function a(µ,w) is determined via the first
derivatives of c:
da =
qcv
s− cµ
dw +
(
p− rq
s− cµ
)
dµ. (34)
In order to compute the function c(µ, v) we integrate the second equation in (18). Indeed,
the equation in question (recall that we have B1 = 1 in this normalization)
∂vcµ = c2µ +B2cµ +B3
On Integrability of a Special Class of Hydrodynamic-Type Systems 9
coincides with (31) up to the replacement cµ ↔ s. Since D1 = 1, the compatibility condition
(cv)µµ = (cµµ)v implies b(v) = 1 in (22). Thus, the derivative cv can also be found. Finally,
substituting (33) and just obtained expressions for cµ, cv into (34) yields the corresponding
dispersionless Lax representation (see (12), (14), (18)) which is now determined by means of the
formulas
cµ = ξ
V k1−1(1− V )k3
1− εV
+ (1− k1)ξ
∫
V k1−2(1− V )k3−1dV,
cV = −ξ
ξ̃
ε1−k2(1− ε)1−k3
V k1−1(1− V )k3−1
1− εV
,
aW = − ξ̃
ξ
ε1−k1(1− ε)1−k3
W k2−1(1−W )k3−1
1− εW
,
aµ = ξ̃
W k2−1(1−W )k3
1− εW
+ (1− k2)ξ̃
∫
W k2−2(1−W )k3−1dW,
where the auxiliary variable ε(µ) is determined by the formula (cf. (32))
ε′(µ) = ξ0ε
k2(1− ε)k3 .
4 Conclusion
As it was mentioned in the Introduction, the integrable (2+1)-dimensional quasilinear sys-
tems of the nonlinear first-order equations (see [5] for details) are determined by the com-
patibility condition (ψt)y = (ψy)t , where in general ψt = a(µ, u1, u2, . . . , uN ) and ψy =
c(µ, u1, u2, . . . , uN ), cf. (8) and (9). An open problem is whether it is possible to construct
hydrodynamic chains (see, for instance, [3]) associated with such (2+1)-dimensional quasilinear
systems. The theory of integrable hydrodynamic chains is much simpler than the theory of inte-
grable (2+1)-dimensional quasilinear equations, because the former still is a theory of integrable
(1+1)-dimensional hydrodynamic-type systems with just one nontrivial extension – allowing for
infinitely many components. Thus, an integrable hydrodynamic chain possesses the properties
that are well-known in the theory of finite-component systems (dispersive or dispersionless),
such as infinite series of conservation laws, infinite series of commuting flows, and infinite series
of Hamiltonian structures.
At least, we can answer the above question regarding the construction of the associated
hydrodynamic chain for the case of (9), but this will be the subject of a separate paper. Moreover,
if we fix the first equation in (9), then an associated integrable hierarchy can be found from the
dispersionless Lax representations (cf. (8) and (9))
ψt = a(ψx, w), ψyN = c
(
ψx, v
1, v2, . . . , vN
)
,
where the first member of this hierarchy is given by (6) and uniquely determined by the disper-
sionless Lax representation (9). This means that infinitely many commuting flows (numbered
by the “times” yk) will be determined. It would be interesting to find the associated hydrody-
namic chains for the case when, instead of the above dispersionless Lax representation, one has
a more general ansatz
ψt = a
(
ψx, w
1, w2, . . . , wM
)
, ψyN = c
(
ψx, v
1, v2, . . . , vN
)
,
where M and N are arbitrary positive integers.
10 M.V. Pavlov and Z. Popowicz
Acknowledgements
We thank Eugeni Ferapontov, Sergey Tsarev and Sergey Zykov for their stimulating and clari-
fying discussions. M.V.P. would like to thank the Institute of Theoretical Physics of Wroc law
University for the hospitality and the Kasa Mianowski Foundation for the financial support
of MVP’s visit to Wroc law making this collaboration possible. MVP is grateful to profes-
sor Boris Dubrovin for a hospitality in SISSA in Trieste (Italy) where part of this work has
been done. MVP was partially supported by the Russian-Italian Research Project (Consortium
E.I.N.S.T.E.IN and RFBR grant 06-01-92053).
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http://arxiv.org/abs/nlin.SI/0310021
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http://arxiv.org/abs/0710.2012
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1 Introduction
2 System in involution
3 Dispersionless Lax representation
4 Conclusion
References
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