Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions
A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of them. Examples of zero curvature representations with 4 × 4 ma...
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| Cite this: | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions / A.G. Meshkov, M.Ju. Balakhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 45 назв. — англ. |
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| author | Meshkov, A.G. Balakhnev, M.Ju. |
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| citation_txt | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions / A.G. Meshkov, M.Ju. Balakhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 45 назв. — англ. |
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| description | A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of them. Examples of zero curvature representations with 4 × 4 matrices are presented.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 4 (2008), 018, 29 pages
Two-Field Integrable Evolutionary Systems of
the Third Order and Their Differential Substitutions
Anatoly G. MESHKOV and Maxim Ju. BALAKHNEV
Orel State Technical University, Orel, Russia
E-mail: a meshkov@orel.ru, maxibal@yandex.ru
Received October 04, 2007, in final form January 17, 2008; Published online February 09, 2008
Original article is available at http://www.emis.de/journals/SIGMA/2008/018/
Abstract. A list of forty third-order exactly integrable two-field evolutionary systems is
presented. Differential substitutions connecting various systems from the list are found. It
is proved that all the systems can be obtained from only two of them. Examples of zero
curvature representations with 4× 4 matrices are presented.
Key words: integrability; symmetry; conservation law; differential substitutions; zero cur-
vature representation
2000 Mathematics Subject Classification: 37K10; 35Q53; 37K20
1 Introduction
We use the term “integrability” in the meaning that a system or equation under consideration
possesses a Lax representation or a zero curvature representation. Such systems can be solved
by the inverse spectral transform method (IST) [1, 2]. Exactly integrable evolution systems are
of interest both for mathematics and applications. In particular, systems of the following form
ut = uxxx + F (u, v, ux, vx, uxx, vxx), vt = a vxxx +G(u, v, ux, vx, uxx, vxx), (1.1)
where a is a constant, excite great interest since about 1980. The paper [3] is devoted to construc-
tion of systems of the form (1.1) among others. Nine integrable systems of the form (1.1) and
their Lax representations have been obtained in the paper. In particular, it contains a complete
list of three integrable systems (1.1) satisfying the conditions a(a − 1) 6= 0 and ord(F,G) < 2.
Here ord = order, ord f < n means that f does not depend on un, vn, un+1, vn+1, . . . . Here and
in what follows, the notations un = ∂nu/∂xn, vn = ∂nv/∂xn are used.
Two of the three mentioned systems can be written in the following form
ut = u3 + v u1, vt = −1
2v3 + uu1 − v v1, (1.2)
ut = u3 + v u1, vt = −1
2v3 − v v1 + u1 (1.3)
and the third system is presented below (see (3.46a)). System (1.2) was found independently
in [4] and the soliton solutions were constructed there. This system is called as the Drinfeld–
Sokolov–Hirota–Satsuma system.
This paper contains two results: (i) a list of integrable systems of the form (1.1) with smooth
functions F , G and a = −1/2; (ii) differential substitutions that allow to connect any equation
from the list with (1.2) or (1.3).
There are many articles dealing with integrable systems, but some of them (see, e.g., [5, 6, 7])
consider multi-component systems. Other papers (see, e.g., [8, 9, 10, 11, 12]) contain two-
component systems reducible to a triangular form. The triangular form is briefly considered
below. There was possibly only one serious attempt [13] to classify integrable systems of the
form (1.1) using the Painlevé test. Unfortunately, fifteen systems presented in [13] contain a large
mailto:a_meshkov@orel.ru
mailto:maxibal@yandex.ru
http://www.emis.de/journals/SIGMA/2008/018/
2 A.G. Meshkov and M.Ju. Balakhnev
number of constants some of which can be removed by scaling and linear transformations. Note
that there are two-field integrable evolutionary systems ut = Au3 + H(u,u1,u2) with a non-
diagonal main matrix A. For example, an integrable evolutionary system with the Jordan main
matrix is found in [14].
Moreover, about 50 two-field integrable systems of the form (1.1) with a = 1 can be extracted
from papers [15, 16, 17, 18, 19] that deal with vector evolutionary equations.
Partial solutions of the classification problem for a = 0 and ordG 6 1 have been obtained
in [20], and in [21] for divergent systems with a 6= 1. A complete list of integrable systems of the
form (1.1) does not exist today because the problem is too cumbersome and the set of integrable
systems is very large.
Our tool is the symmetry method presented in many papers. We shall point out pioneer or
review papers only. In [22] the notions of formal symmetry and canonical conserved density for
a scalar evolution equation are introduced. These tools were applied to classification of the KdV-
type equations in [23]. A complete theory of formal symmetries and formal conservation laws for
scalar equations has been presented in [24]. A generalized theory was developed for evolutionary
systems in [25]. Review paper [26] contains both general theorems of the symmetry method
and classification results on integrable equations: the third and fifth order scalar equations,
Schrödinger-type systems, Burgers-type equations and systems. Review paper [27] is devoted to
higher symmetries, exact integrability and related problems. Peculiarities of systems (1.1) have
been discussed in [21]. For the sake of completeness, the main points of the symmetry method
and some results necessary for understanding of this paper are considered in the Sections 2–4.
Briefly speaking, the symmetry method deals with the so-called canonical conservation laws
Dtρn = Dxθn, Dtρ̃n = Dxθ̃n, n = 0, 1, 2, . . . , (1.4)
where Dt is the evolutionary derivative and Dx is the total derivative with respect to x. In
particular, ρ0 = −Fu2/3, ρ̃0 = −Gv2/(3a). The recursion relations for the canonical conserved
densities ρn and ρ̃n are presented in Section 2. All canonical conserved densities are expressed
in terms of functions F and G. That is why equations (1.4) impose great restrictions on the
forms of F and G. Equations (1.4) are solvable in the jet space iff
EαDtρn = 0, EαDtρ̃n = 0, α = 1, 2, n = 0, 1, 2, . . . (1.5)
(see [28], for example). Here
Eα ≡
δ
δuα
=
∞∑
n=0
(−Dx)n
∂
∂uαn
, (u1 = u, u2 = v),
is the Euler operator.
Conservation law with ρ = Dxχ, θ = Dtχ is called trivial and the conserved density of the
form ρ = Dxχ is called trivial too. This can be written in the form ρ ∈ ImDx, where Im = Image.
If ρ1 − ρ2 ∈ ImDx, then the densities ρ1 and ρ2 are said to be equivalent.
There are a lot of systems in the following form
ut = uxxx + F (u, ux, uxx), vt = a vxxx +G(u, v, ux, vx, uxx, vxx),
satisfying the integrability conditions (1.4). Such systems containing one independent equation
are said to be triangular. It follows from the integrability conditions that the equation for u must
be one of the known integrable equations (KdV, mKdV etc). The second equation is usually
linear with respect to v, vx and vxx. Triangular systems do not possess any Lax representations
and are not integrable in this sense. Therefore triangular systems and those reducible to the
triangular form have been omitted as trivial.
Integrable Evolutionary Systems and Their Differential Substitutions 3
The system of two independent equations
ut = uxxx + F (u, ux, uxx), vt = a vxxx +G(v, vx, vxx),
will be called disintegrated. It is obvious that the disintegrated form is a partial case of the
triangular form. Therefore the disintegrated systems and those reducible to them have been
omitted.
System (1.1) will be called reducible if it is triangular or can be reduced to triangular or
disintegrated form. Otherwise, the system will be called irreducible.
Our computations show that for irreducible integrable systems (1.1) parameter a must belong
to the following set:
A =
{
0, −2, −1
2 , −
7
2 + 3
2
√
5, −7
2 −
3
2
√
5
}
.
These values were found first in [3] and were repeated in [29]. The value of a is always defined
at the end of computations when functions F and G have been found and only some coefficients
are to be specified from the fifth or seventh integrability conditions (see example in Section 3.1).
This means that it is enough to verify conditions (1.5) for n = 0, . . . , 7 and α = 1, 2 to obtain F ,G
and a. But for absolute certainty we have verified conditions (1.5) for n = 8, 9 and α = 1, 2 for
each system.
The presented set A consists of zero and two pairs (a, a−1). The transformation t′ = at,
u′ = v, v′ = u changes the parameter a 6= 0 in (1.1) into a−1. That is why one ought to
consider the values
{
0, −1
2 , −
7
2 + 3
2
√
5
}
of the parameter a. Integrable systems with a = 0
were mentioned above, see also [30]. This paper is devoted to investigation of the case a = −1/2
only. The case a = −7
2 + 3
2
√
5 will be presented in another paper.
Section 2 contains recursion formulas for the canonical densities. The origin of the notion,
some examples and a preliminary classification are considered.
A list of forty integrable systems and an example of computations are presented in Section 3.
Section 4 contains differential substitutions that connect all systems from the list. The
method of computations and an example are considered. It is shown that all systems from the
list presented in Section 3 can be obtained from (1.2) and (1.3) by differential substitutions.
Section 5 is devoted to zero curvature representations. The zero curvature representations
for systems (1.2) and (1.3) are obtained from the Drinfeld–Sokolov L-operators. A method of
obtaining zero curvature representations for other systems is demonstrated.
2 Canonical densities
One of the main objects of the symmetry approach to classification of integrable equations is the
infinite set of the canonical conserved densities. Let us demonstrate how canonical conserved
densities can be obtained the from the asymptotic expansions for eigenfunctions of the Lax
operators. The simplest Lax equations concerned with the KdV equation
ut = 6uux − uxxx
take the following form
ψxx − uψ − µ2ψ = 0, (2.1)
ψt = −4ψxxx + 6uψx + 3uxψ + 4µ3ψ. (2.2)
Here u is a solution of the KdV equation and µ is a parameter. The standard substitution
ψ = exp
(∫
ρ dx
)
4 A.G. Meshkov and M.Ju. Balakhnev
reduces equations (2.1) and (2.2) to the Riccati form
ρx + ρ2 − u− µ2 = 0, (2.3)
∂t
∫
ρ dx = −4(∂x + ρ)2ρ+ 6uρ+ 3ux + 4µ3. (2.4)
Differentiating temporal equation (2.4) with respect to x one can rewrite it, using (2.3), as the
continuity equation:
ρt = ∂x[(2u− 4µ2)ρ− ux]. (2.5)
To construct an asymptotic expansion one ought to set
ρ = µ+
∞∑
n=0
ρn(−2µ)−n. (2.6)
Then equation (2.3) results in the following well known recursion formula [2]
ρn+1 = Dxρn +
n−1∑
i=1
ρiρn−i, n = 1, 2, . . . , ρ0 = 0, ρ1 = −u, (2.7)
and (2.5) results in infinite sequence of conservation laws:
Dtρn = Dx(2uρn − ρn+2), n > 0. (2.8)
We change here ∂t → Dt and ∂x → Dx because u is a solution of the KdV equation. The
obtained conservation laws are canonical. It is easy to obtain several first canonical densities:
ρ2 = −u1, ρ3 = u2 − u2, ρ4 = Dx(2u2 − u2), . . . .
It is shown in [2] that all even canonical densities are trivial. Note that if one chooses an-
other asymptotic expansion, for example, in powers of µ−1 instead of (2.6), then another set of
canonical densities is obtained, which is equivalent to the previous set.
The canonical densities that follow from (2.7) can also be obtained by using the temporal
equation (2.4) only. Indeed, setting ∂t
∫
ρ dx = θ one obtains from (2.4)
−4(∂x + ρ)2ρ+ 6uρ+ 3ux + 4µ3 = θ. (2.9)
Using the same expansions as above
ρ = µ+
∞∑
n=0
ρn(−2µ)−n, θ =
∞∑
n=0
θn(−2µ)−n,
one can obtain from (2.9) the following recursion relation:
ρn+2 = 2uρn + 2
n+1∑
i=0
ρiρn−i+1 − 4
3
n∑
i,j=0
ρiρjρn−i−j − 1
3θn
+ 2Dx
(
ρn+1 −
n∑
i=0
ρiρn−i
)
− 4
3D
2
xρn − uδn,−1 + u1δn0, n = −2,−1, 0, . . . ,
where δi,k is the Kronecker delta. The obtained relation provides ρ0 = 0, ρ1 = −u, ρ2 =
−u1 − θ0/3, etc. As Dtρ0 = Dxθ0 and ρ0 = 0, then θ0 = 0. The higher canonical densities ρn,
Integrable Evolutionary Systems and Their Differential Substitutions 5
n > 2 depend on θn−2. The fluxes θn must be defined now from equations (1.4). For example,
θ1 = u2 − 3u2.
The traditional method to obtain the canonical densities for an evolution system [25]
ut = K(u,ux, . . . , un), u(t, x) ∈ Rm, m > 1, uαk = ∂kxu
α. (2.10)
consists, briefly, in the following. The main idea is to use the linearized equation
(Dt −K∗)ψ = 0 (2.11)
or its adjoint
(Dt + K+
∗ )ϕ = 0 (2.12)
as the temporal Lax equation. Here
(K∗ψ)α =
∑
n,β
∂Kα
∂uβn
Dn
xψ
β, (K+
∗ ϕ)α =
∑
n,β
(−Dx)n
∂Kβ
∂uαn
ϕβ,
Dt =
∂
∂t
+
∑
n,α
Dn
x(Kα)
∂
∂uαn
, Dx =
∂
∂x
+
∑
n,α
uαn+1
∂
∂uαn
.
The spatial Lax operator (formal symmetry) was introduced in [25] as the infinite operator
series
R =
N∑
k=−∞
RkD
k
x, N > 0, (2.13)
commuting with Dt − K∗. Rk are matrix coefficients depending on u,ux, . . . . It was shown
that Tr resR (resR = R−1) is the conserved density for system (2.10). Canonical densities have
been defined by the formulas
ρn = Tr resRn, n = 1, 2, . . . ,
see [26] for details.
Operations with operator series (2.13) are not so simple, therefore we use an alternative
method for obtaining the canonical densities. It was proposed in [32] heuristically and we
present the following explanation (see also [33]).
Observation. One can obtain equation (2.9) from (2.2) by the following substitution
ψ = eω, ω =
∫
ρ dx+ θ dt, (2.14)
where ρ dx+θ dt is the smooth closed 1-form, that is, Dtρ = Dxθ. This implies e−ωDte
ω = Dt+θ,
e−ωDxe
ω = Dx + ρ and so (2.9) follows. Another way to obtain the same equation is to prolong
the operators Dt → ∂t + θ, Dx → ∂x + ρ in (2.2) formally and to set ψ = 1. For systems, one
must set ψα = 1 for a fixed α only.
We shall apply this method to system (1.1) now.
The linearized system (1.1) with prolonged operators Dx → Dx + ρ, Dt → Dt + θ takes the
following form:[
(Dx + ρ)3 + Fu + Fu1(Dx + ρ) + Fu2(Dx + ρ)2 −Dt − θ
]
Ψ1
+
[
Fv + Fv1(Dx + ρ) + Fv2(Dx + ρ)2
]
Ψ2 = 0,
6 A.G. Meshkov and M.Ju. Balakhnev[
Gu +Gu1(Dx + ρ) +Gu2(Dx + ρ)2
]
Ψ1 + a (Dx + ρ)3Ψ2
+
[
Gv +Gv1(Dx + ρ) +Gv2(Dx + ρ)2 −Dt − θ
]
Ψ2 = 0. (2.15)
If one sets here Ψ1 = 1, then the first equation takes the following form
(Dx + ρ)2ρ+ Fu + Fu1ρ+ Fu2(Dx + ρ)ρ− θ
+
[
Fv + Fv1(Dx + ρ) + Fv2(Dx + ρ)2
]
Ψ2 = 0.
It is obvious from this equation that the following forms of the asymptotic expansions are
acceptable:
ρ = µ−1 +
∞∑
n=0
ρn µ
n, θ = µ−3 +
∞∑
n=0
θn µ
n, Ψ2 =
∞∑
n=0
ρn µ
n.
Here µ is a complex parameter. Then, after some simple calculations, the following recursion
relations are obtained (n > −1):
ρn+2 = 1
3θn −
n+1∑
i=0
ρiρn−i+1 − 1
3
n∑
i+j=0
ρiρjρn−i−j − 1
3 Fu1(δn,−1 + ρn)− 1
3 Fu δn,0
− 1
3(Fv + Fv1Dx + Fv2D
2
x)ϕn − 1
3 Fu2
(
Dxρn + 2 ρn+1 +
n∑
i=0
ρiρn−i
)
− 1
3 Fv2
ϕn+2 + 2
n∑
i=0
ρiϕn−i+1 +
n∑
i+j=0
ρiρjϕn−i−j
− 1
3 Fv2
(
2Dx ϕn+1 +
n∑
i=0
ρiDx ϕn−i +Dx
n∑
i=0
ρiϕn−i
)
−Dx
[
ρn+1 + 1
3 Dx ρn + 1
2
n∑
i=0
ρiρn−i
]
− 1
3 Fv1
(
ϕn+1 +
n∑
i=0
ρiϕn−i
)
,
(1− a)ϕn+3 = Guδn,0 +Gu2(Dxρn + 2 ρn+1 +
n∑
i=0
ρiρn−i) +Gu1(δn,−1 + ρn)
−
n∑
i=0
θiϕn−i +Gv ϕn +Gv1
(
Dxϕn + ϕn+1 +
n∑
i=0
ρiϕn−i
)
−Dt ϕn
+Gv2
(
2Dxϕn+1 +
n∑
i=0
ρiDx ϕn−i +Dx
n∑
i=0
ρiϕn−i
)
+Gv2
ϕn+2 +D2
xϕn + 2
n+1∑
i=0
ρiϕn−i+1 +
n∑
i+j=0
ρiρjϕn−i−j
+ aD3
xϕn + 3 aD2
xϕn+1 + 6 a
n+1∑
i=0
ρiDx ϕn−i+1 + 3 a
n+2∑
i=0
ρiϕn−i+2
+ 3 aDxϕn+2 + 3 a
n∑
i+j=0
ρiρj Dx ϕn−i−j + 3 a
n∑
i=0
ϕn−i+1Dx ρi
+ 3
2 a
n∑
i+j=0
ϕn−i−j Dx(ρiρj) + 3 a
n+1∑
i+j=0
ρiρjϕn−i−j+1
Integrable Evolutionary Systems and Their Differential Substitutions 7
+ 3 aDx
n∑
i=0
ρiϕn−i + a
n∑
i=0
ϕn−iD
2
xρi + a
n∑
i+j+k=0
ρiρjρk ϕn−i−j−k.
Here δi,k is the Kronecker delta, Fu1 = ∂F/∂u1 and so on. From the recursion relations it is
obvious why the value a = 1 is singular. Some of initial elements of the sequence {ρn, ϕn} read
ρ0 = −1
3 Fu2 , ϕ0 = 0, ϕ1 =
1
1− a
Gu2 ,
others are introduced via the δ-symbols.
If one sets in (2.15) Ψ2 = 1 and a 6= 0, then one more pair of recursion relations for {ρ̃n, ϕ̃n}
is obtained. These recursion relations give us any desired number of canonical densities. As an
example, we present here some more canonical densities:
ρ0 = −1
3
Fu2 , ρ1 =
1
9
F 2
u2
− 1
3
Fu1 +
1
3 b
Fv2Gu2 +
1
3
Dx Fu2 ,
ρ̃0 = − 1
3 a
Gv2 , ρ̃1 =
1
9 a2
G2
v2 −
1
3 a
Gv1 −
1
3 a b
Fv2Gu2 +
1
3 a
DxGv2 , (2.16)
where b = a − 1. The tilde denotes another sequence of canonical densities. Further canonical
densities are too cumbersome, therefore we do not present them here.
To simplify investigation of the integrability conditions, an additional requirement is always
imposed. This is the existence of a formal conservation law [25, 26]. A formal conservation law
is an operator series N in powers of D−1
x . An equation for the formal conservation law can be
written in the following operator form
(Dt −K∗)N = N (Dt +K∗
+). (2.17)
The form of this equation coincides with the form of the equation for the Noether operator [34].
That is a formal conservation law may be called a formal Noether operator.
If (Dt − K∗, L) is the Lax pair for an equation, then (Dt + K∗
+, L+) is obviously the Lax
pair for the same equation. Hence, canonical densities obtained from (2.11) must be equivalent
to canonical densities obtained from (2.12).
It was shown in [21] that the first sequence of the canonical densities ρn for system (1.1)
obtained from (2.11) is equivalent to the first sequence of the canonical densities τn obtained
from (2.12) and the second sequence of the canonical densities ρ̃n is equivalent to the second
sequence of the canonical densities τ̃n. Hence, ρn − τn ∈ ImDx and ρ̃n − τ̃n ∈ ImDx, or
Eα(ρn − τn) = 0, Eα(ρ̃n − τ̃n) = 0, α = 1, 2, n = 0, 1, 2, . . . . (2.18)
Equations (1.4) (or (1.5)) and (2.18) are said to be the necessary conditions of integrability. We
shall refer to it simply as the integrability conditions for brevity.
Our computations have shown that
τ0 = −ρ0, τ̃0 = −ρ̃0, τ1 = ρ1, τ̃1 = ρ̃1. (2.19)
Other “adjoint” canonical densities τi and τ̃k essentially differ from the “main” canonical den-
sities ρi and ρ̃k. All canonical densities can be obtained using the Maple routines cd and acd
from the package JET (see [36]). These routines generate the “main” and the “adjoint” canon-
ical densities, correspondingly, for almost any evolutionary system (an exclusion is the case of
multiple roots of the main matrix of the system under consideration).
Thus, according to (2.16) and (2.19) we have Fu2 ∈ ImDx and Gv2 ∈ ImDx (a 6= 0). This
implies the following lemma.
8 A.G. Meshkov and M.Ju. Balakhnev
Lemma 1. System (1.1) with a(a − 1) 6= 0 satisfying the zeroth integrability conditions (2.18)
reads
ut = u3 −
3
2f
u2Dx f +
3
4f
fu1u
2
2 + F1(u, v, u1, v1, v2),
vt = av3 −
3a
2g
v2Dx g +
3a
4g
gv1v
2
2 +G1(u, v, u1, v1, u2). (2.20)
where ord(f, g) 6 1.
Indeed, one may set Fu2 = −3/2Dx ln f and Gv2 = −3/2aDx ln g, where ord(f, g) 6 1 because
ord(F,G) 6 2. Then equations (2.20) follow.
From higher integrability conditions one more lemma follows.
Lemma 2. Suppose system (2.20) is irreducible and satisfies the following eight integrability
conditions ρ2−τ2 ∈ ImDx, ρ̃2− τ̃2 ∈ ImDx and Dtρn ∈ ImDx, Dtρ̃n ∈ ImDx, where n = 1, 3, 5.
Then the system must have the following form
ut = u3 −
3
2f
u2Dx f +
3
4f
fu1u
2
2 + f1 v
2
2 + f2 v2 + f3,
vt = av3 −
3a
2g
v2Dx g +
3a
4g
gv1v
2
2 + g1 u
2
2 + g2 u2 + g3, a 6= 0, (2.21)
where ord(f, g, fi, gj) 6 1.
A scheme of the proof has been presented in [21].
3 List of integrable systems
As it is shown in Section 2 the problem of the classification of integrable systems (1.1) is
reduced to investigation of system (2.21). That is why it is necessary to start by investigating
its symmetry properties.
Lemma 3. System (2.21) are invariant under any point transformation of the form
(a) t′ = α3t+ β, x′ = αx+ γt+ δ, α 6= 0, u′ = u, v′ = v,
(b) u′ = h1(u), v′ = h2(v),
and under the following permutation transformation
(c) t′ = at, u′ = v, v′ = u,
where α, β, γ and δ are constants, hi are arbitrary smooth functions.
The classification of systems of type (2.21) has been performed by modulo of the presented
transformations.
Moreover, some systems (2.21) admit invertible contact transformations. An effective tool
for searching such contact transformations is investigation of the canonical conserved densities.
For example, system (3.24) from the next section has the first canonical conserved density of
the following form:
ρ1 =
(
v1 − 2
3ue
v
)2 + 2c21e
−2v.
It is obvious that the best variables for that system are
U = e−v and V = v1 − 2
3ue
v.
Integrable Evolutionary Systems and Their Differential Substitutions 9
This is an invertible contact transformation. In terms of U and V the system takes the following
simple form:
Ut = Dx
(
U2 + 3
2UV1 − 3
4UV
2 + 1
2c
2
1U
3
)
,
Vt = 1
4Dx(V 3 − 2V2)− 3
2c
2
1Dx(2UU1 + U2V ).
If c1 6= 0 this system can be reduced to (3.10) by scaling, otherwise the system is triangular: the
equation for V will be independent single mKdV. Moreover, the equation for U becomes linear.
That is why c1 6= 0 in (3.24).
Canonical densities for the triangular systems contain only one highest order term in the
second power as in the considered example ρ = V 2 or ρ = V 2
x + · · · , or ρ = V 2
xx + · · · etc. Trian-
gular systems and those reducible to the triangular form have been omitted in the classification
process as trivial.
To classify integrable systems (1.1) with a(a − 1) 6= 0 one must solve a huge number of
large overdetermined partial differential systems for eight unknown functions of four variables.
This work has required powerful computers and has taken about six years. All the calculations
have been performed in the interactive mode of operation because automatic solving of large
systems of partial differential equations is still impossible. The package pdsolve from the
excellent system Maple makes errors solving some single partial differential equations. The
package diffalg cannot operate with large systems because its algorithms are too cumbersome.
Thus, one has to solve complicated problems in the interactive mode. Hence, to obtain a true
solution one must enter true data! Under such circumstances errors are probable. The longer
the computations the more probable are errors. This is the reason why we cannot state with
confidence that all computations have been precise all these six years. That is why the statement
on completeness of the obtained set of integrable systems is formulated as a hypothesis.
In this and in the following sections c, ci, k, ki are arbitrary constants.
Hypothesis. Suppose system (2.21) with a = −1/2 is irreducible. If the system has infinitely
many canonical conservation laws, then it can be reduced by an appropriate point transformation
to one of the following systems:
ut = u3 + v u1, vt = −1
2v3 + uu1 − v v1; (3.1)
ut = u3 + v1 u1, vt = −1
2v3 + 1
2(u2 − v2
1); (3.2)
ut = u3 + v u1, vt = −1
2v3 − v v1 + u1; (3.3)
ut = u3 + v u1 + v1 u, vt = −1
2v3 − v v1 + u; (3.4)
ut = u3 + v1 u1, vt = −1
2v3 −
1
2v
2
1 + u; (3.5)
ut = u3 + uu1 + v1, vt = −1
2v3 + 3
2u1u2 − u v1; (3.6)
ut = u3 + v2 + k u1, vt = −1
2v3 + 3
2uu2 + 3
4u
2
1 + 1
3u
3 + k
(
u2 − v1
)
; (3.7)
ut = u3 + 3
2vv2 + 3
4v
2
1 + 1
3v
3 − k
(
v2 + u1
)
, vt = −1
2v3 + u2 + k v1; (3.8)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1, vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1; (3.9)
ut =
(
u2 − 3
2 uv1 −
3
4 uv
2 + 1
4 u
3
)
x
, vt =
(
−1
2 v2 + 3
2uu1 − 3
4u
2v + 1
4v
3
)
x
; (3.10)
ut = u3 − 3
2 v2 −
3
2 u1v1 − 1
2 u
3
1, vt = −1
2 v3 + 3
2
(
v1 − u2 + 1
2 u
2
1
)2 − 3
4 v
2
1; (3.11)
ut =
(
u2 − 3
2 v1 −
3
2 uv −
1
2 u
3
)
x
, vt =
(
−1
2 v2 + 3
2
(
v − u1 + 1
2 u
2
)2 − 3
4 v
2
)
x
; (3.12)
ut = u3 − 3gv2 − 3u1(u1 + v1)− 3
2 v
2
1 − 6v1g2 − c1g
3 − 3g4,
vt = −1
2v3 −
3
4 c1u2 + 3u2
1 − 3
2 v
2
1 − 6u1g
2 + c1g
3 + 3g4, g = u+ v; (3.13)
ut = u3 − 3u1v1 + (u− 3v2)u1, vt = −1
2v3 + 1
2u2 − u1v − (u− 3v2)v1; (3.14)
ut = u3 − 3u1v2 + uu1 − 3u1v
2
1, vt = −1
2v3 + 1
2u1 − u v1 + v3
1; (3.15)
10 A.G. Meshkov and M.Ju. Balakhnev
ut = u3 +
(
k +
√
u2 + v1
)
u1,
vt = −1
2 v3 + 3
8
(2uu1 + v2)2
u2 + v1
− 3uu2 − k(2u2 + v1)− 2
3(u2 + v1)3/2; (3.16)
ut = u3 − 3
4
(2 v v1 + u2)2
v2 + u1
+ 3v v2 + 3
2v
2
1 + 2
3v
3 − k(2v2 + u1),
vt = −1
2v3 + 1
2u2 + k v1; (3.17)
ut = u3 + u1(u1+v2)√
u+v1
− 4
3u1v1 + c1u1
√
u+ v1,
vt = −1
2v3 −
3
2u2 + 3
8
(u1 + v2)2
u+ v1
+ 2
3v
2
1 − 4
3u
2 − 2u1
√
u+ v1 − 2
3c1(u+ v1)3/2; (3.18)
ut = u3 + u1
√
u+ v1 − k u1,
vt = −1
2v3 −
3
2u2 + 3
8
(u1 + v2)2
u+ v1
− 2
3(u+ v1)3/2 + 2ku+ kv1; (3.19)
ut = u3 + uv1 + (u2 + v)u1, vt = −1
2v3 + 3u1u2 − (u2 + v)v1; (3.20)
ut = u3 + 3(u+ k)v2 + 3u1(v1 + u2),
vt = −1
2v3 −
3
2uu2 − 3
2(v1 + u2)2 − 3
4u
2
1 + ku3 + 3
4u
4; (3.21)
ut = u3 − 3
2 v2 −
3
2 u1v1 − 1
2u
3
1 − 3u1(c1eu + 2c2e2u),
vtv = −1
2v3 + 3
2
(
1
2u
2
1 − u2 + v1 + c1e
u + 2c2e2u
)2
− 3
4 v
2
1 − 3
2 c1u2e
u + 3
4 c
2
1e
2u + 2c1c2e3u, c1 6= 0 or c2 6= 0; (3.22)
ut = u3 − 3
2 u1v2 − 3
4 u1v
2
1 + uu1 − c2u1e
−2v,
vt = −1
2v3 + 1
4 v
3
1 + u1 − u v1 + c2v1 e
−2v; (3.23)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + u1 e
v(u1 + 2u v1)− 1
3 u
2u1e
2v − 3
2c
2
1u1 e
−2v,
vt = −1
2v3 + v31
4 + u2 e
v + 1
3u e
2v(2u1 + u v1) + 3
2c
2
1v1 e
−2v, c1 6= 0; (3.24)
ut = u3 + 3u2v1 + 3
2u1v2 + 9
4u1v
2
1 − uu1e
2v − e−3v,
vt = −1
2v3 + 1
4v
3
1 + (u1 + uv1)e2v; (3.25)
ut = u3 + 3u2v1 + 3
2u1v2 + 9
4u1v
2
1 − uu1e
2v − 1
4u1e
−2v,
vt = −1
2v3 + 1
4v
3
1 + (u1 + uv1)e2v + 1
4v1e
−2v; (3.26)
ut = u3 + 3
2u1v2 + 3u2v1 + 9
4u1v
2
1 − c21 u1e
−2v − 1
2 u1e
2v(u2 + c2),
vt = −1
2v3 + 1
4v
3
1 + c21v1e
−2v + 1
2 e
2v(2uu1 + u2v1 + c2v1); (3.27)
ut = u3 + 3
2u1v2 + 3u2v1 + 9
4u1v
2
1 − 1
3 e
2v
(
u1(6u2 + c1) + 4uv1(2u2 + c1)
)
+ ev
(
v2(2u2 + c1) + (u1 + 2u v1)2 + 2c1v2
1
)
,
vt = −1
2v3 + 1
4v
3
1 + 1
3 e
2v
(
4uu1 + (6u2 + c1)v1
)
+ ev (u2 + 2u1v1) ; (3.28)
ut = u3 + 3
2 u1v2 + 3u2v1 + 9
4 u1v
2
1 + 3uv2(c1uev + c2) + c1(c21 − 1)u4e3v
− 3
4 u
2e2v
(
u1(1 + 5c21) + 8c21uv1 + 2c2u(1− 3c21)
)
− 3c22(u1 + 2uv1)
+ 3
2c1e
v(u1 + 2uv1 − 2c2u)2 + 3
2c2v1(2u1 + 3uv1),
vt = −1
2v3 + 1
4v
3
1 + 3
2c1e
v(u2 + 2u1v1) + c1(1− c21)u
3e3v + 6c1c2uev(v1 − c2)
+ 3
4ue
2v
(
2u1(1 + c21) + uv1(1 + 5c21) + 2c2u(1− 3c21)
)
+ 3
2c2v1(2c2 − v1); (3.29)
ut = u3 + 3
2 u1v2 + 3u2v1 + 9
4 u1v
2
1 + 3ev(u2 + c)(v2 + 2v2
1) + 3
2 e
vu1(u1 + 4uv1)
− 3
2e
2v
(
(3u2 + c)u1 + 4(u2 + c)uv1
)
,
vt = −1
2v3 + 1
4 v
3
1 + 3
2 e
v(u2 + 2u1v1) + 3
2e
2v
(
2uu1 + (3u2 + c)v1
)
; (3.30)
Integrable Evolutionary Systems and Their Differential Substitutions 11
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 − c1 e
−2vu1 − c2 (u1 + 2v1) e2(u+v)
+ c3 (u1 − 2v1) e2(v−u),
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + c1 e
−2vv1 +
(
c2 e
2(u+v) − c3 e
2(v−u))v1; (3.31)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 +
(
c2 e
u + c3 e
−u − 3c21 e
−2v
)
u1,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 +
(
c2 e
u − c3 e
−u)u1
+
(
3c21 e
−2v − c2 e
u − c3 e
−u) v1; (3.32)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 + 3 k(u2
1 − 2v2) e−v − 3(c2 − 3k2)u1 e
−2v
+ 8 k(k2 − c2) e−3v,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + 3 k(u2 − 2u1v1) e−v + 3(c2 − 3k2)v1 e−2v; (3.33)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 − 3c21u1e
2(u+v) + 3c1u1(u1 + 2v1)eu+v
+ (c2 e−u − 3c23 e
−2v)u1,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + 3c21(2u1 + v1) e2(u+v) + 3c1(u2 + u2
1)e
u+v
− c2 (u1 + v1)e−u + 3c23v1e
−2v; (3.34)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 + 3c2 u1(u1 + 2v1) eu+v − 4c1c2 e3(u+v)
+ 3[(c1 − c22)u1 + 2c1v1] e2(u+v),
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + 3c2 (u2 + u2
1) e
u+v + 4c1c2 e3(u+v)
+ 3[2c22u1 − (c1 − c22)v1] e
2(u+v); (3.35)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 + 2
3 c
2
1u1 e
−2v + c1(2v2 − u2
1) e
−v
− 2c1c2(u1 + 2v1) eu + 3c2 u1(u1 + 2v1) eu+v − 3c22 u1 e
2(u+v),
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 − 2
3 c
2
1v1 e
−2v + c1(2u1v1 − u2) e−v
− 2c1c2(u1 − v1) eu + 3c2 (u2 + u2
1) e
u+v + 3c22 (2u1 + v1) e2(u+v); (3.36)
ut = u3 − 3
2u1v2 − 3
4 u1v
2
1 + 1
4 u
3
1 − 3u1
[
c21 e
2(u+v) + c22 e
2(v−u) + 2c1c2 e2v
]
− 3c23 u1 e
−2v + 3 c1u1(u1 + 2v1) eu+v − 3 c2u1(u1 − 2v1) ev−u,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + 3c21 (2u1 + v1) e2(u+v) + 6c1c2 v1 e2v
+ 3c22 (v1 − 2u1) e2(v−u) + 3 c1(u2 + u2
1) e
u+v + 3c2 (u2
1 − u2) ev−u + 3c23 v1 e
−2v; (3.37)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 − 6c31 e
3(u+v) − 3
4 c
2
1(5u1 − 8v1) e2(u+v)
+ 9
2 c1u1(u1 + 2v1) eu+v + 2c21c2 e
2u+v + 2
3 c1c
2
2 e
u−v − 1
2 c1c2(7u1 + 12v1) eu
+ c2(2v2 − u2
1) e
−v + 11
12 c
2
2 u1 e
−2v − 2
9 c
3
2 e
−3v,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + 6c31 e
3(u+v) + 3
4 c
2
1(18u1 + 5v1) e2(u+v)
+ 9
2 c1(u2 + u2
1) e
u+v − 4c21c2 e
2u+v + 2
3 c1c
2
2 e
u−v − 7
2 c1c2(u1 − v1) eu
+ c2(2u1v1 − u2) e−v − 11
12 c
2
2 v1 e
−2v; (3.38)
ut = u3 − 3
2u1v2 − 3
4u1v
2
1 + 1
4u
3
1 + c3e
−v(u2
1 − 2v2) + 2
3c
2
3u1e
−2v
+ c1
(
3u1e
u+v + 2c3eu
)
(u1 + 2v1)− c2(3u1e
v−u + 2c3e−u)(u1 − 2v1)
− 3u1(c1eu + c2e
−u)2e2v,
vt = −1
2v3 + 3
2u1u2 − 3
4u
2
1v1 + 1
4v
3
1 + c3e
−v(u2 − 2u1v1)− 2
3c
2
3v1e
−2v
+ 3c21e
2(u+v)(2u1 + v1)− 3c22e
2(v−u)(2u1 − v1) + 3c1eu+v(u2 + u2
1)
− 3c2ev−u(u2 − u2
1) + 2c2c3e−u(u1 + v1) + 2c1c3eu(u1 − v1) + 6c1c2v1e2v; (3.39)
12 A.G. Meshkov and M.Ju. Balakhnev
ut = u3 − 3
4
(2g3 − u2 + 2gv1)2
u1 − g2
+ 3g(u2 − v2)− 6u2
1 − 9u1v1 − 3
2v
2
1
− 3(5g2 + 4cg + c2)u1 − 6g2v1 + 2cg2(8g + 3c) + 9g4,
vt = −1
2v3 + 3
4
(2g3 − u2 + 2gv1)2
u1 − g2
− 3(3g + c)u2 − 3
2v
2
1 + 3(9g2 + 8cg + 2c2)u1
+ 3(6g2 + 4cg + c2)v1 − 2cg2(8g + 3c)− 9g4, g = u+ v; (3.40)
Remark 1. Systems (3.1), (3.3), (3.6) and (3.20) were proposed in [3], where system (3.20) is
given with a misprint. System (3.10) was presented in [29].
Remark 2. Ten pairs of integrability conditions (for ρ0–ρ9 and ρ̃0–ρ̃9) have been verified for
each system (3.1)–(3.40), and nontrivial higher conserved densities with orders 2, 3, 4 and 5
have been found.
Remark 3. It is shown in [21] that systems (3.10) and (3.12) are unique divergent systems of
the form (2.21) that satisfy the integrability conditions.
Remark 4. System (3.22) is a modification of (3.11), systems (3.31)–(3.39) are modifications
of (3.9).
Remark 5. Canonical densities for system (3.25) depend on the nonlocal variable w = D−1
x e−v.
Remark 6. Many of the systems possess discrete symmetries. They are:
u→ −u for (3.1), (3.2), (3.9), (3.10), (3.16), (3.20) and (3.27);
u→ −u, v → v + πi for (3.24);
u→ iu, v → v − i
2π, c1 → −c1 for (3.28);
{u→ −u, v → v + πi} ∪ {v → v + πi, c1 → −c1} ∪ {u→ −u, c1 → −c1} for (3.29);
{u→ −u, v → v + πi} ∪ {u→ iu, v → v − i
2π} for (3.30);
u→ −u, c2 → c3, c3 → c1 for (3.32);
u→ −u, k → −k for (3.33);
u→ −u, c1 → c2, c2 → c1 for (3.37);
u→ −u, c3 → −c3, c2 → c1, c1 → c2 for (3.39).
Also, systems (3.1), (3.2), (3.9), (3.10) and (3.27) preserve the real shape under the transfor-
mation u→ iu. System (3.33) keeps the real shape under the transformation u→ iu, k → ik.
3.1 Example of computations
Let us consider the simplest case of system (1.1):
ut = u3 + f1(u, v)u1 + f2(u, v)v1, vt = av3 + g1(u, v)u1 + g2(u, v)v1, (3.41)
where a(a− 1) 6= 0. Formulas (2.16) are reduced now to the following
ρ0 = 0, ρ̃0 = 0, ρ1 = −1
3
f1, ρ̃1 = − 1
3a
g2. (3.42)
The further canonical densities read
ρ2 = −1
3
(f1,uu1 + f2,uv1) +
1
3
Dxf1, ρ̃2 = − 1
3a
(g1,vu1 + g2,vv1) +
1
3a
Dxg2,
τ2 =
1
3
(f1,uu1 + f2,uv1), τ̃2 =
1
3a
(g1,vu1 + g2,vv1), (3.43)
where indices after commas denote derivatives.
Integrable Evolutionary Systems and Their Differential Substitutions 13
The first integrability condition (1.5) for ρ1 can be split with respect to u3, v3, u2 and v2.
This provides the following equations
f1,uv = 0, f1,uuu = 0, f1,vvv = 0, or f1(u, v) = c1u
2 + c2u+ c3v
2 + c4v + c5.
Analogously, condition (1.5) for ρ̃1 implies g2(u, v) = b1u
2 + b2u+ b3v
2 + b4v + b5. It is obvious
from (3.43) that the second integrability conditions (2.18) are τ2 ∈ ImDx and τ̃2 ∈ ImDx. These
conditions provide f2,uu = g1,vv = 0 or f2(u, v) = uf3(v) + f4(v), g1(u, v) = vg3(u) + g4(u).
Thus, system (3.41) takes the following form:
ut = u3 + (c1u2 + c2u+ c3v
2 + c4v + c5)u1 + (uf3(v) + f4(v))v1,
vt = av3 + (vg3(u) + g4(u))u1 + (b1u2 + b2u+ b3v
2 + b4v + b5)v1. (3.44)
Now one can obtain θ2 = D−1
x Dtρ2 and θ̃2 = D−1
x Dtρ̃2 in an explicit form. The expres-
sions Dtρ1 and Dtρ̃1 are not the total derivatives yet:
Dtρ1 = Dxh1(ui, vj) +R1(u, v, u1, v1) = Dxθ1,
Dtρ̃1 = Dxh̃1(ui, vj) + R̃1(u, v, u1, v1) = Dxθ̃1.
Therefore, we have set θ1 = h1(ui, vj) + q1(u, v), θ̃1 = h̃1(ui, vj) + q̃1(u, v), where q1 and q̃1 are
unknown functions and Dxq1 = R1, Dxq̃1 = R̃1. This trick allows us to evaluate ρ4, τ4, ρ̃4, τ̃4
and verify the fourth integrability conditions (2.18). These conditions imply f ′′3 = g′′3 = 0, hence
f3 = a1v + a2, g3 = a3u+ a4.
To simplify the further analysis one must list all irreducible cases of f1 (or g2). Let us take
f1 = c1u
2 + c2u+ c3v
2 + c4v + c5 for definiteness.
Lemma 4. Using complex dilatations of u and v, translations u→ u+ λ1, v → v + λ2 and the
Galilei transformation ut → ut + αux, vt → vt + αvx one can reduce f1 to one of the following
forms:
1) u2 + v2; 2) u2 + αv; 3) v2 + αu; 4) u+ v; 5) u; 6) v; 7) f1 = 0,
where α is any constant. Moreover, in the cases 4–7 the function g2 must be linear (b1 = b3 = 0)
because otherwise the permutation u↔ v gives one of the cases 1–3.
In cases 1 and 3 contradictions follow from the integrability conditions (1.5) with n = 1, 3
and (2.18) with n = 2, 4. In case 2 the integrability conditions (1.5) with n = 1, 3, 5 and (2.18)
with n = 2, 4 are satisfied iff system (3.44) is reduced to a pair of independent equations. Thus,
a nontrivial integrable system (3.41) must belong to the following class:
ut = u3 + (c2u+ c4v)u1 + (u(a1v + a2) + f4(v))v1,
vt = av3 + (v(a3u+ a4) + g4(u))u1 + (b2u+ b4v + b5)v1, (3.45)
and only the following cases are possible:
4) c2 = c4 = 1; 5) c2 = 1, c4 = 0; 6) c2 = 0, c4 = 1; 7) c2 = c4 = 0.
In case 4 the integrability conditions (1.5) with n = 1, . . . , 5 provide the functions g4 = k1u+k2,
f4 = k3v + k4, the coefficients a3 = 0, a4 = −1 + a2 + b2, b4 = b2 = (a+ 1)a2 − 2a− 1 and the
following equations:
(a+ 1)(2a2 + aa2 − 2a− 1) = 0, a2(a2 − 2)− a(4a2
2 − 5− 7a2) + 16a2 − 14a2
2 − 3 = 0,
14 A.G. Meshkov and M.Ju. Balakhnev
a5(a2 − 2)− a4(4a2
2 − 7a2 − 6)− 2a3(16a2
2 − 5a2 − 37)
− a2(177a2
2 − 224a2 − 41)− a(236a2
2 − 353a2 + 102)− a2(37a2 − 53)− 17 = 0.
Using the package Groebner in Maple, one can obtain a2 = (1 − a)/3, a2 + 7a + 1 = 0 or
a = (3c− 7)/2, c2 = 5. Then, the remaining coefficients are also determined and we obtain
ut = u3 + (u+ v)u1 + 1
2
(
(3− c)u+ (5c− 11)v
)
v1, c2 = 5,
vt = 1
2v3(3c− 7) +
(
(c+ 2)u− v
)
u1 + 1
2(c− 3)(u+ v)v1. (3.46)
The following substitution
V = 1
6(c+ 1)(v − u), U = 1
12(c+ 3)u+ 1
6v
reduces system (3.46) to the third Drinfeld–Sokolov system
Ut = −8U3 + 3V3 + 6(V − 8U)U1 + 12UV1,
Vt = 12U3 − 2V3 + 48V U1 + 12(2U − V )V1 (3.46a)
that has been presented first in [3]. Scaling
t→ −1
2 t, U =→ 1
6U, V =→ −1
3V
gives more symmetric form of system (3.46a)
Ut = 4U3 + 3V3 + (4U + V )U1 + 2UV1,
Vt = 3U3 + V3 − 4V U1 − 2(V + U)V1 (3.46b)
that was found in [14].
In case 5 the equations a1 = a2 = 0, g′4f
′
4 = 0, g′′′4 = 0 follow from the integrability condi-
tions (1.5) with n = 1, . . . , 5. This implies f4 6= 0 because otherwise the first equation of (3.45)
will be independent. Hence, there are two branches (1) f4 = 1 and (2) f ′4 6= 0, g′4 = 0. Along
the first branch, if one use additionally the integrability conditions (1.5) with n = 7 and solves
a large polynomial system for constants, one can obtain the following system:
ut = u3 + uu1 + v1, vt = −2v3 − uv1,
that can be transformed to (1.3) by a scaling.
Along the second branch the integrability conditions (1.5) with n = 1, . . . , 5 provide the
following system
ut = u3 + uu1 − vv1, vt = −2v3 − uv1,
that is equivalent to (1.2).
Case 6 is symmetric to case 5: one can obtain f ′′′4 = 0, g′′4 = 0, a3 = a4 = b2 = 0 from the
integrability conditions (1.5) with n = 1, . . . , 5. Hence g4 6= 0 and we have two branches g4 = 1
or g4 = u. Using the additional integrability conditions (1.5) with n = 6, 7 one can obtain
equations (1.2) and (1.3).
There are many branches in case 7 but all of them provide linear or triangular systems only.
As one can see, classification of integrable systems of the form (3.41) is a sufficiently laborious
task. System (2.21) contains eight unknown functions depending on four variables, therefore
classification of these systems is much more difficult.
Integrable Evolutionary Systems and Their Differential Substitutions 15
4 Differential substitutions
A differential substitution is a pair of equations
u = f(U, V, Ux, Vx, . . . , Un, Vn), v = g(U, V, Ux, Vx, . . . , Un, Vn), (4.1)
where f and g are some smooth functions.
Definition 1. If for any solution (U, V ) of a system (Σ) formulas (4.1) provide a solution (u, v)
of system (1.1), then one says that system (1.1) admits substitution (4.1).
In all cases that we know, the new systems (Σ) belong to the same class (1.1)
Ut = Uxxx + P (U, V, Ux, Vx, Uxx, Vxx), Vt = aVxxx +Q(U, V, Ux, Vx, Uxx, Vxx), (S)
with some smooth functions P and Q. There exist some group-theoretical explanation of this
fact for KdV type equations [35]. Our attempts to introduce another parameter a′ 6= a in (S)
had no success.
Substituting (4.1) into (1.1) one obtains the following equations(
D3
xf + F (f, g,Dxf,Dxg,D
2
xf,D
2
xg)− ∂tf
)
S
= 0,(
aD3
xg +G(f, g,Dxf,Dxg,D
2
xf,D
2
xg)− ∂tg
)
S
= 0. (4.2)
It is obvious that transition to the manifold (S) in (4.2) is equivalent to a replacement of ∂t by
the evolutionary differentiation Dt performed in accordance with (S):
Dtf = D3
xf + F (f, g,Dxf,Dxg,D
2
xf,D
2
xg),
Dtg = aD3
xg +G(f, g,Dxf,Dxg,D
2
xf,D
2
xg), (4.3)
where
Dtf =
n∑
i=1
∂f
∂Ui
Di
x(U3 + P ) +
n∑
i=1
∂f
∂Vi
Di
x(aV3 +Q).
Another way to obtain (4.3) is to differentiate equations (4.1) with respect to t in accordance
with (1.1) and (S) and exclude u and v by using (4.1). This algorithm and many others are
coded in Maple, see for example [36].
To find the admissible functions f , g, P , Q from (4.3) one can use the following easily provable
formula:
∂
∂Uk
Dm
x f =
m∑
s=0
(
m
s
)
Dm−s
x
∂f
∂Uk−s
,
∂f
∂U−i
≡ 0 for i > 0,
and the analogous formula for ∂/∂Vk. Differentiating (4.3) with respect to Un+3 and Vn+3, one
obtains
∂f
∂Vn
= 0,
∂g
∂Un
= 0. (4.4)
Other corollaries of (4.3) are too cumbersome to consider them in the general form.
Let us consider, as an example, the first order differential substitutions for system (1.2)
ut = u3 + v u1, vt = −1
2v3 + uu1 − v v1. (4.5)
16 A.G. Meshkov and M.Ju. Balakhnev
According to (4.4) one has f = f(U, V, U1), g = g(U, V, V1), hence equations (4.3) now read
D3
xf − fU (U3 + P )− fV (Q− V3/2)− fU1(U4 +DxP ) + g Dxf = 0,
gU (U3 + P ) + gV (Q− V3/2) + gV1(DxQ− V4/2) + 1
2 D
3
xg − f Dxf + g Dxg = 0. (4.6)
Differentiating (4.6) with respect to U3 and V3 one can obtain four equations:
∂f
∂U1
∂P
∂U2
= 3Dx
∂f
∂U1
,
∂f
∂U1
∂P
∂V2
=
3
2
∂f
∂V
,
∂g
∂V1
∂Q
∂U2
= −3
2
∂g
∂U
,
∂g
∂V1
∂Q
∂V2
= −3
2
Dx
∂g
∂V1
. (4.7)
Let us consider some corollaries of these equations.
1. If ∂f/∂U1 = 0 and ∂g/∂V1 = 0, then u = f(U), v = g(V ) is a trivial point transformation.
2. If ∂f/∂U1 = 0, then u = f(U) and one can set f(U) = U by modulo of the point
transformation. In this case P = gU1 from the first of equations (4.6).
3. If ∂g/∂V1 = 0, then v = g(V ) and one can set g(V ) = V by modulo of the point transfor-
mation. In this case Q = fDxf − V V1 from the second of equations (4.6).
4. If (∂f/∂U1)(∂g/∂V1) 6= 0, then one can find P and Q as polynomials of U2 and V2 from
equations (4.7).
Investigation of cases 2–4 provides seven nontrivial solutions of equations (4.6) (see below
(3.1) → (3.2), . . . , (3.1) → (3.17)).
Note that integrable system (3.6) admits strange differential substitutions that generate non-
integrable systems. For example, system (3.6) admits the following differential substitution:
u = 3
2V2 − 3
4V
2
1 − 3
2U1e
V ,
v = 9
4
(
−V4 + V1V3 + V 2
1 V2 − 1
4V
4
1 − U2
1 e
2V + eV (U3 + 2U2V1 + 3U1V2)
)
, (4.8)
so that the functions U and V satisfy the following system:
Ut = U3 + 3
2U2V1 + 3
4U1V
2
1 − U1e
−V f ′′(U) + f(U),
Vt = −1
2V3 + 1
4V
3
1 + 3
2e
V (U2 + U1V1)− f ′(U)
with arbitrary function f . This system does not satisfy the integrability conditions (1.4). To
comprehend this unusual phenomenon we evaluate V2, V3 and V4 from the first equation (4.8)
V2 = 2
3u+ U1e
V + 1
2V
2
1 , V3 = 2
3u1 +Dx(U1e
V ) + V1V2,
V4 = 2
3u2 +D2
x(U1e
V ) +Dx(V1V2),
and substitute them into the second one. The result is
v = −u2 − 3
2u2.
It is easily verified that the obtained constraint is a reduction of system (3.6) into the single
KdV equation ut = −1/2u3−uu1. This means that using a substitution like (4.8) we are trying
to construct an integrable system from the single KdV equation. There are other such examples
for system (3.6). Note that the reduction obtained above follows from the reduction u = const
for system (3.3) (see (3.3) and (3.3) → (3.6)).
Integrable Evolutionary Systems and Their Differential Substitutions 17
To organize the presented list of systems we have computed admissible differential substitu-
tions for each system and present the results in this section. The formula
u′ = f(u, v, ux, vx, . . . ), v′ = g(u, v, ux, vx, . . . ) (A) → (B)
will denote that if u′ and v′ are substituted into system (A), then system (B) follows for u and v.
We say in this case that system (B) is obtained from system (A) by the differential substitution.
Substitution (A) → (B) establishes an interrelation between the sets of solutions of sys-
tems (A) and (B): (u, v) 7→ (u′, v′) is a single valued map. And conversely, if for some solu-
tion (u′, v′) of system (A) one solves the system of two ordinary differential equations (A) → (B)
for u and v, then one or more solutions of system (B) are obtained. Of course, explicit solutions
can be obtained very rarely when the substitution is linear or invertible (see below).
Let us consider the following simple example:
u′ = u1, v′ = v1. (3.10) → (3.9)
This substitution is possible for any divergent system
ut =
(
u2 + F (u, v, u1, v1)
)
x
, vt =
(
−1
2v2 +G(u, v, u1, v1)
)
x
.
It produces the system ut = u3 + F (u1, v1, u2, v2), vt = −v2/2 + G(u1, v1, u2, v2) without u0
and v0. The inverse transformation is quasi-local u = D−1
x u′, v = D−1
x v′. This is a well known
fact, that is why the substitutions (u, v) → (u1, v1) are not written for the divergent systems
below. In some cases analogous substitutions are not so obvious and we present them likewise
(3.1) → (3.2), for example.
Theorem 1. Differential substitutions presented below connect all systems from the list of Sec-
tion 3 with systems (1.2) and (1.3). Systems (1.2) and (1.3) are also implicitly connected with
each other.
The proof can be obtained by a direct verification.
List of the substitutions:
u′ = u, v′ = v1; (3.1) → (3.2)
u′ = 3√
2
(u2 − u1v1), v′ = 3
2v2 −
3
4(u2
1 + v2
1); (3.1) → (3.9)
u′ = 3√
2
(u1 − u v), v′ = 3
2v1 −
3
4(u2 + v2); (3.1) → (3.10)
u′ = 3
4
√
2
(
u2
1 − 2u2 + 2v1
)
, v′ = 3
2 v1; (3.1) → (3.11)
u′ = 3
4
√
2
(
u2 − 2u1 + 2v
)
, v′ = 3
2 v; (3.1) → (3.12)
u′ = 3
√
2
(
(u+ v)2 − u1
)
+ 3
16
√
2 c21, v′ = 3v1 + 3
2c1(u+ v); (3.1) → (3.13)
u′ =
√
2u, v′ = −3v1 + u− 3v2; (3.1) → (3.14)
u′ = u, v′ = k +
√
u2 + v1; (3.1) → (3.16)
u′ =
√
v2 + u1, v′ = v − k; (3.1) → (3.17)
u′ =
√
2
(
4
3u+ 3
16c
2
1
)
, v′ =
u1 + v2√
u+ v1
− 4
3v1 + c1
√
u+ v1; (3.1) → (3.18)
u′ = 3
2
√
2
(
1
2u
2
1 − u2 + v1 + c1e
u + 2c2e2u
)
, v′ = 3
2 (v1 + c1e
u); (3.1) → (3.22)
u′ =
√
2u, v′ = −3
2 v2 −
3
4 v
2
1 + u− c2e−2v; (3.1) → (3.23)
u′ = 2c1u, v′ = −3
2v2 −
3
4v
2
1 − 1
3u
2e2v − 3
2c
2
1 e
−2v + ev(u1 + 2uv1); (3.1) → (3.24)
u′ =
√
3u1e
v + 2c1u,
18 A.G. Meshkov and M.Ju. Balakhnev
v′ = 3
2 v2 −
3
4 v
2
1 + 2
√
3 c1v1e−v − 1
2 e
2v(u2 + c2)− c21e
−2v; (3.1) → (3.27)
u′ = 1
3
√
2 ev(3u1 + c1e
v + 2u2ev),
v′ = 3
2v2 −
3
4 v
2
1 − u1e
v + 1
3 e
2v(c1 − 2u2); (3.1) → (3.28)
u′ = 3
2
√
2ev(u1 + 2c2u+ c1u
2ev),
v′ = 3
2 v2 −
3
4 v
2
1 − 3
2c1u1e
v + 3c2v1 − 3
4(2c2 + c1ue
v)2 − 3
4u
2e2v; (3.1) → (3.29)
u′ = 3
2
√
2ev
(
u1 + (c+ u2)ev
)
, v′ = 3
2v2 −
3
4 v
2
1 − 3
2u1e
v + 3
2(c− u2)e2v; (3.1) → (3.30)
u′ = 3
2
√
2 (u2 − u1v1) +
√
2
(
c2 e
u − c3 e
−u − 3c1u1 e
−v) ,
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1) + c2 e
u + c3 e
−u − 6c1v1 e−v − 3c21 e
−2v; (3.1) → (3.32)
u′ = 3
2
√
2 (u2 − u1v1)− 3
√
2(cu1 + 2kv1) e−v − 6ck
√
2 e−2v,
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1)− 3(ku1 + 2cv1) e−v − 3(c2 + k2) e−2v; (3.1) → (3.33)
u′ = 3√
2
(u2 − u1v1)−
√
2
(
c2 e
−u + 6c1c3 eu − 3c1u1 e
u+v + 3c3u1 e
−v) , (3.1) → (3.34)
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1) + c2 e
−u − 3c21 e
2(u+v) − 3c1u1 e
u+v − 6c3v1 e−v − 3c23 e
−2v;
u′ = 3√
2
(u2 − u1v1) + 3
√
2
(
c1 e
2(u+v) + c2u1 e
u+v
)
,
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1) + 3(c1 − c22) e
2(u+v) − 3c2u1 e
u+v; (3.1) → (3.35)
u′ =
√
2
(
3
2 (u2 − u1v1)− 2
3 kc
2
1 e
−2v + 3c2u1 e
u+v + 2kc1c2 eu
)
+ c1
√
2 (ku1 + 2v1)e−v, k2 = 1,
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1)− 2
3 c
2
1 e
−2v − 3c22 e
2(u+v) − 3c2u1 e
u+v
+ 2c1c2 eu + c1(u1 + 2kv1)e−v; (3.1) → (3.36)
u′ = 3
2
√
2 (u2 − u1v1) + 3
√
2
(
c1 e
u+v + c2 e
v−u − c3 e
−v)u1
+ 6c3
√
2 (c2 e−u − c1 e
u),
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1)− 3c21 e
2(u+v) − 3c22 e
2(v−u)
+ 3u1(c2 ev−u − c1 e
u+v)− 6c3v1 e−v − 6c1c2 e2v − 3c23 e
−2v; (3.1) → (3.37)
u′ = 3
2
√
2 (u2 − u1v1) + 1
3
√
2 c22 e
−2v − 1
2
√
2 c2(u1 − 4v1) e−v
− 2
√
2 c1c2 eu + 3
2
√
2 c1(2c1 e2(u+v) + 3u1 e
u+v),
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1)− 15
4 c
2
1 e
2(u+v) − 9
2 c1u1 e
u+v + 5
2 c1c2 e
u
− 5
12 c
2
2 e
−2v + c2(u1 − v1)e−v; (3.1) → (3.38)
u′ = 3
2
√
2 (u2 − u1v1) + 3
√
2u1
(
c1 e
u+v + c2 e
v−u)− 2
3
√
2 c23 e
−2v
+ 2
√
2 c3(c2 e−u − c1 e
u)−
√
2 c3 (u1 + 2v1)e−v,
v′ = 3
2 v2 −
3
4 (u2
1 + v2
1)− 3e2v(c1 eu + c2 e
−u)2 − c3(u1 + 2v1)e−v
+ 2c3(c2 e−u − c1 e
u)− 2
3 c
2
3 e
−2v + 3u1
(
c2 e
v−u − c1 e
u+v
)
; (3.1) → (3.39)
u′ = 3√
2
u2 − 3gv1 − 2g3√
u1 − g2
− 3
√
2 (3g + 2c)
√
u1 − g2,
v′ = 3(v1 − 2cg − c2), g = u+ v; (3.1) → (3.40)
u′ = u, v′ = v1; (3.3) → (3.5)
u′ = 3
2u2 + u2 + v, v′ = u, (3.3) → (3.6)
this substitution is invertible, see below (3.6) → (3.3);
u′ = u1 + 1
2v
2, v′ = v − k; (3.3) → (3.8)
Integrable Evolutionary Systems and Their Differential Substitutions 19
u′ = 4u2 + u1
(
16
3
√
u+ v1 − 2c1
)
+ 16
9 u
2 + 1
2c
2
1u,
v′ = − u1+v2√
u+v1
− 4
3v1 + c1
√
u+ v1; (3.3) → (3.18)
u′ = u1, v′ = v; (3.4) → (3.3)
u′ = u1, v′ = v1; (3.4) → (3.5)
u′ = 3
2u3 + 2uu1 + v1, v′ = u, (3.4) → (3.6)
this substitution has the quasi-local inverse substitution, see below (3.6) → (3.4);
u′ = u2 + vv1, v′ = v − k; (3.4) → (3.8)
u′ = 3u3 − 6(u1v1 + vu2) + 2uu1, v′ = 3v1 − 3v2 + u; (3.4) → (3.14)
u′ = e−v, v′ = 3
2v2 −
3
4v
2
1 − ue2v; (3.4) → (3.25)
u′ = u1 +
√
3 ev(u2 + u1v1),
v′ = 3
2v2 −
3
4v
2
1 − ue2v +
√
3 v1e−v − 1
4e
−2v; (3.4) → (3.26)
u′ = v, v′ = −3
2v2 − v2 + u; (3.6) → (3.3)
u′ = v, v′ = −3
2v2 − v2 + w, wx = u, wt = u2 + uv; (3.6) → (3.4)
u′ = v1, v′ = −3
2v3 − v2
1 + u; (3.6) → (3.5)
u′ = u+ k, v′ = v1 − 1
2v
2; (3.6) → (3.7)
u′ = v − k, v′ = u1 − 3
2v2 −
1
2(v2 + 3k2) + 2kv; (3.6) → (3.8)
u′ = 3u1 − 3
2v1 −
3
4(u2 + v2),
v′ = 9
4v3 −
9
4u(u2 + 2v2) + 9
4vv2 −
9
2u
2
1 + 9
4u
2(2u1 + v1)
− 9
4vv1(2u+ v)− 9
16(u2 − v2)2; (3.6) → (3.10)
u′ = −3u2 + 3
2v1,
v′ = 9
4v3 + 9
2u1v2 − 9
16(u2
1 − 2u2 + 2v1)2 + 9
4(u2
1 + v1)2 − 9
8u
4
1; (3.6) → (3.11)
u′ = −3u1 +
3
2
v,
v′ = 9
4v2 + 9
2uv1 −
9
16(u2 − 2u1 + 2v)2 + 9
4(u2 + v)2 − 9
8u
4; (3.6) → (3.12)
u′ = −3(v1 + 2u1) + 3
2c1g, g = u+ v,
v′ = 9
2v3 + 9
4c1(u2 − v2) + 18gv2 − 9u2
1 + 9v2
1
+ 9(c1g + 2g2)u1 − 9
8c
2
1(u1 + g2) + 36g2v1 + 9g4; (3.6) → (3.13)
u′ = 3v1 + u− 3v2,
v′ = −9
2v3 + 3
2u2 + 9vv2 − 6u1v + 6(3v2 − u)v1 + 3v2(2u− 3v2); (3.6) → (3.14)
u′ = − u1 + v2√
u+ v1
+ c1
√
u+ v1 − 4
3v1,
v′ = 3
2
u3 + v4√
u+ v1
− 3
4c1
u2 + v3√
u+ v1
+ 8
3u
u1 + v2√
u+ v1
+ 9
8
(u1 + v2)3
(u+ v1)5/2
− (u1 + v2)2
u+ v1
− 3
8
u1 + v2
(u+ v1)3/2
(
6(u2 + v3)− c1(u1 + v2)
)
+ 2(v3 + 2u2 + c1v2)
+ 16
9 (u2 − v2
1)− 8
3(v2 − u1 − c1v1)
√
u+ v1 − 1
2c
2
1(u+ 2v1); (3.6) → (3.18)
u′ =
√
u+ v1 − k,
v′ = −3
4
u2 + v3√
u+ v1
+ 3
8
(u1 + v2)2
(u+ v1)3/2
+ 2k
√
u+ v1 − v1 −
u
2
; (3.6) → (3.19)
u′ = i
√
6u1 + u2 + v, v′ = 3
2(u2
1 − v2) + i
√
6uv1 − v2; (3.6) → (3.20)
20 A.G. Meshkov and M.Ju. Balakhnev
u′ = 3(u1 + v1 − ku),
v′ = −9
2v3 −
9
2uu2 + 9(u+ k)v2 − 27
4 u
2
1 − 9v2
1
+ 9u(u+ k)u1 − 9(u2 + k2)v1 − 9
4u
2(u2 + 2k2); (3.6) → (3.21)
u′ = −3u2 + 3
2v1 + 3
2e
u(c1 + 4
√
c2 u1),
v′ = 9
4v3 −
9
4
(
1
2u
2
1 − u2 + v1 + c1e
u + 2c2e2u
)2 + 9
4c1u2e
u + 9
2u1v2 + 9
4v
2
1
+ 9
8u
2
1(u
2
1 + 4v1) + 9
4e
u
(
c1(5u2
1 + 2v1)− 4
√
c2 (v2 + 2u1v1 + u3
1)
)
+ 18c22e
4u
+ 9
4e
2u
(
c1(c1 − 12
√
c2 u1) + 4c2(3u2
1 + 2v1)
)
+ 18c2e3u(c1 − 2
√
c2 u1); (3.6) → (3.22)
u′ = −3u2 − 3
2v2 −
3
4(u2
1 + v2
1)− 2
√
6c2 (u1 + v1)eu+v − c1e
−2v
− c2e
2(u+v) + c3e
2(v−u),
v′ = 9
4v4 −
9
4u1u3 + 9
4v3(2u1 + v1)− 9
2u2(u2 + u2
1)− 9
16(u2
1 − v2
1)
2
+ 9
4v2(u
2
1 + 2u1v1 − v2
1) + 4
√
6c2 v1
(
2c3e3v−u − c1e
u−v)+ c
+ 3
√
6c2
(
v3 − u1u2 + v2(2u1 + v1) + 2u1v1(u1 + v1)
)
eu+v
+ 6v2
(
c2e
2(u+v) + c3e
2(v−u) − c1e
−2v
)
+ 3
2c1(u1 − v1)(u1 − 3v1)e−2v
− c21e
−4v + 4c2c3e4v − 3
2c2(u1 − v1)2e2(u+v) + 3
2c3(u1 − 3v1)2e2(v−u); (3.6) → (3.31)
u′ = 3u2 − 3
2v2 −
3
4(u2
1 + v2
1)− 3ke−v(u1 + 2v1)− 3(c1 + 2k2)e−2v,
v′ = 9
4v4 −
9
4u1u3 + 9
4v3(v1 − 2u1) + 9
2u2(u2
1 − u2)− 9
16(u2
1 − v2
1)
2
+ 9
4v2(u
2
1 − 2u1v1 − v2
1)− 9c21e
−4v + 18kc1e−3v(u1 + 2v1)
+ 9
2e
−2v
(
4k2u2 − 4c1v2 − 2k2(u1 + v1)2 + c1(u1 + v1)(u1 + 3v1)
)
− 9
2ke
−v(u3 + 2v3 − 4u2(u1 + v1) + v2(2v1 − 3u1) + u2
1(u1 + 2v1)
)
,
c1 = c2 − k2; (3.6) → (3.33)
u′ = −3u2 − 3
2v2 −
3
4(u2
1 + v2
1)− 3c21e
2(u+v) − 3c1(3u1 + 2v1)eu+v
+ c2e
−u − ke−2u, k = 3c23,
v′ = 9
4(v4 − u1u3) + 9
4v3(2u1 + v1)− 9
2u2(u2 + u2
1) + 9
4v2(u
2
1 + 2u1v1 − v2
1)
− 9
16(u2
1 − v2
1)
2 − 9c41e
4(u+v) − 18c31(3u1 + 2v1)e3(u+v) + 6c21c2e
u+2v
+ 6kc21e
2u + 2kc2e−u−2v + 6kc1(u1 − 2v1)eu−v + 3
2c2e
−u(u2 + 2v2 + v2
1)
− 9
2c1e
u+v
(
u3 − 2v3 + u2(9u1 + 4v1)− v2(u1 + 2v1) + 4u2
1(u1 + v1)
)
− 3
2ke
−2v
(
4v2 + u1(4v1 − u1)− 3v2
1
)
− k2e−4v + 6c1c2ev(u1 + 2v1); (3.6) → (3.34)
u′ = 3u2 − 3
2v2 −
3
4(u2
1 + v2
1) + 3(c1 − c22)e
2(u+v) + 3c2(u1 + 2v1)eu+v,
v′ = 9
4v4 −
9
4u1u3 + 9
4v3(v1 − 2u1) + 9
2u2(u2
1 − u2)− 9
16(u2
1 − v2
1)
2
+ 9
4v2(u
2
1 − 2u1v1 − v2
1)− 9kc22e
4(u+v) + 18kc2e3(u+v)(u1 + 2v1)
+ 9
2e
2(u+v)
(
2c22u2 + 4c1v2 + c1(u1 + 3v1)2 − c22(u
2
1 + 4u1v1 + 5v2
1)
)
− 9
2c2e
u+v
(
u3 + 2v3 + u2(u1 + 4v1) + v2(2v1 − u1)
)
, k = c22 − 2c1; (3.6) → (3.35)
u′ = 3u2 − 3
2v2 −
3
4(u2
1 + v2
1)− 3c22e
2(u+v) + 3c2(u1 + 2v1)eu+v
+ 2c1c2eu + c1e
−v(u1 + 2v1)− 2
3c
2
1e
−2v,
v′ = 9
4v4 −
9
4u1u3 + 9
4v3(v1 − 2u1) + 9
2u2(u2
1 − u2)− 9
16(u2
1 − v2
1)
2
+ 9
4v2(u
2
1 − 2u1v1 − v2
1)− 9c42e
4(u+v) + 18c32e
3(u+v)(u1 + 2v1)
+ 12c1c32e
3u+2v + 9
2c
2
2e
2(u+v)(2u2 − u2
1 − 4u1v1 − 5v2
1)
− 4c21c
2
2e
2u + c21e
−2v
(
2u2 − (u1 + v1)2
)
Integrable Evolutionary Systems and Their Differential Substitutions 21
− 9
2c2e
u+v
(
u3 + 2v3 + u2(u1 + 4v1) + v2(2v1 − u1)
)
− 6c1c22e
2u+v(u1 + 2v1)− 3c1c2eu(3u2 + 2v2 + 2u1v1 + 3v2
1)
+ 3
2c1e
−v(u3 + 2v3 − 4u2(u1 + v1) + v2(2v1 − 3u1) + u2
1(u1 + 2v1)
)
; (3.6) → (3.36)
u′ = 3u2 − 3
2v2 −
3
4(u2
1 + v2
1)− 15
4 c
2
1e
2(u+v) + 9
2c1(u1 + 2v1)eu+v
+ 5
2c1c2e
u + c2e
−v(u1 + 2v1)− 5
12c
2
2e
−2v,
v′ = 9
4v4 −
9
4u1u3 + 9
4v3(v1 − 2u1) + 9
2u2(u2
1 − u2)− 9
16(u2
1 − v2
1)
2
+ 9
4v2(u
2
1 − 2u1v1 − v2
1)− 81
16c
4
1 e
4(u+v) + 27
4 c
3
1 e
3(u+v)(u1 + 2v1)
+ 27
4 c
3
1c2e
3u+2v + 9
8c
2
1e
2(u+v)(18u2 + 16v2 − 5u2
1 − 12u1v1 − 9v2
1)
− 27
8 c
2
1c
2
2 e
2u + 3
4c1c
3
2 e
u−2v − 9
4c1c
2
2 e
u−v(u1 + 2v1)
− 27
4 c1e
u+v
(
u3 + 2v3 + u2(u1 + 4v1) + v2(2v1 − u1)
)
− 3
4c1c2e
u(17u2 + 14v2 + 12u1v1 + 19v2
1) + 1
2c
3
2e
−3v(u1 + 2v1)
+ 1
8c
2
2e
−2v
(
16u2 + 12v2 − 11u2
1 − 28u1v1 − 17v2
1
)
− 1
16c
4
2e
−4v
+ 3
2c2e
−v(u3 + 2v3 − 4u2(u1 + v1) + v2(2v1 − 3u1) + u2
1(u1 + 2v1)
)
; (3.6) → (3.38)
u′ = 3
2g3 − u2 + 2gv1√
u1 − g2
+ 6g
√
u1 − g2 − 6u1 − 3v1 − 3c(2g + c), g = u+ v,
v′ = 9
2v3 + 9(3g + c)u2 + 9(2g + c)v2 − 9
4
(2g3 − u2 + 2gv1)2
u1 − g2
+ 9(2u1 + v1)2
− 18u2
1 + 18
√
u1 − g2
(
v2 + 2(2g + c)(v1 + g2)
)
+ 36(2g + c)(u1 − g2)3/2
− 9g2u1 − 18
(
(g + c)2 + 2g2
)
v1 − 36cg
(
(g + c)2 + g2
)
− 45g4; (3.6) → (3.40)
u′ = v − 2k, v′ = u− 3
2v1 + 4
3k
3t; (3.7) → (3.8)
this substitution is invertible:
u′ = 3
2u1 + v − 4
3k
3t, v′ = u+ 2k; (3.8) → (3.7)
u′ = −1
2v, v′ =
√
u+ v1; (3.8) → (3.19)
u′ = c1
√
2 e−v, v′ = −v1 + 2
3u e
v; (3.10) → (3.24)
u′ = uev, v′ = v1 − c1ue
v − 2c2; (3.10) → (3.29)
u′ = (u+
√
−c )ev, v′ = v1 + (
√
−c− u)ev; (3.10) → (3.30)
u′ = u1 + 2k e−v, v′ = v1 − 2c e−v; (3.10) → (3.33)
u′ = u1 + ceu+v, v′ = v1 − keu+v,
where c and k are roots of z2 − 2c2z + 2c1 = 0; (3.10) → (3.35)
u′ = u1 − 2
3c1 e
−v, v′ = v1 − 2c2 eu+v + 2
3c1 e
−v; (3.10) → (3.36)
u′ = u1 + 2c1 eu+v − 2
3 c2 e
−v, v′ = v1 − c1 e
u+v + 1
3 c2 e
−v; (3.10) → (3.38)
u′ = 2
√
u1 − (u+ v)2, v′ = 2(u+ v + c); (3.10) → (3.40)
u′ = −2v, v′ = 2
3u− 2(v1 + v2); (3.12) → (3.14)
this substitution is invertible:
u′ = 3
2(v − u1) + 3
4u
2, v′ = −1
2u; (3.14) → (3.12)
u′ = 3
2(u1 − uv), v′ = 1
2(u− v); (3.14) → (3.10)
u′ = 3(u+ v)2 − 3u1 + 3
16c
2
1, v′ = 1
4c1 − u− v; (3.14) → (3.13)
u′ = u, v′ = v1; (3.14) → (3.15)
u′ = 4
3u+ 3
16c
2
1, v′ = 1
4c1 −
2
3
√
u+ v1; (3.14) → (3.18)
22 A.G. Meshkov and M.Ju. Balakhnev
u′ = u, v′ = 1
2 v1 + 1√
3
ce−v; (3.14) → (3.23)
u′ = c1
√
2u, v′ = 1
2v1 −
1
3u e
v −
√
2
2 c1e
−v; (3.14) → (3.24)
u′ = −u1e
v − 1
3 e
2v
(
2u2 + c1
)
, v′ = −1
2v1 + 1
3
√
−2c1 ev; (3.14) → (3.28)
u′ = u1e
v + 1
3 e
2v
(
2u2 + c1
)
, v′ = −1
2v1 + 2
3 u e
v; (3.14) → (3.28)
u′ = 3
2e
v(u1 + 2c2u+ c1u
2 ev), v′ = −1
2v1 + 1
2(c1 + 1)u ev + c2; (3.14) → (3.29)
u′ = 9
4(u+ v)2 − 9
4 u1, v′ = 9
4 u; (3.18) → (3.13)
this substitution is invertible:
u′ = 4
9 v, v′ = 2
3
√
u+ v1 − 4
9 v. (3.13) → (3.18)
The graph of the substitutions is very cumbersome, therefore we show the most interesting
subgraph only.
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3.1
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3.13 �
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3.18
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@
@
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3.63.3
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@
@@I
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Q
Q
QQs�
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6
3.5
3.4
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A
A
A
A
A
A
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3.8
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Fig. 1. A subgraph of the differential substitutions.
Comments. (1) System (3.1) coincides with (1.2) and (3.3) coincides with (1.3).
(2) It was simpler to obtain some systems from (3.6) than (3.3) or vice versa. These systems
are connected by the second order invertible substitution, see (3.6) → (3.3) and (3.3) → (3.6).
Hence, each system obtained from (3.6) can be obtained from (3.3) and vice versa. (3) Fifteen
systems (3.9)–(3.15), (3.18), (3.22), (3.33)–(3.36), (3.38) and (3.40) can be obtained from both
(3.1) and (3.6) by the presented differential substitutions. Twelve systems (3.2), (3.16), (3.17),
(3.23), (3.24), (3.27)–(3.30), (3.32), (3.37) and (3.39) can be obtained from (3.1). The remaining
eleven systems (3.3)–(3.5), (3.7), (3.8), (3.19)–(3.21), (3.25), (3.26) and (3.31) can be obtained
from (3.6) or (3.3).
Remark 7. As the systems (3.1) and (3.3) have the Lax representations, then all systems from
the list have the Lax representations in a generalized meaning (see Section 5).
Remark 8. Some of the presented substitutions are superpositions of lower order substitutions,
other substitutions are irreducible.
Remark 9. System (3.1) admits the first and second order substitutions and does not admit the
third and fourth order substitutions. Probably it does not admit any higher order substitutions,
Integrable Evolutionary Systems and Their Differential Substitutions 23
either. Systems (3.4) and (3.6) admit the substitutions from the first till fourth orders. We
do not present higher order substitutions for (3.4) because simpler substitutions exist for (3.6).
Fifth and higher order substitutions for systems (3.4) and (3.6) have not been computed because
the computations are extremely cumbersome.
Remark 10. There are some additional differential substitutions under the constraints for
constants in the systems. For example, there are substitutions (3.1) → (3.31) for c1 = 0 and
(3.6) → (3.32) for c3 = 0 or (3.6) → (3.39) for c2 = 0 and so on. These substitutions are not so
important and we do not present them here.
Remark 11. Unexpectedly, the well known systems (3.1) and (3.3) are implicitly connected as
it is shown in Fig. 1.
5 Examples of zero curvature representations
The IST method for nonlinear equations with two independent variables is based on investigation
of a linear overdetermined system
L(u, λ, ∂x)ψ = 0, ψt = A(u, λ, ∂x)ψ, (5.1)
where L and A are ordinary linear operators, u is a smooth (vector) function satisfying a non-
linear partial differential equation E(u) = 0 and λ is a parameter. The operators L, A may be
both scalar and matrix. The operators L, A and E may be also pseudodifferential or integro-
differential. The compatibility condition of system (5.1) reads(
∂L
∂t
+ LA
)
ψ
Lψ=0
= 0.
There are two ways to satisfy this condition. The first operator condition was introduced by
P.D. Lax [37]:
∂L
∂t
+ LA = AL, or
∂L
∂t
= [A,L]. (5.2)
The second more general operator condition was introduced in [38], see also [39]:
∂L
∂t
+ LA = BL. (5.3)
If an equation E(u) = 0 is equivalent to equation (5.2), then (5.2) is said to be the Lax
representation of the equation E(u) = 0. The pair of operators (L,A) is said to be the (L,A)-
pair or the Lax pair.
If an equation E(u) = 0 is equivalent to equation (5.3), then (5.3) is said to be the (L,A,B)
representation of the equation E(u) = 0 or the triad representation.
In all cases, operator L must essentially depend on the parameter λ. This parameter cannot
be removed by a gauge transformation L→ f−1Lf with some smooth function f , in particular.
If system (5.1) is differential, then the standard substitution ψ = Ψ1, ψx = Ψ2 and so on,
provides the following first order system:
Ψx = UΨ, Ψt = VΨ, (5.4)
where U and V are square matrices depending on u and λ. The compatibility condition of linear
system (5.4) reads
Ut − Vx + [U, V ] = 0 (5.5)
24 A.G. Meshkov and M.Ju. Balakhnev
if E(u) = 0. Usually a stronger condition is required: (5.5) is valid iff E(u) = 0. In this
case equation (5.5) is said to be the zero curvature representation. For an evolutionary system
ut = K(u,ux, . . . ,un), u = {uα} the matrix U usually depend on u only, but it may depend
on u, ux, uxx, and so on. Let us consider the general case.
If some smooth functions F = F (u,ux, . . . ,ur) and Φ = Φ(u,ux, . . . ,up) satisfy the condi-
tion (
∂tF + Φ
)
ut=K
= 0, (5.6)
then one obtains
Φ
ut=K
= Φ, ∂tF
ut=K
=
∂F
∂uαi
∂tu
α
i
ut=K
=
∂F
∂uαi
∂ixu
α
t
ut=K
=
∂F
∂uαi
Di
xK
α,
where the summation over i = 0, . . . , r and α = 1, . . . ,m is implied. This implies
Φ = − ∂F
∂uαi
Di
xK
α
according to (5.6). Using this result one obtains the following identity:
∂tF + Φ ≡ ∂F
∂uαi
Di
x(u
α
t −Kα), ∀u (5.7)
for any F and Φ satisfying (5.6).
Let us apply identity (5.7) to equation (5.5). If the matrix U depends on u only then
Ut − Vx + [U, V ] =
∂U
∂uα
(uαt −Kα), ∀u. (5.8)
It is obvious now that equation (5.5) is equivalent to ut = K iff the matrices ∂U/∂uα, α =
1, . . . ,m are linearly independent. Suppose now that the matrix U depends on u and ux then
one obtains
Ut − Vx + [U, V ] =
∂U
∂uα
(uαt −Kα) +
∂U
∂uαx
Dx(uαt −Kα), ∀u. (5.9)
If the matrices ∂U/∂uα, ∂U/∂uβx, α, β = 1, . . . ,m are linearly independent, then equation (5.5) is
equivalent to ut = K again. Otherwise, equation (5.5) would be equivalent to some differential
consequence of the system ut = K that is a more general system than the original one. In this
case we call equation (5.5) the generalized zero curvature representation.
It is well known that equations (5.4) and (5.5) are covariant under the following transforma-
tion
Ψ̄ = S−1Ψ, Ū = S−1(US − Sx), V̄ = S−1(V S − St), (5.10)
where S is any non-degenerate matrix. This transformation is called a gauge one. Any gauge
transformation is invertible and preserves compatibility of system (5.4).
Two (L,A)-pairs were proposed for system (1.2) in [3]. One of these (L,A)-pairs coincides
with the (L,A)-pair that was presented in [4]. The L-operator of the common (L,A)-pair takes
the form L = (∂2
x + f)(∂2
x − g), where
f = 1
6(u
√
2− 2v), g = 1
6(u
√
2 + 2v).
Integrable Evolutionary Systems and Their Differential Substitutions 25
The temporal Lax equation reads ψt = Aψ, where A is a fractional degree of L. The spatial
Lax equation Lψ = λ2ψ can be transformed into the system (∂2
x − g)ψ = λϕ, (∂2
x + f)ϕ = λψ
and then into the normal form (5.4), were
U =
0 1 0 0
(u
√
2− 2v)/6 0 λ 0
0 0 0 1
λ 0 −(u
√
2 + 2v)/6 0
,
V =
(u1
√
2 + v1)/6 −(u1
√
2 + v1)/3 0 −2λ
f1 + f2 −(u1
√
2 + v1)/6 λv/3 0
0 −2λ −(u1
√
2− v1)/6 (u1
√
2− v1)/3
λv/3 0 f1 − f2 (u1
√
2− v1)/6
. (5.11)
Here f1 and f2 take the following form:
f1 = 1
18(3v2 − 2u2 + 2v2)− 2λ2, f2 =
√
2
18 (3u2 + uv).
Matrices (5.11) realize the zero curvature representation of system (1.2).
System (1.3) also has two Lax representations (see [3]). Using the simpler L-operator, we
have found, similarly to the previous case, the following matrices that realize the zero curvature
representation of system (1.3):
Ũ =
0 1 0 0
−v/3− λ 0 1 0
0 0 0 1
u/9 0 λ− v/3 0
,
Ṽ =
v1/6 2λ− v/3 0 −2
h− λv/3 −v1/6 v/3 0
u1/9 −2u/9 v1/6 −2λ− v/3
uv/27 + u2/9 −u1/9 h+ λv/3 −v1/6
, (5.12)
where
h = 1
6v2 + 1
9(v2 − 2u)− 2λ2.
Let us consider an admissible differential substitution u = f(ũi, ṽj), v = g(ũi, ṽj) of sys-
tem (1.1). Substituting u and v in the matrices U(u, v) and V (ui, vj) one obtains
Û(ũi, ṽj) = U(f, g), V̂ (ũi, ṽj) = V (Dk
xf,D
l
xg).
As Û depends on derivatives of ũ or ṽ, then one has a generalized zero curvature representation.
To obtain an ordinary zero curvature representation one can try to remove higher order
derivatives from the matrix Û using the gauge transformation (5.10). But this is not always
possible (see example B below).
A. Performing the substitution (3.1) → (3.9) into the matrix U from (5.11) one obtains
Û =
0 1 0 0
(u2 − v2)/2 + h1 0 λ 0
0 0 0 1
λ 0 −(u2 + v2)/2 + h2 0
,
26 A.G. Meshkov and M.Ju. Balakhnev
where
h1 = 1
4(u1 − v1)2, h2 = 1
4(u1 + v1)2.
One can easily verify that matrices A = Ûu2 and B = Ûv2 are commutative, hence the system
Sx = (Auxx + Bvxx)S has the following solution S = exp(Aux + Bvx). The matrix U1 = ¯̂
U
evaluated according to (5.10) takes the following form
U1 =
1
2
ux − vx 2 0 0
0 vx − ux 2λ 0
0 0 −ux − vx 2
2λ 0 0 ux + vx
.
Now another gauge transformation is possible with the following diagonal matrix:
S1 = exp
(∫
diag (U1)dx
)
,
where diag (U1) is the main diagonal of U1. This gauge transformation provides the following
matrix
U2 =
0 ev−u 0 0
0 0 λe−v 0
0 0 0 eu+v
λe−v 0 0 0
.
A corresponding V -matrix can be obtained by solving equation (5.5) directly or by the previous
twofold gauge transformation. This matrix takes the following form
V2 =
0 ev−uf1 λe−u(u1 + v1) −2λev
−2λ2eu−v 0
λ
4
e−v(2v2 − 3u2
1 + v2
1) λeu(u1 − v1)
λeu(v1 − u1) −2λev 0 eu+vf2
λ
4
e−v(2v2 − 3u2
1 + v2
1) −λe−u(u1 + v1) −2λ2e−u−v 0
,
where
f1 = −u2 − 1
2v2 + u1v1 + 1
4(u2
1 + v2
1), f2 = u2 − 1
2v2 − u1v1 + 1
4(u2
1 + v2
1).
The matrices U2 and V2 realize the zero curvature representation of system (3.9).
B. Substitution (3.1) → (3.17) reduces matrix U from (5.11) to the following form
Û =
0 1 0 0
(k − v)/3 +R 0 λ 0
0 0 0 1
λ 0 (k − v)/3−R 0
,
where R =
√
2(u1 + v2)/6. It is obvious that one cannot remove u1 from Û by a gauge trans-
formation. It is clear from the structure of the matrix Û that
Ut − Vx + [U, V ] = A
(
kv1 + 1
2u2 − 1
2v3 − vt
)
+BDx
(
u3 − 3
4
(2 v v1 + u2)2
v2 + u1
+ 3v v2 + 3
2v
2
1 + 2
3v
3 − k(2v2 + u1)− ut
)
,
Integrable Evolutionary Systems and Their Differential Substitutions 27
where A and B are some linearly independent matrices. Thus, this zero curvature representation
for system (3.17) is generalized. Of course, one may introduce here the new variable u′ = u1 to
obtain an ordinary zero curvature representation. But we do not know if it is always possible.
C. Performing the substitution (3.3) → (3.6) into the matrix U from (5.12) one obtains
ˆ̃U =
0 1 0 0
−u/3− λ 0 1 0
0 0 0 1
u2/6 + (u2 + v)/9 0 λ− u/3 0
.
The first gauge transformation is performed using S1 = exp(u1(∂
ˆ̃U/∂u2)):
S1 =
1 0 0 0
0 1 0 0
0 0 1 0
u1/6 0 0 1
.
The transformed U -matrix is
Ũ1 =
0 1 0 0
−u/3− λ 0 1 0
u1/6 0 0 1
(u2 + v)/9 −u1/6 λ− u/3 0
.
The second gauge transformation is performed using S2 = exp(u (∂Ũ1/∂u1)):
S2 =
1 0 0 0
0 1 0 0
u/6 0 1 0
0 −u/6 0 1
.
The result of the twofold gauge transformation is
Ũ2 =
0 1 0 0
−u/6− λ 0 1 0
0 −u/3 0 1
u2/36 + v/9 0 λ− u/6 0
,
Ṽ2 =
−u1/6 2λ 0 −2
f3 − λu/3 −u1/6 u/3 0
(uu1 − v1)/18− λu1/3 −u2/3− u2/6− 2v/9 u1/6 −2λ
f4 (v1 − uu1)/18− λu1/3 f3 + λu/3 u1/6
,
where
f3 = −1
6u2 − 1
18u
2 − 2
9v − 2λ2, f4 = 1
18(uu2 − v2 + u2
1) + 1
108u
3 − 1
27uv −
2
3uλ
2.
Matrices Ũ2 and Ṽ2 realize the zero curvature representation of system (3.6).
6 Conclusion
The examples in Section 5 illustrate the fact that some systems possess ordinary zero curva-
ture representation while others possess generalized zero curvature representation. All these
28 A.G. Meshkov and M.Ju. Balakhnev
representations are obtained from the Drinfeld–Sokolov L, A operators by using corresponding
differential substitutions listed in Section 4. Matrices U and V that realize all zero curvature
representations have the size 4 × 4. Thus, the two-field evolutionary systems presented above
are integrable in principle by the inverse spectral transform method. But the fact is that the
inverse scattering problem for differential equations with order more than two is extremely
difficult. That is why other methods for solution of equations may be useful [40]. They may be
Bäcklund transformations [41], Darboux transformations [42, 43], Hirota method [44] or numeric
simulating (see [45], for example).
Acknowledgments
We are grateful to Professor V.V. Sokolov for helpful discussions. This work was supported by
Federal Agency for Education of Russian Federation, project # 1.5.07.
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1 Introduction
2 Canonical densities
3 List of integrable systems
3.1 Example of computations
4 Differential substitutions
5 Examples of zero curvature representations
6 Conclusion
References
|
| id | nasplib_isofts_kiev_ua-123456789-148971 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-11-30T16:38:06Z |
| publishDate | 2008 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Meshkov, A.G. Balakhnev, M.Ju. 2019-02-19T12:15:45Z 2019-02-19T12:15:45Z 2008 Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions / A.G. Meshkov, M.Ju. Balakhnev // Symmetry, Integrability and Geometry: Methods and Applications. — 2008. — Т. 4. — Бібліогр.: 45 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37K10; 35Q53; 37K20 https://nasplib.isofts.kiev.ua/handle/123456789/148971 A list of forty third-order exactly integrable two-field evolutionary systems is presented. Differential substitutions connecting various systems from the list are found. It is proved that all the systems can be obtained from only two of them. Examples of zero curvature representations with 4 × 4 matrices are presented. We are grateful to Professor V.V. Sokolov for helpful discussions. This work was supported by Federal Agency for Education of Russian Federation, project # 1.5.07. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions Article published earlier |
| spellingShingle | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions Meshkov, A.G. Balakhnev, M.Ju. |
| title | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions |
| title_full | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions |
| title_fullStr | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions |
| title_full_unstemmed | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions |
| title_short | Two-Field Integrable Evolutionary Systems of the Third Order and Their Differential Substitutions |
| title_sort | two-field integrable evolutionary systems of the third order and their differential substitutions |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148971 |
| work_keys_str_mv | AT meshkovag twofieldintegrableevolutionarysystemsofthethirdorderandtheirdifferentialsubstitutions AT balakhnevmju twofieldintegrableevolutionarysystemsofthethirdorderandtheirdifferentialsubstitutions |