Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide.
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irk-123456789-1773052021-02-15T01:26:54Z Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation Prykarpatsky, Y.A. The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide. Розглянуто одновимiрне нелiнiйне рiвняння Дiрака та описано множину його iнварiантiв за допомогою деформованого лiнiйного рiвняння Дiрака з використанням того факту, що два звичайних диференцiальних рiвняння є еквiвалентними, якщо множини їх iнварiантiв збiгаються мiж собою. 2017 Article Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation / Y.A. Prykarpatsky // Нелінійні коливання. — 2017. — Т. 20, № 2. — С. 267-273 — Бібліогр.: 11 назв. — англ. 1562-3076 http://dspace.nbuv.gov.ua/handle/123456789/177305 517.9 en Нелінійні коливання Інститут математики НАН України |
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The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent if their sets of invariants coincide. |
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Article |
| author |
Prykarpatsky, Y.A. |
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Prykarpatsky, Y.A. Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation Нелінійні коливання |
| author_facet |
Prykarpatsky, Y.A. |
| author_sort |
Prykarpatsky, Y.A. |
| title |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation |
| title_short |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation |
| title_full |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation |
| title_fullStr |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation |
| title_full_unstemmed |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation |
| title_sort |
steen – ermakov – pinney equation and integrable nonlinear deformation of one-dimensional dirac equation |
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Інститут математики НАН України |
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2017 |
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http://dspace.nbuv.gov.ua/handle/123456789/177305 |
| citation_txt |
Steen – Ermakov – Pinney equation and integrable nonlinear deformation of one-dimensional Dirac equation / Y.A. Prykarpatsky // Нелінійні коливання. — 2017. — Т. 20, № 2. — С. 267-273 — Бібліогр.: 11 назв. — англ. |
| series |
Нелінійні коливання |
| work_keys_str_mv |
AT prykarpatskyya steenermakovpinneyequationandintegrablenonlineardeformationofonedimensionaldiracequation |
| first_indexed |
2025-07-15T15:21:03Z |
| last_indexed |
2025-07-15T15:21:03Z |
| _version_ |
1837726814948032512 |
| fulltext |
UDC 517.9
STEEN – ERMAKOV – PINNEY EQUATION
AND INTEGRABLE NONLINEAR DEFORMATION
OF ONE-DIMENSIONAL DIRAC EQUATION
РIВНЯННЯ СТIНА – ЄРМАКОВА – ПIННЕЯ
ТА IНТЕГРОВНI НЕЛIНIЙНI ДЕФОРМАЦIЇ
ОДНОВИМIРНОГО РIВНЯННЯ ДIРАКА
Ya. Prykarpatskyy
Agricultural Univ. Krakow, Poland
Inst. Math. Nat. Acad. Sci. Ukraine
and
Drohobych Ivan Franko State Ped. Univ.
e-mail: yarpry@gmail.com
The paper deals with nonlinear one-dimensional Dirac equation. We describe its invariants set by means of
the deformed linear Dirac equation, using the fact that two ordinary differential equations are equivalent
if their sets of invariants coincide.
Розглянуто одновимiрне нелiнiйне рiвняння Дiрака та описано множину його iнварiантiв за до-
помогою деформованого лiнiйного рiвняння Дiрака з використанням того факту, що два зви-
чайних диференцiальних рiвняння є еквiвалентними, якщо множини їх iнварiантiв збiгаються
мiж собою.
1. Introduction. In 1874 Danish mathematician Adolph Steen wrote a paper [1] where he intro-
duced the system of two equations
r′′ + qr =
1
r3
, (1.1)
and
g′′ + qg = 0, (1.2)
where q = q(u) is a continuous function on a real interval. He discovered that these two equati-
ons above are in some sense equivalent, that is the general solution to the second equation
(1.2) gives rise to that to the first one (1.1) and vise-versa. Unluckily, the paper was published in
Danish and his research was lost. Later many authors have been rediscovering these equations
and mentioning too the property above. In 1880 V. Yermakov [2] gave a novel derivation and
generalization of the Steen’s equations. This generalization was actively studied and developed
by others researches. Later, in 1950 Edmund Pinney [3] showed that the solution of the first
equation (1.1) is
r(t) =
√
Au2 + 2Buv + Cv2, (1.3)
c© Ya. Prykarpatskyy, 2017
ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2 267
268 YA. PRYKARPATSKYY
where u(t) and v(t) are two arbitrary linearly independent solutions of the second equation of
(1.1), A, B and C are constants which satisfy the equality
B2 −AC =
1
W 2
, (1.4)
with W being the constant Wronskian of the two independent solutions u, v to (1.2).
Nowadays the Steen’s contribution is forgotten too and nobody calls systems of equati-
ons (1.1) Steen’s equations. The name of Ermakov is commonly used or they also often called
Ermakov – Pinney equations. Raymond Redheffer and Irene Redheffer wrote [4] an historical
survey on the Steen’s equations with English translation of the original Steen’s paper.
It is worth highlighting the following quite simple fact. Let us take the two equations
y′′ = ω(t)y (1.5)
and
z′′ = ω(t)z +
k
z3
, (1.6)
which are, in fact, rewritten equations from (1.1). These equations can be easily transformed to
the more generalized form (py′)′ = ω(t)y and (pz′)′ = ω(t)z +
k
z3
for some smooth function
p(z) by changing the variables on which we will not stop here.
Let us denote α := y1y2, where y1 and y2 are two arbitrary solutions to the equation (1.5),
and β := z2. After differentiation and substituting into (1.5) and (1.6) one can obtain the
following expressions:
α′′′ − 2ω′α− 4ωα′ = 0, (1.7)
or [
d3
dt3
−
(
2ω
d
dt
+ 2
d
dt
ω
)]
α = 0, (1.8)
and
β′′′ − 2ω′β − 4ωβ′ = 0, (1.9)
or [
d3
dt3
−
(
2ω
d
dt
+ 2
d
dt
ω
)]
β = 0. (1.10)
It is worth mentioning that the differential expression η =
[
d3
dt3
−
(
2ω
d
dt
+ 2
d
dt
ω
)]
is the
second Poisson operator [5, 6] for the KdV equation ut = −uxxx − 6uux = K[u] in its Hamil-
tonian form ut = −ηgradH, where H is the corresponding [7] Hamiltonian functional.
ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2
STEEN – ERMAKOV – PINNEY EQUATION AND INTEGRABLE NONLINEAR DEFORMATION . . . 269
Taking into account that the equations (1.8) and (1.10) are linear and the same, one can infer
that their sets of the solutions coincide to each other. It means that the solution to the equation
(1.10) is the sum of the solutions to the equation (1.8), whereas the expression
z2 = Ay21 +By1y2 + Cy22
is the solution of the (1.8), and then the function
z =
√
Ay21 +By1y2 + Cy22
is the solution of the (1.6). Taking into account that the Wronskian of any two different solutions
to (1.5) is constant, one easily obtains the relationship (1.4).
In the next section we will use the above mentioned trick in finding solutions to the one-di-
mensional Dirac equations [5].
2. Dirac equation, its invariants and integrable nonlinear deformation. Let us consider the
one-dimensional Dirac equation
df
dx
= l(λ; q)f, l(λ; q) :=
(
λ q1(x)
q2(x) −λ
)
, (2.1)
where x ∈ R, λ ∈ C is a complex parameter, (q1, q2)
ᵀ ∈ M ' C∞(R;C2) is a functional vector
of potentials, and f ∈ (L∞(R;C2) is found solution to (2.1). The whole solution set to the
equation (2.1) is completely described [8] by means of the corresponding fundamental solution
F :=
(
f11 f12
f21 f22
)
∈ C∞(R× R,EndC2), satisfying the matrix equation
d
dx
F (x, x0) = l(λ; q)F (x, x0), F (x, x0)|x=x0 = 1 (2.2)
at any point x0 ∈ R. Then an arbitrary solution f ∈ C∞(R;C2) to (2.2), evidently, can be
represented as
f(x) = F (x, x0)f(x0), (2.3)
where f(x0) ∈ C2 is a suitable Cauchy data vector.
It is well known from the general theory of ordinary differential equations [8] that the soluti-
on set to the equation (2.1) can be equivalently described by means of its complete set of invari-
ants. Moreover, the two ordinary differential equations are then considered to be equivalent if
their sets of invariants coincide. From this point of view one can consider a suitable deformed
Dirac equation
df̃
dx
= l(λ; q)f̃ + δf̃ , (2.4)
where a vector δf̃ ∈ C∞(RC2) can depend on f̃ ∈ C∞(R;C2) and the vector q ∈ M coincides
with that chosen in (2.1).
ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2
270 YA. PRYKARPATSKYY
Now a problem consists in determining the vector δf̃ ∈ C∞(R;C2) in such a form which
guarantees that the set of invariants of (2.4) will contain or coincide with that of the equati-
on (2.1).
To approach a solution to this problem we need first to describe the invariants set to the
equation (2.1). To do that we assume for simplicity that the functional vector q ∈ M is
2π-periodic in x ∈ R : q(x+ 2π) = q(x) for all x ∈ R. Then one can define [9] the monodromy
matrix S(x) ∈ EndC2 as
S(x) := F (x+ 2π, x), (2.5)
satisfying the well known Novikov commutator equation [10]
dS(x)
dx
= [l(λ; q), S(x)]. (2.6)
As a consequence from (2.6) one easily obtains that all functions
γj := trSj(x),
dγj
dx
= 0, (2.7)
where j ∈ Z+, are invariants for (2.1) and form [11] its complete set. In particular, we can
determine only two dependent invariants for (2.1)
γ1 = trS(x), γ2 = detS(x) = 1, (2.8)
and will now try to search for such a deformation vector δf̃ ∈ C∞(R;C2) which will give rise to
the invariants set of (2.4) coinciding with (2.8) of (2.1). For this problem to be solved effectively,
we need to find the determining equations for invariants (2.8) as for the independent gradient
grad γ1 ∈ T ∗(M), q2D
−1
x q2
d
dx
+ λ− q2D−1x q1
d
dx
− λ− q1D−1x q2 q1D
−1
x q1
gradγ1 = 0, (2.9)
where, by definition, we put
D−1x (·) :=
1
2
x∫
x0
(·)dy −
x0+2π∫
x
(·)dy
, x ∈ R.
It is now worth observing that the monodromy matrix S(x) ∈ EndC2 for any x ∈ R allows
the following matrix representation:
S(x) = F (x, x0)C(x0)F
−1(x, x0) (2.10)
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STEEN – ERMAKOV – PINNEY EQUATION AND INTEGRABLE NONLINEAR DEFORMATION . . . 271
for some specially chosen matrix C(x0) ∈ EndC2, x0 ∈ R. As a simple consequence of the
representation (2.10) one easily obtains that
gradγ1 =
c11f21f22 − c12f221 + c21f
2
22 − c22f22f21
c12f
2
11 − c11f11f12 − c21f212 + c22f12f11
, (2.11)
where we used the matrix expression C(x0) =
(
c11 c12
c21 c22
)
∈ EndC2. Taking into account
the arbitrariness of the matrix C(x0) ∈ EndC2 entering (2.10), we can easily obtain from (2.9)
and (2.11) that the equation q2D
−1
x q2
d
dx
+ λ− q2D−1x q1
d
dx
− λ− q1D−1x q2 q1D
−1
x q1
−f221
f211
= 0 (2.12)
holds for all x ∈ R and λ ∈ C.
Now it is easy to infer that if the vector f̃ := (f̃1, f̃2)
τ ∈∞ (C;C2) of the equation (2.4)
satisfies the same equation as (2.12) q2D
−1
x q2
d
dx
+ λ− q2D−1x q1
d
dx
− λ− q1D−1x q2 q1D
−1
x q1
−f̃22
f̃21
= 0, (2.13)
then the corresponding set of invariants of the deformed equation (2.4) will possess that of
invariants for (2.1). In particular, from (2.13) it follows that a partial solution to the deformed
Dirac equation (2.4) can be represented as
f̃ =
(
c12f
2
11 − c21f11f12 − c21f212 + c22f12f11
)1/2
(
c12f
2
21 − c11f21f22 + c22f22f21 − c21f222
)1/2
, (2.14)
depending only on the fundamental matrix F (x, x0) ∈ EndC2 of the equation (2.1) and equi-
valently, on its set of invariants. It is clear that the deformed Dirac equation (2.4) can generate a
new solutions to it, yet the problem of describing this set of invariants is much more complicated
and will not be herewith discussed.
Let us proceed now to describing the deformed Dirac equation (2.4), taking into account
that the vector
(
−f̃22 , f̃21
)
∈ C∞(R;C2) satisfies in general the following equation: q2D
−1
x q2
d
dx
+ λ− q2D−1x q1
d
dx
− λ− q1D−1x q2 q1D
−1
x q1
−f̃22
f̃21
=
=
−δf̃2 f̃2 + q1
d−1
dx
(
δf̃1 f̃2 + δf̃2 f̃1
)
δf̃1 f̃1 − q2
d−1
dx
(
δf̃1 f̃2 + δf̃2 f̃1
)
, (2.15)
ISSN 1562-3076. Нелiнiйнi коливання, 2017, т . 20, N◦ 2
272 YA. PRYKARPATSKYY
which reduces to (2.13) if identically one has
δf̃2 f̃2 = q1D
−1
x
(
δf̃1 f̃2 + δf̃2 f̃1
)
,
δf̃1 f̃1 = q2D
−1
x
(
δf̃1 f̃2 + δf̃2 f̃1
)
.
(2.16)
As a simple consequence of (2.16) one obtains that there exists some function α ∈ C∞(R;C)
for which
δf̃2 =
α
f̃2
q1, δf̃1 =
α
f̃1
q2. (2.17)
Having substituted (2.17) into (2.16) one obtains that
dα
dx
= α
(
q1
f̃1
f̃2
+ q2
f̃2
f̃1
)
, (2.18)
or upon integrating (2.18), one ensues
α = ᾱ exp
[
D−1x
(
q1
f̃1
f̃2
+ q2
f̃2
f̃1
)]
,
where ᾱ ∈ C is an arbitrary constant. Having now summarized the results obtained above one
can formulate the following theorem.
Theorem 2.1. Consider two Dirac type equations: the first one (2.1) linear and the second one
(2.4) nonlinear, where
δf̃1 =
ᾱ
f̃1
q2 exp
[
D−1x
(
q1
f̃1
f̃2
+ q2
f̃2
f̃1
)]
,
δf̃2 =
ᾱ
f̃2
q1 exp
[
D−1x
(
q1
f̃1
f̃2
+ q2
f̃2
f̃1
)] (2.19)
and ᾱ ∈ C is an arbitrary constant. Then a partial solution to the nonlinear Dirac type equation
(2.4) is given by the explicit expression (2.14), represented by means of the fundamental solution
F (x, x0) ∈ EndC2 to the Dirac equation (2.1) and the arbitrary constant matrix C ∈ EndC2.
Thus, the Dirac equation (2.4), deformed by means of the vector components (2.19), is a
nonlinear integro-differential equation depending on the functional element q ∈ M, whose 2π-
periodicity assumed before is not essential, as the main inferences, which are presented above,
were based strictly on local reasonings.
3. Conclusion. Based on the analogy with the oscillator type equations (2.1) and (2.2) we
succeeded in deriving a more general Steen type statement about the relationship between the
solution sets to the linear Dirac equation (2.1) and its nonlinear deformation (2.4), specified by
the expressions (2.20).
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STEEN – ERMAKOV – PINNEY EQUATION AND INTEGRABLE NONLINEAR DEFORMATION . . . 273
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Received 20.02.17
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