Table Of ContentEXACT SOLUTIONS FOR NONLOCAL NONLINEAR
FIELD EQUATIONS IN COSMOLOGY
S. Yu. Vernov
Skobeltsyn Institute of Nuclear Physics, Moscow State University,
1
Leninskie Gory, GSP-1, 119991, Moscow, Russia
1
0
E-mail: [email protected]
2
n
a
J
Abstract
7
2
A method for the search of exact solutions for equation of a nonlocal scalar field in a
] non-flat metric is considered. In the Friedmann–Robertson–Walker metric the proposed
h
t method can be used in the case of an arbitrary potential, with the exception of linear
-
p and quadratic potentials, and allows to get in quadratures solutions, which depend on two
e arbitrary parameters. Exact solutions have been found for an arbitrary cubic potential,
h
which consideration is motivated by the string field theory, as well as for exponential,
[
logarithmic and power potentials. It has been shown that one can add the k-essence field
2
v to the model to get exact solutions for all Einstein equations.
7
0
0 1 GRAVITATIONAL MODELS WITH A NONLOCAL
5
.
5 SCALAR FIELD
0
0
1 Nonlocal cosmological models, based on the string field theory (SFT) [1, 2] and p-adic string
:
v theory[3]areactivelydeveloped[4]–[22]. Characteristicpropertiesofnonlocalmodelsarethenull
Xi energy condition violation and arising of phantom fields, which is connected with high derivative
r terms. Local models with phantom fields are considered as physically unacceptable ones because
a
of a problem quantum instability [23, 24]. In papers [25, 26] the instability problem is reduced
to such choosing of effective theory parameters that the instability turns out to be essential only
at times that are not described in the framework of the effective theory approximation. It is
mathematically expressed in that the terms, which result to instability, are treated as corrections
essential only at small energies below the physical cutoff. The given approach allows considering
such effective theories as physically acceptable with the presumption that an effective theory
admits immersion in a fundamental theory, for example, the string field theory.
The interest in cosmological models related to open string field theory [4] is provoked by the
possibility of obtaining solutions describing transitions from a perturbed vacuum to the true
vacuum (the so-called rolling solutions). After all massive fields (or some of the lower massive
fields) are integrated out using the equations of motion, the open string tachyon gets a potential
with a nontrivial vacuum corresponding to the minimal value of energy. In the dark energy
1
model [4] (see also [27, 28, 29, 30]) it is implied that the Universe is a slowly decomposing
D3–brane, whose decay is described by an open string tachyon mode. According to the Sen
conjugation [1], the tachyon motion from an unstable vacuum to the stable vacuum describes
the D-brane transition to the true vacuum. In fact one obtains a nonlocal potential with a
nonlocality parameter determined by the string scale. Using a suitable redefinition of the fields,
one can make the potential local, at that the kinetic term becomes nonlocal. This nonstandard
kinetic term leads to a behavior similar to the behavior of a phantom field, and it can be
approximated with a phantom kinetic term. Hence, the behavior of an open string tachyon can
be effectively simulated by a scalar field with a negative kinetic term [31].
The string field theory also gives asymptotic conditions on the solutions [27, 28, 29, 30].
Special interest is represented with solutions, which are bounded on the whole of real axis and
have a nonzero asymptotic at t + . In this paper such solutions of the field equation in the
→ ∞
Friedmann–Robertson–Walker metric have been found for cubic and logarithmic potentials.
Let us consider the gravitational model with a nonlocal scalar field φ, which is described by
the following action:
R 1 1
S = d4x√ gα′ + φ ((cid:3) )φ V(φ) Λ , (1)
− 16πG g2 2 F g − −
Z (cid:18) N o (cid:18) (cid:19) (cid:19)
where G is the Newtonian gravitational constant (8πG = 1/M2, M is the Planck mass),
N N P P
′
α is the string length squared, g is the open string coupling constant. We use the signature
o
( ,+,+,+), the matrix g is the metric tensor, g is the determinant of g , R is the scalar
µν µν
−
curvature. The potential V(φ) is a twice continuously differentiable function, the cosmological
constant is considered as a part of the potential V(φ). The d’Alembert operator (cid:3) is applied
g
to scalar functions and can be written as follows:
1
(cid:3) = ∂ √ ggµν∂ . (2)
g µ ν
√ g −
−
The scalar field φ is dimensionless, while [α′] = cm2 and [g ] = cm. It is convenient to
o
introduce dimensionless coordinates x¯ = x /√α′, the dimensionless gravitational constant
µ µ
G = G /α′, and the dimensionless coupling constant g¯ = g /√α′. The curvature scalar
N N o o
calculated in dimensionless coordinates is denoted as R¯, the corresponding d’Alembert operator
(cid:3)
is marked as . We get action (1) in the following form:
g
¯
R 1 1
S = d4x¯√ g + φ ((cid:3) )φ V(φ) ,
− 16πG g¯2 2 F g −
Z (cid:18) N o(cid:18) (cid:19)(cid:19)
In the following formulae we always use dimensionless coordinates and parameters and omit bars
over them.
The function is assumed to be an analytic function on whole complex plane (i.e. an entire
F
function), therefore, one can represent it by the convergent series expansion:
∞
((cid:3) ) = f (cid:3)n. (3)
F g n g
n=0
X
2
From action (1) we obtain the following equations
G = 8πG T 8πG Λg , (4)
µν N µν N µν
−
dV
((cid:3) )φ = , (5)
g
F dφ
where G is the Einstein tensor.
µν
The energy–momentum (stress) tensor T is calculated by the standard formula
µν
2 δS 1
T = = E +E g (gρσE +W) , (6)
µν − √ gδgµν g2 µν νµ − µν ρσ
− o(cid:16) (cid:17)
where
∞ n−1 ∞ n−1
1 1 f
E f ∂ (cid:3)lφ∂ (cid:3)n−1−lφ, W f (cid:3)lφ(cid:3)n−lφ 0φ2 +V(φ). (7)
µν ≡ 2 n µ g ν g ≡ 2 n g g − 2
n=1 l=0 n=2 l=1
X X X X
The case of a quadratic potential has been studied in papers [14, 15, 17, 19]. In paper [14] the
algorithm of localization of the Einstein equations has been proposed and exact solutions in the
spatially flat Friedmann–Robertson–Walker metric have been found. This algorithm can be used
only in the cases of linear and quadratic potentials. At the same time from the string theory one
gets cubic and the fourth degree potentials, so, a quadratic potential can be considered only as
an approximation [17]. In this connection the search of solutions for equation (5) with a cubic
potential in the Friedmann–Robertson–Walker metric is actively conducted [16, 18].
In this paper we propose the method of the search of solutions for (5) in the Friedmann–
Robertson–Walker metric, which gives exact two-parameter solutions in the close form or in
quadratures. This method allows to find solutions for an arbitrary potential V(φ), with the
exception of the cases of linear and quadratic potentials. In the paper we consider the case
of cubic potential, which is connected with the string theory (Section 3), as well as cases of
logarithmic, exponential and power potentials (Section 4).
Note that in distinguish to the localization method [19], which allows to localize all Einstein
equations, this method solves only equation (5), whereas equations (4), after the substitution
of the obtained solution, are inconsistent in the general case. In Section 5 we show that the
supplement of a scalar k-essence field gives an exact solution of the system of all nonlocal
Einstein equations. The question about possible types of the additional matter and ability to
obtain an exact solution for all Einstein equations in modified gravitation models, for example,
in f(R) gravitational model, without additional matter requires distinct investigations.
For cubic and exponential potentials approximate solutions for equation (5) with the string
(cid:3)
field theory inspired form of ( ) have been found by G. Calcagni and G. Nardelli [18] as a
F
generalization of their solutions in the Minkowski space [32]. In distinguish from [18] in this
paper we obtain the exact solutions for equation (5), in addition, the Hubble parameter H(t) is
a solution of equation, it is not given a priori.
2 SOLUTIONS FOR EQUATIONS OF MOTION
Let us consider nonlocal Klein–Gordon equation in the case of an arbitrary potential:
((cid:3) )φ = V′(φ), (8)
g
F
3
where a prime denotes the derivative with respect to φ. A particular solution of (8) is a solution
of the following system of local equations:
N−1
f (cid:3)nφ = V′(φ) C, f (cid:3)Nφ = C, (9)
n g − N g
n=0
X
where N 1 is a natural number, C is an arbitrary constant.
−
In the case f = 0 we can choose N = 2. In the spatially flat Friedmann–Robertson–Walker
1
6
metric with the interval:
ds2 = dt2 +a2(t) dx2 +dx2 +dx2 , (10)
− 1 2 3
where a(t) is the scale factor, we obtain from (9)(cid:0)the following sy(cid:1)stem:
f (cid:3) φ = f φ¨+3Hφ˙ = V′(φ) f φ C, f (cid:3)2φ = C. (11)
1 g − 1 − 0 − 2 g
(cid:16) (cid:17)
ThefunctionH(t)istheHubbleparameter: H a˙(t)/a(t),andadotdenotesthetimederivative.
≡
It is easy to see that at f = 0 solutions can exist only for C = 0. Let us consider the case
2
f = 0. The first equation of (11) can be rewritten in the following form:
2
6
1 C
¨ ˜′
H = φ+V (φ) , (12)
− 3φ˙ − f
(cid:18) 1(cid:19)
where
1
V˜′(φ) (V′(φ) f φ). (13)
0
≡ f −
1
Equation
C
(∂2 +3H∂ ) φ¨+3Hφ˙ = , (14)
t t f
2
(cid:16) (cid:17)
transfigures to the following form
C
(∂2 +3H∂ )V˜′ = V˜′′′φ˙2 +V˜′′(φ¨+3Hφ˙) = . (15)
t t − f
2
We eliminate H and obtain
1 C C
φ˙2 = V˜′′V˜′ V˜′′ . (16)
V˜′′′ − f − f
(cid:18) 1 2(cid:19)
Equation (16) can be solved in quadratures. The obtained solution depends on two arbitrary
parameters C and t , the latter papameter corresponds to the time shift.
0
At C = 0 we get the following equation
˜′˜′′ ′ ′′
V V (V f φ)(V f )
φ˙2 = − 0 − 0 , (17)
V˜′′′ ≡ f V′′′
1
using which one can get solutions at f = 0 as well.
2
Remark that, the proposed method can not be used in the cases of linear and quadratic
′′′
potentials, since for them V 0. Consequently this way of the search of solutions is suited
≡
only for nonlinear in φ equation (8). In the case of linear in φ equation (8) the search of solutions
is possible due to the localiezation method [14, 15, 19].
4
3 CUBIC POTENTIAL
The case of cubic potential is actively studied, since it is connected with the bosonic string field
theory [16, 18]. Let us find solutions for equation (8) at
V(φ) = B φ3 +B φ2 +B φ+B , (18)
3 2 1 0
where B , B , B , and B are arbitrary constants, but B = 0. We get (16) in the following
0 1 2 3 3
6
form
φ˙2 = 4C φ3 +6C φ2 +4C φ+C , (19)
3 2 1 0
where
(B C)(2B f ) Cf2 2B f
C = 1 − 2 − 0 1 , C = 2 − 0, (20)
0 2
6f B − 6f f B 4f
1 3 1 2 3 1
6B (B C)+(2B f )2 3B
3 1 2 0 3
C = − − , C = . (21)
1 3
24f B 4f
1 3 1
Note, that C = 0 since B = 0. Constants B and f appear in equation (19) only in the
3 3 2 0
6 6
combination 2B f . Using the transformation
2 0
−
1
φ = (2ξ C ), (22)
2
2C −
3
we get the following equation
ξ˙2 = 4ξ3 g ξ g , (23)
2 3
− −
where
(2B f )2 12B (B C)
g = 3C2 4C C = 2 − 0 − 3 1 − , (24)
2 2 − 1 3 16f2
1
3B C
g = 2C C C C3 C C2 = 3 . (25)
3 1 2 3 − 2 − 0 3 − 32f f
2 1
A solution of equation (23) is either the Weierstrass elliptic function
ξ(t) = ℘(t t ,g ,g ), (26)
0 2 3
−
where t is anarbitrarynumber, or a degenerate elliptic function. As known [33], theWeierstrass
0
elliptic function is a double periodic meromorphic function, which has one double pole in the
fundamental parallelogram of periods. If φ(t) is an elliptic function, then the Hubble parameter
H(t) is an elliptic function as well.
Let us consider degenerated cases. At g = 0 and g = 0 the general solution for equation
2 3
(23) is
1
ξ = , (27)
(t t )2
0
−
therefore,
1 C 4f 2B f
2 1 2 0
φ = = − . (28)
1 C (t t )2 − 2C 3B (t t )2 − 6B
3 0 3 3 0 3
− −
5
Substituting φ into (12), we get
1
5
H = . (29)
1
3(t t )
0
−
From conditions g = 0 and g = 0 it follows that
2 3
(2B f )2
2 0
C = 0, B = − . (30)
1
12B
3
Solutions, which are bounded on the whole real axis and tends to a finite limit at t ,
→ ∞
attract the special interest. Such a solution is the following function
φ = D tanh(β(t t ))2 +D , (31)
2 2 0 0
−
4 1
D = f β2, D = 3(f 2B ) 16f β2 , (32)
2 1 0 0 2 1
3B 18B − −
3 3
where β is a root of the following equation (cid:0) (cid:1)
1024f f β6 +576f2β4 +324B B 27(2B f )2 = 0. (33)
2 1 1 3 1 − 2 − 0
Bounded real solutions for equation (19) correspond to real roots of equation (33). Pure
imaginary roots of equation (33) correspond to unbounded real solutions for equation (19),
because tanh(βt)2 = tan(iβt)2. The solution φ exists at
2
−
1
C = 64f2β4 3(2B f )2 +36B B . (34)
36B 1 − 2 − 0 3 1
3
(cid:0) (cid:1)
The Hubble parameter has the following form:
β(2cosh(βt)2 3)
H = −
2
3cosh(βt)sinh(βt) −
3B (D tanh(βt)2 +D )2 +(2B f )(D tanh(βt)2 +D )+B
3 2 0 2 0 2 0 1
− .
− 6f D βtanh(βt)(1 tanh(βt)2)
1 2
−
The parameter t is an arbitrary complex number, so, using the equality tanh(t+ iπ/2) =
0
coth(t), one gets the following real solutions
φ˜ = D coth(β(t t ))2 +D . (35)
2 2 0 0
−
Note that solutions in terms of hyperbolic functions exist only at C = 0, since g = 0 at
3
6
C = 0, and solution (28) is the unique solution in terms of elementary functions.
4 EXACT SOLUTIONS FOR OTHER TYPES OF PO-
TENTIAL
4.1 Logarithmic Potential
Note that one can get equation (16) in the form (19), staring from a nonpolynomial potential
as well. Indeed, let
V(φ) = C ln(αφ), (36)
1
6
where C and α are arbitrary constants. The parameter α does not enter into equation (16):
1
f2 C(f f f2) C C
φ˙2 = 0 φ4 + 2 0 − 1 φ3 + φ 1 . (37)
2f C 2f f C 2f − 2f
1 1 1 2 1 1 1
At f = 0 in the general case the Jacobi elliptic functions are solutions for (37). At f = 0 and
0 0
6
C = 0 equation (37) is a particular case of equation (19), which solutions are the Weierstrass
6
elliptic functions.
Let us analyse real solutions in terms of the elementary functions. Such solutions have been
found only at f = 0.
0
At C = 0 one gets the following solutions
√ 2C f 2 f
1 1 0
φ (t) = − (t t ), H = + (t t ). (38)
0 0 0 0
− 2f − 3(t t ) 3f −
1 0 1
−
At C = 0 the following solution exists
6
φ = D˜ tanh2(A(t t ))+D˜ , (39)
ln 2 0 0
−
where A is an arbitrary number,
C (32f A2 9f ) C C1
D˜ = 1 2 − 1 , D˜ = 1 , ς = ,
2 − 18Af1(16f2A2 3f1)ς 0 12Af1ς ±s144f2A2 27f1
− −
at that C = 64A3f ς. The Hubble parameter
2
A(2cosh(At)2 3) C cosh(At)2
1
H = − . (40)
ln 3cosh(At)sinh(At) − 6Af D˜ ((D˜ tanh(At)2 +D˜ )tanh(At)
1 2 2 0
corresponds to this solution.
4.2 Exponential Potential
In paper [18] the exponential potential has been consider in addition to the cubic potential and
approximate solutions for equation (8) have been obtained.
Let V(φ) = C eαφ. At f = 0 and C = 0 solutions for (9) are elementary functions and have
1 0
the following form:
1 4f 1
1
φ (t) = ln , H (t) = . (41)
exp α C α2(t t )2 exp t t
(cid:18) 1 − 0 (cid:19) − 0
Note that the obtained Hubble parameter H (t) is proportional to the Hubble parameter,
exp
which has been used in paper [18], and to the Hubble parameter, obtained in the case of cubic
potential (formula (29)).
7
4.3 Power Potential
Let us consider solutions in the case of the potential V(φ) = C φn. At f = 0 equation (16) is
1 0
as follows:
C2f n2(n 1)φn C Cf n(n 1)φ Cf2φ3−n
φ˙2 = 1 2 − − 1 2 − − 1 (42)
f f C n(n 1)(n 2)
1 2 1
− −
At C = 0 this equation is equivalent to
C nφn
φ˙2 = 1 (43)
f (n 2)
1
−
and has a solution in the form of the elementary function:
1/(n−2)
f
φ (t) = 22/(n−2) 1 . (44)
n C n(n 2)(t t )2
(cid:18) 1 − − 0 (cid:19)
The corresponding Hubble parameter is equal to
3n 4
H (t) = − . (45)
n
3(n 2)(t t )
0
− −
At n = 4/3 we get a particular solution to equation (8) in the Minkowski space:
2C √ 2f C
φ (t) = 1 − 1 1(t t )3. (46)
m ± 27f2 − 0
1
NotethatintheMinkowski spaceexactboundedatallvaluesoftsolutionsfornonlocalequations
with power potentials have been found in paper [28].
5 COSMOLOGICAL MODEL WITH A NONLOCAL
SCALAR FIELD AND A k-ESSENCE FIELD
The goal of this Section is to show, that the supplement of the k-essence scalar field Ψ allows
to obtain a system of the Einstein equations, which has an exact solution, at that the Hubble
parameter and the nonlocal field are given by formulae (12) and (16) respectively. The k-essence
models are considered in cosmology both as inflation models [34, 35, 36], and as dark energy
models [37, 38, 39, 40, 26], (see also [41] and references therein).
Let us consider the following action
R 1 1
S = d4x√ g + φ ((cid:3) )φ V(φ) (Ψ,X) , (47)
2 − 16πG g2 2 F g − −P
Z (cid:18) N o(cid:18) (cid:19) (cid:19)
where X gµν∂ Ψ∂ Ψ. In the Friedmann–Robertson–Walker metric the function Ψ depends
µ ν
only on ti≡me−, so X = Ψ˙2.
Following paper [26], we select the pressure in the following form
1 1 1
(Ψ,X) = (p (Ψ) ̺ (Ψ))+ (p (Ψ)+̺ (Ψ))X + M4(Ψ)(X 1)2. (48)
q q q q
P 2 − 2 2 −
8
We consider functions p (Ψ), ̺ (Ψ), and M4(Ψ) as arbitrary differentiable functions. The
q q
energy density of the k-essence field is equal to
(Ψ,X) = (p (Ψ)+̺ (Ψ))X +2M4(Ψ)(X2 X) (Ψ,X). (49)
q q
E − −P
TheEinsteinequations, whichhavebeenobtainedusingthevariationofS ,havethefollowing
2
form:
3H2 = 8πG (̺+ ),
N
E (50)
2H˙ +3H2 = 8πG (p+ ).
N
− P
Varying the action S , we also get equation (8) and equation for the k-essence scalar field Ψ
2
˙ = 3H( + ). (51)
E − E P
In the Friedmann–Robertson–Walker metric the energy–momentum tensor (6) has the fol-
lowing form
T = g diag( ̺,p,p,p), (52)
µν µν
−
where
∞ n−1 n−1
1 f f
̺ = n ∂ (cid:3)lφ∂ (cid:3)n−1−lφ+ (cid:3)lφ(cid:3)n−lφ 0φ2 +V(φ) ,
t t
g2 2 − 2
o(cid:18)n=1 (cid:18)l=0 l=1 (cid:19) (cid:19)
X X X
∞ n−1 n−1
1 f f
p = n ∂ (cid:3)lφ∂ (cid:3)n−1−lφ (cid:3)lφ(cid:3)n−lφ + 0φ2 V(φ) .
t t
g2 2 − 2 −
o(cid:18)n=1 (cid:18)l=0 l=1 (cid:19) (cid:19)
X X X
Let φ is a solution to system (9) at N = 2. Using (cid:3)2φ = C/f , we get
2 g 2 2
̺(φ ) = E (φ )+W(φ ), p(φ ) = E (φ ) W(φ ), (53)
2 00 2 2 2 00 2 2
−
where
1
E (φ ) = f (∂ φ )2 +2f ∂ φ ∂ (cid:3) φ +f (∂ (cid:3) φ )2 ,
00 2 1 t 2 2 t 2 t g 2 3 t g 2
2g2
o
1 f(cid:0) f C f C2 f (cid:1)
W(φ ) = 2((cid:3) φ )2 + 3 (cid:3) φ + 4 0φ2 +V(φ ) .
2 g2 2 g 2 f g 2 2f2 − 2 2 2
o(cid:18) 2 2 (cid:19)
Thek-essence models(without additionalfields) have oneuseful property. Foranyreal differ-
entiable function H (t) there exist such differentiable functions ̺ (Ψ) and p (Ψ), that functions
0 q q
H (t t ) and Ψ(t) = t t are a particular solution to system (50)–(51). This property can
0 0 0
− −
be generalized on the case of the models with an additional nonlocal scalar field, which are
described by action (47). Indeed, at Ψ(t) = t t we obtain
0
−
= ̺ (Ψ) = ̺ (t t ), = p (Ψ) = p (t t ). (54)
q q 0 q q 0
E − P −
Substituting into (50) expression of ̺ and p , we get
q q
3
̺ (Ψ) = ̺ (t t ) = H2(t t ) ̺(t t ),
q q 0 0 0
− 8πG − − −
N (55)
1
˙
p (Ψ) = p (t t ) = ̺ (t t ) ̺(t t ) p(t t ) H(t t ).
q q 0 q 0 0 0 0
− − − − − − − − 4πG −
N
9
It is easy to see that system (50)–(51) and equation (8) have the exact particular solution, at
that functions H(t t ) and φ(t t ) are the obtained solution of equation (8) and Ψ(t) = t t .
0 0 0
− − −
So, the algorithm to obtain exact solutions is as follows: for the given potential V(φ) we
find H(t) and φ(t), calculate the energy–momentum tensor and substitute the obtained values
into (55). The obtained values of ̺ and p give the exact solvable model with a nonlocal scalar
q q
field and the k-essence field. The function M(Ψ) can be selected arbitrarily.
Let us illustrate this scheme on the simplest example, connected with a cubic potential, and
consider solution (28)–(29). Conditions (30) of existence of this solution leave constants B
3
and B arbitrary. Using the arbitrariness of B , we can, without loss of generality, put f = 0.
2 2 0
Also, for compactness of the record we put t = 0. For solution (28)–(29) we get
0
16f
(cid:3) φ = 1 , (56)
g 1 3B t4
3
therefore,
32f2(f t4 +16f t2 +64f ) 128f f2 1
E = 1 1 2 3 , W = 2 1 + V(φ ). (57)
00 9B2t10 9g2B2t8 g2 1
3 o 3 o
Consequently we get
B3 27B B2 25 160f3 640f f2 2048f f2
̺ (Ψ) = 2 − 0 3 + 1 2 1 3 1 ,
q 27g2B2 24πG Ψ2 − 27g2B2Ψ6 − 9g2B2Ψ8 − 9g2B2Ψ10
o 3 N o 3 o 3 o 3
B3 27B B2 5 32f3 128f f2 2048f f2
p (Ψ) = 2 − 0 3 1 2 1 3 1 .
q − 27g2B2 − 8πG Ψ2 − 27g2B2Ψ6 − 3g2B2Ψ8 − 9g2B2Ψ10
o 3 N o 3 o 3 o 3
Thus, we can make a conclusion that the adding of the k-essence field allows to construct a
self-consistent system of the Einstein equations for the obtained solution to equation (8). The
choice of the k-essence scalar field as an additional field is reasoned by the goal to get a self-
consistent system of the Einstein equations with no restriction on the type of the potential and
solutions.
6 CONCLUSIONS
In this paper the method of the search of exact solutions to the nonlocal field equation (8) in
the Friedmann–Robertson–Walker metric has been proposed. It has been demonstrated that for
an arbitrary potential besides of linear or quadratic ones there exists a particular two-parameter
solution, which can be found in quadratures.
The string field theory inspired case of a cubic potential has been consider in detail. It is
shown that solutions are either Weierstrass elliptic functions or degenerate elliptic functions, for
example, solutions in terms of hyperbolic tangent have been obtained. Exact solutions in terms
of elementary functions have been found for logarithmic, exponential and power potentials as
well.
Note that the proposed method allows to find a solution only for the field equation, but not
for the whole system of the Einstein equations. It has been shown that the supplement of the
k-essence scalar field allows to construct a self-consistent system of the Einstein equations for
the obtained solution to equation (8). The localization of the Einstein equations in cosmological
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