On Integration of One Class of Systems of Lax-Type Equations
A nonlinear system of Lax-type equations is studied. The system is the basis of the construction of triangular models for commutative systems of linear non-selfadjoint bounded operators. Some of its solutions for n = 4 are described. In one of the cases, the general solution is explicitly expressed...
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irk-123456789-1179832017-05-28T10:24:17Z On Integration of One Class of Systems of Lax-Type Equations Lunyov, A.A. Oliynyk, E.V. A nonlinear system of Lax-type equations is studied. The system is the basis of the construction of triangular models for commutative systems of linear non-selfadjoint bounded operators. Some of its solutions for n = 4 are described. In one of the cases, the general solution is explicitly expressed in terms of special (elliptic) functions. Исследуется нелинейная система уравнений типа Лакса, которая лежит в основе построения треугольных моделей для коммутативных систем линейных несамосопряженных ограниченных операторов. Описаны некоторые ее решения при n = 4. В одном из случаев общее решение явно выражается в терминах специальных (эллиптических) функций. Досліджується нелінійна система рівнянь типу Лакса, яка лежить в основі побудови трикутних моделей для комутативних систем лінійних несамоспряжених обмежених операторів. Описано деякі її розв'язки при n = 4. В одному з випадків загальний розв'язок явно виражається в термінах спеціальних (еліптичних) функцій. 2015 Article On Integration of One Class of Systems of Lax-Type Equations / A.A. Lunyov, E.V. Oliynyk // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 45-62— Бібліогр.: 7 назв. — англ. 1812-9471 MSC2000: 47A48, 47N20, 34G20 DOI: 10.15407/mag11.01.045 http://dspace.nbuv.gov.ua/handle/123456789/117983 en Журнал математической физики, анализа, геометрии Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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A nonlinear system of Lax-type equations is studied. The system is the basis of the construction of triangular models for commutative systems of linear non-selfadjoint bounded operators. Some of its solutions for n = 4 are described. In one of the cases, the general solution is explicitly expressed in terms of special (elliptic) functions. |
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Lunyov, A.A. Oliynyk, E.V. |
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Lunyov, A.A. Oliynyk, E.V. On Integration of One Class of Systems of Lax-Type Equations Журнал математической физики, анализа, геометрии |
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Lunyov, A.A. Oliynyk, E.V. |
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Lunyov, A.A. |
| title |
On Integration of One Class of Systems of Lax-Type Equations |
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On Integration of One Class of Systems of Lax-Type Equations |
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On Integration of One Class of Systems of Lax-Type Equations |
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On Integration of One Class of Systems of Lax-Type Equations |
| title_full_unstemmed |
On Integration of One Class of Systems of Lax-Type Equations |
| title_sort |
on integration of one class of systems of lax-type equations |
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Фізико-технічний інститут низьких температур ім. Б.І. Вєркіна НАН України |
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On Integration of One Class of Systems of Lax-Type Equations / A.A. Lunyov, E.V. Oliynyk // Журнал математической физики, анализа, геометрии. — 2015. — Т. 11, № 1. — С. 45-62— Бібліогр.: 7 назв. — англ. |
| series |
Журнал математической физики, анализа, геометрии |
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AT lunyovaa onintegrationofoneclassofsystemsoflaxtypeequations AT oliynykev onintegrationofoneclassofsystemsoflaxtypeequations |
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2025-07-08T13:06:58Z |
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Journal of Mathematical Physics, Analysis, Geometry
2015, vol. 11, No. 1, pp. 45–62
On Integration of One Class of Systems of Lax-Type
Equations
A.A. Lunyov
Institute of Applied Mathematics and Mechanics of the NAS of Ukraine
74, R. Luxemburg Str., Donetsk 83114, Ukraine
E-mail: A.A.Lunyov@gmail.com
E.V. Oliynyk
V.N. Karazin Kharkiv National University
4, Svobody Sq., Kharkiv 61022, Ukraine
E-mail: elenaoliynik@gmail.com
Received February 12, 2013, revised May 15, 2014
A nonlinear system of Lax-type equations is studied. The system is the
basis of the construction of triangular models for commutative systems of
linear non-selfadjoint bounded operators. Some of its solutions for n = 4 are
described. In one of the cases, the general solution is explicitly expressed in
terms of special (elliptic) functions.
Key words: triangular models, nonlinear differential equations, commu-
tative systems of linear non-selfadjoint operators.
Mathematics Subject Classification 2010: 47A48, 47N20, 34G20.
1. Introduction
As is well known from [1, 5], an approach based on Lax’s idea to write the
initial nonlinear equation in the form L′(x) = i[L(x), A(x)], where L,A are some
differential operators, is the main method of integration of nonlinear equations.
It is established that if the Lax pair {L,A} is found for a nonlinear equation,
then this equation can be ’integrated’.
In this paper, we study the system of equations
d
dx
(σ2 + λγ(x)) = i [a(x), σ2 + λγ(x)] , γ(0) = γ+, x ∈ [0, l], λ ∈ C, (1.1)
c© A.A. Lunyov and E.V. Oliynyk, 2015
A.A. Lunyov and E.V. Oliynyk
which is equivalent to the special system of the Lax-type equations
[a(x), γ(x)] = 0, x ∈ [0, l],
γ′(x) = i[a(x), σ2], x ∈ [0, l],
γ(0) = γ+
(1.2)
where a(x) is a spectral matrix measure, γ(x), σ2, γ
+ are selfadjoint n×n matrices,
and
a(x) > 0, tr a(x) ≡ 1, x ∈ [0, l]. (1.3)
System (1.2) appears in the construction of triangular models for commutative
systems of non-selfadjoint bounded operators.
The purpose of this paper is to describe and study all the pairs of matrix
functions {a(x), γ(x)}, the solutions of system (1.2) (when n = 4), for the given
selfadjoint n× n matrices γ+ and σ2 such that
γ(x) ∈ AC([0, l];Cn×n), a(x) ∈ L1([0, l];Cn×n), (1.4)
and (1.3) takes place.
In [7], general qualities of the solutions of system (1.2) for the case n = 3 are
obtained and the descriptions of all solutions of this system for different cases are
given. The idea of paper [7] is used in this paper. This idea lies in the fact that in
the case when γ+ (and so γ(x) also) has a simple spectrum, a(x) is a polynomial
of γ(x) of no higher degree than n− 1 (with scalar coefficients depending on x).
When n = 4 and the matrices σ2, γ+ have a simple spectrum, the explicit
form of the solution in terms of elliptic functions is obtained (see Theorem 2.2
and Corollary 2.3). In Example 2.4, the solutions expressed by trigonometric
functions are found.
When studying cubic dependency of a(x) from γ(x) (n = 4), the explicit
form of the solution is also expressed in terms of special (elliptic) functions (see
Theorem 3.2 and Corollary 3.3).
2. Description of the Solutions of System (1.2)
Proposition 2.1. Let
σ2 = diag(b1, . . . , bn), γ+ = α1σ2 + α0I + iC, (2.1)
where α1, α0 ∈ R, the matrix C = (cjk)n
j,k=1 = −C∗ and cjj = 0 when j ∈
{1, . . . , n}.
Further, let κ0, κ1, κ2 ∈ L1[0, l] be real functions. Then the pair {a(·), γ(·)},
where a(x) = κ2(x)γ(x)2 + κ1(x)γ(x) + κ0(x), x ∈ [0, l] and γ(·) = (γjk(·))n
j,k=1,
46 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
is the solution of system (1.2) if and only if the equalities
γjj(x) = γ+
jj , j ∈ {1, . . . , n}, (2.2)
γjk(x) = iei(bj−bk)(K1(x)+(γ+
jj+γ+
kk)K2(x))yjk(x), j 6= k, (2.3)
hold as x ∈ [0, l], where
Kj(x) :=
x∫
0
κj(t)dt, j ∈ {1, 2}, (2.4)
and the functions yjk(·), j 6= k, satisfy the system
y′jk(x) = (bk − bj)κ2(x)
n∑
s=1
s6=j,k
yjs(x)ysk(x), x ∈ [0, l], j 6= k,
ykj(x) = −yjk(x), x ∈ [0, l], j 6= k,
yjk(0) = cjk, j 6= k.
(2.5)
Besides, if cjk ∈ R, j 6= k, every solution of system (2.5) is real.
P r o o f. Since a(x) commutes with γ(x), then system (1.2) has the form
{
γ′(x) = i[κ2(x)γ(x)2 + κ1(x)γ(x), σ2], x ∈ [0, l],
γ(0) = γ+.
(2.6)
By Lemma 3.12 from [7], in view of the diagonal form of σ2, (2.23) is true for
every solution of system (2.6). Taking this into account, system (2.6) takes the
form
γ′jk(x) = i(bj − bk)(κ1(x) + κ2(x)(γjj + γkk))
+ i(bj − bk)κ2(x)
n∑
s=1
s6=j,k
γjs(x)γsk(x), x ∈ [0, l], j 6= k,
γkj(x) = γjk(x), x ∈ [0, l], j 6= k,
γjk(0) = γjk, j 6= k.
(2.7)
We search the solution of this system in the form of (2.24). In view of (2.1),
γjj = α1bj + α0, j ∈ {1, . . . , n}, holds. Therefore formula (2.24) becomes
γjk(x) = iEj(x)/Ek(x)yjk(x), Ej(x) := eibj(K1(x)+(α1bj+2α0)K2(x)). (2.8)
After substituting formula (2.8) into system (2.7), it is easy to check that it is
equivalent to (2.5).
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 47
A.A. Lunyov and E.V. Oliynyk
Now let the matrix C be real. We are to prove that every solution of sys-
tem (2.5) is real. Let {yjk(·)}j 6=k be some (complex-valued) solution of sys-
tem (2.5). Let
ujk(·) := Re yjk(·), vjk(·) := Im yjk(·), j 6= k. (2.9)
By separating the imaginary part from the equations of problem (2.5) and taking
into account that cjk ∈ R, j 6= k, we obtain the following system on the functions
vjk(·):
v′jk = (bk − bj)κ2(x)
n∑
s=1
s 6=j,k
(uskvjs + ujsvsk), x ∈ [0, l], j 6= k,
vjk(0) = 0, j 6= k.
(2.10)
System (2.10) is a Cauchy problem for the system of linear ordinary differential
equations with zero initial data. Therefore, by the uniqueness theorem, vjk(·) = 0,
j 6= k, which signifies the reality of the solution {yjk(·)}j 6=k.
Theorem 2.2. Let n = 4,
b4 < b1 < b2 < b3, b1 + b2 = b3 + b4, (2.11)
α3 :=
b3 − b2
b3 − b1
, α4 :=
b4 − b2
b4 − b1
. (2.12)
Further, let
cjk ∈ R, cjk = −ckj , j, k ∈ {1, 2, 3, 4}, (2.13)
c1k > 0, k ∈ {2, 3, 4}, (2.14)
c23 =
√
α3 · c13, c24 = −√α4 · c14, c34 = 0. (2.15)
Next suppose
β3 :=
b2 − b1
b3 − b1
, β4 :=
b2 − b1
b4 − b1
, (2.16)
α :=
c14
c13
, β := β3 + β4α
2, (2.17)
F (y) :=
y∫
c13
du
u
√
c2
12 + β(u2 − c2
13)
, (2.18)
ρ :=
√
(b3 − b1)(b3 − b2), (2.19)
v(x) := F−1 (ρK2(x)) , (2.20)
48 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
where F−1(·) is the function inverse to the function F (·). Let (y−0 , y+
0 ) ⊂ R be
the largest by inclusion interval containing the number c13, and the inequality
c2
12 + β(y2 − c2
13) > 0, y−0 < y < y+
0 , (2.21)
be true. Further, let κ0, κ1κ2 ∈ L1[0, l] be real functions and the functions K1(x),
K2(x) be given by the equalities Kj(x) :=
∫ x
0 κj(t)dt, j = 1, 2, and
F (y−0 ) < ρK2(x) < F (y+
0 ), x ∈ [0, l). (2.22)
Then the pair {a(·), γ(·)}, where a(x) = κ2(x)γ(x)2+κ1(x)γ(x)+κ0(x), x ∈ [0, l],
is the solution of system (1.2) if and only if the equalities
γjj(x) = γ+
jj , j ∈ {1, . . . , n}, (2.23)
γjk(x) = iei(bj−bk)(K1(x)+(γ+
jj+γ+
kk)K2(x))yjk(x), j 6= k, (2.24)
hold as x ∈ [0, l], where the functions yjk, j 6= k are given by
y12(x) =
√
c2
12 + β(v2(x)− c2
13), (2.25)
y13(x) = v(x), (2.26)
y14(x) = αv(x), (2.27)
y23(x) =
√
α3 v(x), (2.28)
y24(x) = −√α4 αv(x), (2.29)
y34(x) = 0, (2.30)
ykj(x) = yjk(x), 1 6 j < k 6 4. (2.31)
P r o o f. First check that the set {y12(·), y13(·), y14(·), y23(·), y24(·), y34(·)}
is the solution of system (2.5). Inequality (2.22) implies that the function v(·),
given by formula (2.20), is correctly defined on the segment [0, l], besides, v(x) ∈
(y−0 , y+
0 ), x ∈ [0, l). Therefore the equality
F (v(x)) = ρK2(x), x ∈ [0, l],
is true. Differentiating it, we obtain
v′(x)F ′(v(x)) = ρκ2(x), x ∈ [0, l].
Hence, taking into account (2.18), (2.25), (2.26), (2.28), we obtain
y′13(x) = v′(x) = ρκ2(x)v(x)
√
c2
12 + β(v2(x)− c2
13)
= ρκ2(x)v(x)y12(x) (2.32)
=
ρ√
α3
κ2(x)y12(x)y23(x)
= (b3 − b1)κ2(x)y12(x)y23(x), x ∈ [0, l]. (2.33)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 49
A.A. Lunyov and E.V. Oliynyk
Note that accordingly to (2.11), the equality α3α4 = 1 is true. Therefore,
√
α3(b1 − b3)√
α4
= α3(b1 − b3) = b2 − b3 = b4 − b1,
and in virtue of (2.26), (2.27), (2.33),(2.28), (2.29), we have
y′14(x) = αy′13(x) = α(b3 − b1)κ2(x)y12(x)y23(x)
=
√
α3(b1 − b3)√
α4
κ2(x)y12(x)y24(x)
= (b4 − b1)κ2(x)y12(x)y24(x), x ∈ [0, l]. (2.34)
Then we have
y′23(x) =
√
α3 y′13(x) =
√
α3 (b3 − b1)κ2(x)y12(x)y23(x)
= α3(b3 − b1)κ2(x)y12(x)y13(x)
= (b3 − b2)κ2(x)y12(x)y13(x), x ∈ [0, l], (2.35)
y′24(x) = −√α4 y′14(x) = −√α4 (b4 − b1)κ2(x)y12(x)y24(x)
= α4(b4 − b1)κ2(x)y12(x)y14(x)
= (b4 − b2)κ2(x)y12(x)y14(x), x ∈ [0, l]. (2.36)
From (2.25) and (2.32), we obtain
y′12(x) =
βv(x)v′(x)
y12(x)
= ρβκ2(x)v2(x), x ∈ [0, l]. (2.37)
Further, taking into account (2.26)–(2.29), we have
y13(x)y23(x) + y14(x)y24(x) = (
√
α3 −√α4α
2)v2(x), x ∈ [0, l]. (2.38)
Note that in virtue of (2.11) the equality
ρ =
√
(b3 − b2)(b3 − b1) =
√
(b4 − b2)(b4 − b1) (2.39)
is true. Therefore, since b4 < b1 < b3, then
ρβ = ρ(β3 + β4α
2) =
√
(b3 − b2)(b3 − b1)
b2 − b1
b3 − b1
+
√
(b4 − b2)(b4 − b1)
b2 − b1
b4 − b1
α2
= (b2 − b1)
(√
b3 − b2
b3 − b1
−
√
b4 − b2
b4 − b1
α2
)
= (b2 − b1)
(√
α3 −√α4 α2
)
. (2.40)
50 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
Now the relations (2.37)–(2.40) yield
y′12(x) = (b2 − b1)κ2(x)(y13(x)y23(x) + y14(x)y24(x)), x ∈ [0, l]. (2.41)
Since α3α4 = 1, then in virtue of (2.26)–(2.30),
(b4 − b3)κ2(x)(y13(x)y14(x) + y23(x)y24(x))
= (b4 − b3)κ2(x)(y13(x)y14(x)−√α3 y13(x)
√
α4 y14(x))
= 0 = y′34(x). (2.42)
Now the equalities (2.41), (2.33), (2.34), (2.35), (2.36), (2.42) imply that the
functions y12, y13, y14, y23, y24, y34 satisfy the equations of system (2.5). Further,
since c12 > 0, then
y13(0) = v(0) = F−1(ρK2(0)) = F−1(0) = c13, (2.43)
y12(0) =
√
c2
12 + β(v2(0)− c2
13) = c12, (2.44)
y14(0) = αv(0) =
c14
c13
c13 = c14, (2.45)
y23(0) =
√
α3 y13(0) =
√
α3 c13 = c23, (2.46)
y24(0) = −√α4 y14(0) = −√α4 c14 = c24, (2.47)
y34(0) = 0 = c34. (2.48)
Thus the functions y12(·), y13(·), y14(·), y23(·), y24(·), y34(·) satisfy also the initial
data of system (2.5). General uniqueness theorems for the initial problem imply
that this system has the unique solution on [0, l].
We will show the way of finding expressions for yks(·), k, s ∈ 1, . . . , n (2.25)
– (2.31) as solutions of the corresponding system of differential equations (2.5).
Let now the functions y12(·), y13(·), y14(·) y23(·), y24(·), y34(·) ∈ AC[0, l] sat-
isfy problem (2.5). Multiply equation (2.33) by (b3 − b2)y13(x) and equation
(2.35) by (b3− b1)y23(x). After summarizing the results, we arrive at the relation
y′23(x)y23(x)− α3y
′
13(x)y13(x) = 0, integrating which we obtain
y23(x) =
√
c2
23 + α3(y2
13(x)− c2
13). (2.49)
Taking into account (2.15), we have (2.28). Following the same procedure with
(2.34) and (2.36), we get y24(x) =
√
c2
24 + α4(y2
14(x)− c2
14). After using (2.15),
we obtain
y24(x) = −√α4 y14(x). (2.50)
Use (2.49) and (2.50) for the equations (2.33) and (2.34) to get
y′13(x) = κ2(x)(b3 − b1)y12(x)
√
α3 y13(x), (2.51)
y′14(x) = κ2(x)(b4 − b1)y12(x)
√
α4 y14(x). (2.52)
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 51
A.A. Lunyov and E.V. Oliynyk
Express κ2(x)y12(x) from (2.51) and taking into account (2.39) equate the ob-
tained relations
y′13(x)
y13(x)
=
y′14(x)
y14(x)
. (2.53)
Thus (2.53) yields
y14(x) =
c14
c13
y13(x) = αy13(x). (2.54)
Thus the correctness of equality (2.27) is confirmed. After substituting it into (2.50),
the correctness of (2.29) is also confirmed. By using the obtained results, we
transform the equations (2.41) and (2.33), subject to (2.40),
y′12(x) = κ2(x)(b2 − b1)y2
13(x)
(√
α3 − α2√α4
)
= κ2(x)ρβy2
13(x); (2.55)
y′13(x) = κ2(x)ρy12(x)y13. (2.56)
Whence we obtain that
y′12(x)
y13(x)β
=
y′13(x)
y12(x)
(2.57)
or
y12(x) =
√
c2
12 + β(y2
13(x)− c2
13). (2.58)
Thus (2.25) takes place. Substituting the obtained expressions for y12(x) (2.25)
and y23(x) (2.28) into the equation for y′13(x) (2.33), we obtain
y′13(x) = κ2(x)ρy13(x)
√
c2
12 + β(y2
13(x)− c2
13), (2.59)
or, using the notation (2.19),
y′13(x)
y13(x)
√
c2
12 + β(y2
13(x)− c2
13)
= ρκ2(x). (2.60)
It is easy to see that (2.60), subject to notation (2.18), becomes (2.26).
Corollary 2.3. In the conditions of Theorem 2 suppose c12 = 0 which
determines the form of the matrix C in (2.1) in the following way:
iC =
0 0 c13 c14
0 0 c23 c24
c13 c23 0 0
c14 c24 0 0
.
52 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
Then for the functions y12(·), y13(·) the following representations take place:
y12(x) = c13
√
β tan(z(x)), β > 0, (2.61)
y13(x) =
c13
cos(z(x))
, β > 0, (2.62)
or
y12(x) = c13
√
β, β < 0, (2.63)
y13(x) = 0, (2.64)
or
y12(x) = c13
√
β th(z(x)), β < 0, (2.65)
y13(x) =
c13
ch(z(x))
, β < 0, (2.66)
where
z(x) = c13ρ
√
β K2(x). (2.67)
P r o o f. Consider the case β > 0. The definition of the function F (·) and
formula (2.26) imply that
ρK2(x) = F (y13(x)) =
1√
β
y13(x)∫
c13
du
u
√
u2 − c2
13
. (2.68)
Then (2.68) has the form
−c13
√
β rhoK2(x) = arcsin
∣∣∣∣
c13
y13(x)
∣∣∣∣−
π
2
.
Using notation (2.67), we obtain
− sin(z(x)) = sin
(
arcsin
∣∣∣∣
c13
y13(x)
∣∣∣∣−
π
2
)
= sin
(
arcsin
∣∣∣∣
c13
y13(x)
∣∣∣∣
)
cos
π
2
− cos
(
arcsin
∣∣∣∣
c13
y13(x)
∣∣∣∣
)
sin
π
2
= −
√
1− c2
13
y2
13(x)
.
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 53
A.A. Lunyov and E.V. Oliynyk
Suppose z(x) is such that sin(z(x)) and cos(z(x)) are positive, then we ob-
tain (2.62). Substituting (2.62) into (2.25)–(2.30), we will come to the form
of (2.61) for y12(·) and to the corresponding expressions for the functions y14(·),
y23(·), y24(·), y34(·).
In the case β < 0, in virtue of inequality (2.21), we obtain that y2
13(x)−c2
13 < 0.
Thus,
ρK2(x) = F (y13(x)) =
1√
β
y13(x)∫
c13
du
u
√
c2
13 − u2
. (2.69)
Whence it follows that
−z(x) = ln
(√
c2
13 − u2 + c13
y13(x)
)
or √
c2
13 − y2
13(x) = y13(x) exp−z(x)−c13.
As a result of transformations, we have
y13(x)
(
y13(x)(exp−2z(x) +1)− 2 exp−z(x) c13
)
= 0,
i.e., either y13(x) = 0, which results in the solutions of (2.63)–(2.64), or y13(x) =
2c13
expz(x) +exp−z(x)
, which corresponds to the solutions in the forms of (2.65)
and (2.66).
E x a m p l e 2.4. Let
γ+ =
0 0 2i i
0 0
√
2i −√2i
−2i −√2i 0 0
−i
√
2i 0 0
, σ2 =
0 0 0 0
0 b 0 0
0 0 2b 0
0 0 0 −b
, b > 0. (2.70)
Then the pair {a(x), γ(x)}, where a(x) = γ2(x), is the solution of system (1.2) if
and only if
γ(x) =
0
√
2 tg(2bx)
2
cos(2bx)
1
cos(2bx)
−√2 tg(2bx) 0
√
2
cos(2bx)
−
√
2
cos(2bx)
− 2
cos(2bx)
−
√
2
cos(2bx)
0 0
− 1
cos(2bx)
√
2
cos(2bx)
0 0
. (2.71)
54 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
3. Case of Cubic Dependency of a(x) from γ(x)
Remind the statement proved in [7].
Theorem 3.1. Let
γ+ = diag(l1In1 , . . . , lrInr), n1 + . . . nr = n, (3.1)
where l1, . . . , lr are different real numbers.
Then for every solution {a(·), γ(·)} of system (1.2) there exists a unique uni-
tary matrix function U ∈ AC([0, l]; Cn×n) such that
γ(x) = U(x)γ+U∗(x), x ∈ [0, l], U(0) = In, (3.2)
and the matrix function C(·) := −iU−1(·)U ′(·) is self-adjoint and has the zero
block diagonal relative to the decomposition Cn = Cn1 ⊕ . . .⊕ Cnr . Besides,
a(x) = U(x)A(x)U∗(x), x ∈ [0, l], (3.3)
where
A(·) = diag(A1(·), . . . , Ar(·)) = A∗(·), (3.4)
Aj ∈ L1([0, l];Cnj×nj ), j ∈ {1, . . . , r}. (3.5)
Moreover, for B(x) := U∗(x)σ2U(x),
[C(x), γ+] = [A(x), B(x)], x ∈ [0, l], (3.6)
B′(x) = i[B(x), C(x)], x ∈ [0, l], B(0) = σ2 (3.7)
take place.
Conversely, if for the self-adjoint matrix functions
A,C ∈ L1([0, l]; Cn×n), B ∈ AC([0, l]; Cn×n), (3.8)
(3.4), (3.5), (3.6), (3.7) take place, and U ∈ AC([0, l]; Cn×n) is the solution of
the initial problem
U ′(x) = iU(x)C(x), x ∈ [0, l], U(0) = In, (3.9)
then U(x) is unitary for every x ∈ [0, l],
B(x) = U∗(x)σ2U(x), x ∈ [0, l], (3.10)
and the pair {a(·), γ(·)}, given by (3.2), (3.3), is the solution of system (1.2).
Theorem provides the following step-by-step procedure [7] for finding all the
solutions of system (1.2):
1. Choose an orthonormal basis in Cn, in which the matrix γ+ has the diagonal
form (3.1).
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 55
A.A. Lunyov and E.V. Oliynyk
2. Choose an arbitrary matrix function A(·) satisfying conditions (3.4)–(3.5).
3. Solve the Cauchy problem for the nonlinear system of ordinary differen-
tial equations on the matrix B obtained from (3.6), (3.7) in the previous
remark.
4. If it has the global solution on the segment [0, l], then we calculate the
matrix C(x) by the formula
Cjk(x) = (lk − lj)−1(Aj(x)Bjk(x)−Bjk(x)Ak(x)),
x ∈ [0, l], j, k ∈ {1, . . . , r}, j 6= k, (3.11)
supposing that its diagonal blocks are equal to zero.
5. Find U(·) as a unique solution of the Cauchy problem (3.9) for the system
of linear ordinary differential equations.
6. Finally, we obtain the solution {a(·), γ(·)} of system (1.2) by the formu-
las (3.2), (3.3).
Using Theorem 3.1, we will show how to find the explicit form of the matrix B(x)
for the case n = 4 and a(x) = κ(x)γ3(x), where κ(x) is a real function such that
κ ∈ L1[0, l].
Theorem 3.2. Let n = 4, λ1, . . . , λ4 be different real numbers such that λ3
and λ4 are between λ1 and λ2,
γ+ = diag(λ1, . . . , λ4),
σ2 = (b(0)
jk )4j,k=1 = iβjk, βjk > 0, βjk ∈ R as j 6= k,
b
(0)
jj = β1λj + β0, j ∈ {1, 4},
C(x) = (cjk(x))4j,k=1, besides cjj = 0,
a(x) = κ(x)γ3(x),
where a(x) is a matrix function and κ(x) ∈ L1[0, l] is a real function, x ∈ [0, l],
and the matrix B is such that B(x) = (bjk(x))4j,k=1 = B∗(x) as j 6= k.
Let further
αj =
√
λ2 − λj
λj − λ1
, βj =
λ2 − λ1
λj − λ1
, j = 3, 4, (3.12)
α =
√
(λ4 − λ1)(λ2 − λ4) (λ1 + λ2 + λ4)√
(λ3 − λ1)(λ2 − λ3) (λ1 + λ2 + λ3)
, (3.13)
56 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
ψ(y) =
√√√√β2
12 − β3(y2 − β2
13)− β4β2
14
[(
y
β13
)2α
− 1
]
, (3.14)
F (y) =
y∫
β13
dt
tψ(t)
. (3.15)
Then the elements of the matrix B(x) from (3.10) are given by
bjk(x) = i exp
−iβ1·(λ3
j−λ3
k)
x∫
0
κ(t)dt
yjk(x), j 6= k, (3.16)
where the real functions yjk(·) (j 6= k) are given by the equalities
y12(x) = ψ(y13(x)), (3.17)
y13(x) = F−1
√
(λ3 − λ1)(λ2 − λ3)
x∫
0
κ(t)dt
, (3.18)
y14(x) = β14
(
y13(x)
β13
)α
, (3.19)
y2j(x) = αjy1j(x), (j = 3, 4), (3.20)
y34(x) = 0, (3.21)
ykj(·) = yjk(·), (j 6= k). (3.22)
Here F−1(·) is a function inverse to the monotonously increasing function F (·).
P r o o f. (3.6) yields the relation
cjk(x) = −κ(x)djkbjk(x), (3.23)
where djk =
λ3
j−λ3
k
λj−λk
= λ2
j + λjλk + λ2
k. Substitute (3.23) into (3.7), then
b′jk(x) = −iκ(x)djk(b
(0)
jj − b
(0)
kk )bjk(x)−
iκ(x)
4∑
l=1,l 6=k
(dkl − djl)bjl(x)blk(x), (3.24)
where j 6= k, and djk(b
(0)
jj − b
(0)
kk ) = β1(λ3
j − λ3
k). It is easy to see that
dkl − djl = λ2
k + λkλl + λ2
l − λ2
j − λjλl − λ2
l = (λk − λj)(λj + λk + λl).
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 57
A.A. Lunyov and E.V. Oliynyk
Let bjk(x) be given by (3.16). Then
y′jk(x) = κ(x)
4∑
l=1,l 6=j,k
(dkl − djl)yjl(x)ylk(x),
yjk(0) = βjk.
(3.25)
Use the fact that yjk(x) are real and y34(x) = 0. Then β34 = 0, yjk(x) = ykj(x),
as 1 ≤ j < k ≤ 4, and the system has the form
y′12 = κ(x)(d13 − d23)y13y23 + κ(x)(d14 − d24)y14y24, (3.26)
y′13 = κ(x)(d23 − d12)y12y23, (3.27)
y′23 = κ(x)(d12 − d13)y12y13, (3.28)
y′14 = κ(x)(d24 − d12)y12y24, (3.29)
y′24 = κ(x)(d12 − d14)y12y14. (3.30)
The condition y34(x) = 0 implies
0 = κ(x)(d13 − d14)y13y14 + (d23 − d24)y23y24. (3.31)
It follows from (3.27) and (3.28) that
(d23 − d12)(y2
23(x)− β2
23) = (d12 − d13)(y2
13(x)− β2
13) (3.32)
or
(λ3 − λ1)(y2
23(x)− β2
23) = (λ2 − λ3)(y2
13(x)− β2
13). (3.33)
The restriction on λj , where j = 1, 4, follows from the relation
(λ3 − λ1)β2
23 = (λ2 − λ3)β2
13, (3.34)
i.e., (λ3 − λ1)(λ2 − λ3) > 0, thus λ3 is between λ1 and λ2. Therefore,
y23(x) =
√
λ2 − λ3
λ3 − λ1
y13(x). (3.35)
Analogously, (3.29) and (3.30) imply that
y24(x) =
√
λ2 − λ4
λ4 − λ1
y14(x) (3.36)
if (λ4 − λ1)β2
24 = (λ2 − λ4)β2
14, and thus λ4 is between λ1 and λ2. Taking into
account (3.12), we write equality (3.31) in the form
(λ1 + λ3 + λ4 + α3α4(λ2 + λ3 + λ4)) y13(x)y14(x) = 0, (3.37)
58 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
besides, β23 = α3β13, β24 = α4β14. Thus y2j(x) = αjy1j(x) for j = 3, 4. From (3.26),
(3.27), (3.29), we find
y2
12(x)− β2
12 = −β3(y2
13(x)− β2
13)− β4(y2
14(x)− β2
14). (3.38)
So the equations (3.27) and (3.29) become
y′13 = κ(x)(d23 − d12)α3y12y13; (3.39)
y′14 = κ(x)(d24 − d12)α4y12y14. (3.40)
Since
y′14
y14
=
α4
α3
(d24 − d12)
(d23 − d12)
y′13
y13
, then denoting α =
(d24 − d12)α4
(d23 − d12)α3
(which is equi-
valent to (3.13)), we search y14(x) (subject to the initial conditions) in the form
y14(x) = β14
(
y13(x)
β13
)α
. Substitute
y12(x) = ψ(y13(x)) (3.41)
into (3.38), where
ψ(y13(x)) = √√√√β2
12 − β3(y2
13(x)− β2
13)− β4β2
14
((
y13(x)
β13
)2α
− 1
)
. (3.42)
By using (3.39), we can find
y′13(x) = κ(x)ψ(y13(x))y13(x).
Since F (y) is represented in the form (3.15), then
y13(x) = F−1
√
(λ3 − λ1)(λ2 − λ3)
x∫
0
κ(t)dt
.
Thus, in the case n = 4 and cubic dependency of a(x) from γ(x), the elements
of the matrix B(x) are expressed in terms of elliptic functions.
Corollary 3.3. In the conditions of Theorem 3 we suppose λ1 = −a, λ2 = a,
λ3 = −b, λ4 = b, where a, b ∈ R and the condition 0 < a < 3b is fulfilled,
moreover, β2
13 =
1
2
, β2
14 = 2, β2
12 =
3ab− a2
a2 − b2
, then the solutions yjk(x) as j 6= k,
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 59
A.A. Lunyov and E.V. Oliynyk
(3.17)–(3.20), are given by
y12(x) = −i
√
2a
a + b
cn(z(x) + N, k)dn(z(x) + N, k)
sn(z(x) + N)
, (3.43)
y13(x) = sn(z(x) + N, k), (3.44)
y14(x) =
1
sn(z(x) + N, k)
, (3.45)
y23(x) = k sn(z(x) + N, k), (3.46)
y24(x) =
1
k sn(z(x) + N, k)
, (3.47)
where
k2 =
a + b
a− b
, z(x) = −i
√
2a(a− b)
x∫
0
κ(t)dt, (3.48)
N =
1√
2∫
0
dt√
(1− t2) (1− k2t2)
. (3.49)
P r o o f. From (3.18), we obtain
√
(a− b)(a + b)
x∫
0
κ(t)dt = F (y13(x)). (3.50)
Applying (3.14), (3.15) and taking into account that under the given choice of
λj , where j = 1, 4, the value of α = −1 (which is evident from (3.13)), we have
F (y13(x)) =
y13(x)∫
β13
dt√
2a
b− a
t4 +
4a2
a2 − b2
t2 − 2a
b + a
=
i
√
a + b
2a
y13(x)∫
1√
2
dt√
a + b
a− b
t4 +
2a
a− b
t2 + 1
=
i
√
a + b
2a
y13(x)∫
1√
2
dt√
(1− t2)
(
1− a + b
a− b
t2
) . (3.51)
60 Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1
On Integration of One Class of Systems of Lax-Type Equations
Now relation (3.50) can be expressed as
−i
√
2a(a− b)
x∫
0
κ(t)dt =
y13(x)∫
1√
2
dt√
(1− t2)
(
1− a + b
a− b
t2
) , (3.52)
i.e., taking into account (3.48),
z(x) =
y13(x)∫
1√
2
dt√
(1− t2) (1− k2 t2)
(3.53)
or
z(x) +
1√
2∫
0
dt√
(1− t2) (1− k2 t2)
=
y13(x)∫
0
dt√
(1− t2) (1− k2 t2)
. (3.54)
In view of the notation and definition of elliptic functions, we have
z(x) + N =
y13(x)∫
0
dt√
(1− t2) (1− k2 t2)
, (3.55)
namely,
y13(x) = sn(z(x) + N, k). (3.56)
From (3.17) and (3.14), we obtain
y12(x)=
1
sn(z(x) + N)
(
−i
√
2a
a + b
√
(1− sn2(z(x) + N)(1− k2sn2(z(x) + N))
)
= −i
√
2a
a + b
cn(z(x) + N, k)dn(z(x) + N, k)
sn(z(x) + N)
. (3.57)
Substituting the values of λj , as j = 1, 4, we have that α3 = k, and α4 =
1
k
,
and then (3.20) implies that y23(x) = α3y13(x) = k sn(z(x) + N, k) and y24(x) =
α4y14(x) =
1
k sn(z(x) + N, k)
.
Journal of Mathematical Physics, Analysis, Geometry, 2015, vol. 11, No. 1 61
A.A. Lunyov and E.V. Oliynyk
References
[1] V.Ye. Zaharov, S.V. Manakov, S.P. Novikov, and L.P. Pitayevsky, Soliton Theory.
Nauka, Moscow, 1980. (Russian)
[2] V.A. Zolotarev, Spectral Analisys of Non-selfadjoint Commutative Operator Sys-
tems and Nonlinear Differential Equations. — Teor. Funktsij, Funkts. Analiz, i ih
pril. Kharkov, Resp. sb., 40 (1983), 68–71. (Russian)
[3] V.A. Zolotarev, Time Cones and Functional Model on Riemann Surface. — Mat.
Sb. 181 (1990), No. 7, 965–994. (Russian)
[4] V.A. Zolotarev, Analitical Methods of Spectral Representations of Non-selfadjoint
and Non-unitary Operators. KhNU, Kharkov, 2003. (Russian)
[5] P.D. Lax, Integrals of Nonlinear Equations of Evolution and Solitary Waves. —
Commun. Pure Appl. Math. 21 (1968), 467–490.
[6] M.S. Livs̆ic and A.A. Yantsevich, Operator Colligations in Hilbert Spaces. Winston,
Washington, D. C. (distributed by Wiley, New York), 1979.
[7] A.A. Lunyov and E.V. Oliynyk, On One Class of System of Lax-type Equations. —
UMV 10 (2013), No. 4, 507–531. (Russian)
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