Nonlinear Dyn (2016) 86:2115–2126
DOI 10.1007/s11071-016-3020-x
ORIGINAL PAPER
Soliton solutions of nonlinear diffusion–reaction-type
equations with time-dependent coefficients accounting
for long-range diffusion
Houria Triki · Hervé Leblond ·
Dumitru Mihalache
Received: 19 June 2016 / Accepted: 10 August 2016 / Published online: 26 August 2016
© Springer Science+Business Media Dordrecht 2016
Abstract We investigate three variants of nonlinear
diffusion–reaction equations with derivative-type and
algebraic-type nonlinearities, short-range and longrange diffusion terms. In particular, the models with
time-dependent coefficients required for the case of
inhomogeneous media are studied. Such equations are
relevant in a broad range of physical settings and biological problems. We employ the auxiliary equation
method to derive a variety of new soliton-like solutions
for these models. Parametric conditions for the existence of exact soliton solutions are given. The results
demonstrate that the equations having time-varying
coefficients reveal richness of explicit soliton solutions
using the auxiliary equation method. These solutions
may be of significant importance for the explanation
of physical phenomena arising in dynamical systems
described by diffusion–reaction class of equations with
variable coefficients.
H. Triki (B)
Radiation Physics Laboratory, Department of Physics,
Faculty of Sciences, Badji Mokhtar University,
P.O. Box 12, 23000 Annaba, Algeria
e-mail: [email protected]
H. Leblond
Laboratoire de Photonique d’Angers, EA 4464, Université
d’Angers, 2 Bd. Lavoisier, 49045 Angers Cedex 01, France
D. Mihalache
Horia Hulubei National Institute for Physics and Nuclear
Engineering, P.O.B. MG-6, 077125 Bucharest-Magurele,
Romania
Keywords Nonlinear diffusion–reaction equations ·
Time-dependent coefficients · Auxiliary equation
method · Soliton solutions
1 Introduction
Most of the past research interest in propagation of solitons is normally confined to homogeneous nonlinear
media. In such media, the soliton dynamics is described
by nonlinear evolution equations (NLEEs) with constant coefficients. When the medium presents some
inhomogeneities, a more rigorous description of the
wave dynamics should be through NLEEs with variable
coefficients. For instance, in realistic fiber transmission
lines where no fiber is truly homogeneous due to longdistance communication and manufacturing problems,
optical pulse propagation is described by the nonlinear
Schrödinger equation with distributed coefficients [1].
Moreover, the Korteweg–de Vries equation with variable coefficients has also been studied recently in the
context of ocean waves, where the spatiotemporal variability of the coefficients is due to the changes in the
water depth and other physical conditions [2]. Basically, the existence of the inhomogeneities in media
influences the accompanied physical effects giving rise
to spatial or temporal dispersion and nonlinearity variations; see, for example [3–6] and references therein.
A soliton is a self-localized solution of a NLEE
describing the evolution of a nonlinear dynamical system with an infinite number of degrees of freedom
123
2116
[7]. Solitons appear in a large variety of subfields of
physics, such as fluid dynamics [8], plasma physics
[9,10], atomic physics (atomic Bose–Einstein condensates) [11–15], and nonlinear optics [16–28]. The distinction between solitary wave and soliton solutions is
that when any number of solitons interact, they do not
change form, and the only outcome of the interaction
is a phase shift [29].
In recent years, several powerful methods have
been developed to exactly solve nonlinear evolution
equations and their generalizations. These approaches
include the F-expansion technique [30], the solitary
wave ansatz method [31,32], Hirota bilinear method
[33], the homogeneous balance and F-expansion technique [34,35], among others. Having obtained exact
solutions is helpful for realizing properties of the nonlinear equation and for understanding physical phenomena described by the model considered. In addition, exact solutions allow one to calculate certain
important physical quantities analytically as well as
serving as diagnostics for simulations [36].
A particularly important setting for solitons is
described by the nonlinear diffusion–reaction (DR)
equations with derivative-type and/or algebraic-type
nonlinearities. These equations have drawn a great
deal of attention, due to their very wide applications
in physics, chemistry, and biology [37–40]. The most
common version of a DR-type equation contains the
u x x term, which describes the so-called short-range
diffusion. Accounting for this term, Kumar et al. [41]
derived some new soliton-like solutions of certain types
of nonlinear DR equations with quadratic and cubic
nonlinearities with a time-dependent velocity in the
convective flux term, using the method of Zhao et
al. [42] (based on the work of Gao and Tian [43]).
In Ref. [44], the authors derived exact solutions of
certain types of nonlinear DR equations involving
quadratic and quartic nonlinearities with a nonlinear
time-dependent convective flux term, using the same
method.
Recent attention has been drawn on the incorporation of the term u x x x x into the nonlinear wave DR
equations [45]. In this regard, the authors of Ref. [45]
studied three variants of nonlinear wave DR equations with derivative-type and/or algebraic-type nonlinearities having both long-range diffusion through
u x x x x -term and short-range diffusion through u x x term. Importantly, the simplest case when all coefficients in three variants are constant was investigated in
123
H. Triki et al.
[45] using a simplified auxiliary equation of the form
dz/dξ = b + z 2 (ξ ), where b is a constant.
Here we go beyond the previous study on the nonlinear DR equations [45] and include the presence of
time-dependent coefficients for constructing different
types of explicit solutions. In particular, we present a
variety of exact soliton-like solutions of the following
nonlinear DR equations with variable coefficients:
u t + η(t)u x x x x − D(t)u x x
+ α(t)u − β(t)u 3 + δ(t)u x = 0,
u t + η(t)u x x x x − D(t)u x x + α(t)u − β(t)u
+ γ (t)u 5 + δ(t)u x = 0,
(1)
3
(2)
u t + η(t)u x x x x − D(t)u x x + α(t)u
(3)
− β(t) (u x )2 + δ(t)uu x = 0,
where α (t) , β (t) , D (t) , γ (t) , η (t), and δ (t) are
arbitrary functions of t.
To the best of our knowledge, exact analytic solutions to Eqs. (1)–(3) with time-dependent coefficients
have not been reported. In this work, we employ the
method of Zhao et al. [42] (based on the work of Gao
and Tian [43]) to obtain a set of soliton-like solutions
for Eqs. (1) and (2). In particular, we use the auxiliary equation method of Sirendaoreji [46] to solve
the DR equations (1) and (2). For solving Eq. (3), we
use a simplified auxiliary ordinary equation reported in
Ref. [45], which requires a modification for the solution of the DR equations with variable coefficients (3).
We obtain new soliton-like solutions for the models
under consideration. The conditions for the existence
of soliton-like solutions are also presented.
2 Soliton-like solutions of Eqs. (1)–(3)
2.1 Exact solutions of Eq. (1)
We consider the first variant of the nonlinear DR equation with time-dependent coefficients (1):
u t + η(t)u x x x x − D(t)u x x + α(t)u − β(t)u 3
+ δ(t)u x = 0,
(4)
Balancing the highest-order derivative term u x x x x with
the nonlinear term u 3 in (4) gives
M + 4 = 3M,
(5)
so that M = 2, where M is the balance constant.
Accordingly, we suppose that the solution of Eq. (4)
is of the form [42]
Soliton solutions of nonlinear diffusion–reaction-type equations
u = f (t) + g1 (t)ϕ (ξ ) + g2 (t)ϕ 2 (ξ ) ,
(6)
ξ = p (t) x + q(t).
(7)
The function ϕ (ξ ) satisfies an auxiliary ordinary differential equation of the form [46]
dϕ
dξ
2117
with
ρ2 =
120η
.
β
(20)
Then Eq. (14) gives
2
= q4 ϕ + q3 ϕ + q2 ϕ ,
4
3
2
(8)
where q2 , q3 , and q4 are constants. Here, in (6) and
(7), f (t), g1 (t), g2 (t), p(t), and q(t) are functions of
t, which are unknown and to be further determined.
Substituting Eq. (6) into Eq. (4) and making use
of Eq. (8), then setting the coefficients of ϕ i (where
i = 0, . . . , 6), ϕ , and ϕϕ to zero, we arrive at the
following set of equations:
ϕ 0 : f + α f − β f 3 = 0,
(9)
g1 =
(12)
105
q 2 g2 + 120q2 q4 g2
ϕ 4 : ηp 4 30q3 q4 g1 +
2 3
(13)
− 6Dq4 p 2 g2 − 3β g12 g2 + f g22 = 0,
ϕ 5 : ηp 4 24q42 g1 + 168q3 q4 g2 − 3βg1 g22 = 0,
(14)
ϕ 6 : 120ηp 4 q42 g2 − βg23 = 0,
ϕ : g1 p x + q + δg1 p = 0,
ϕϕ : 2g2 p x + q + 2δg2 p = 0,
(15)
f = μp 2 + ν,
t
p x + δp dt.
(22)
with
ν=
−8D
4βρ
(23)
100ηq2 q4 − 15ηq32
.
4βq4 ρ
(24)
and
μ=
If (21), (19), and (22) are reported into Eqs. (10) and
(11), we multiply Eqs. (10) and (11) by q3 and 2q4 ,
respectively, subtract one from the other, and get the
equation
3q3 4q2 q4 − q32 ρp 4
16q4
(25)
× 4q4 D − 20ηq2 q4 p 2 + 15ηq32 p 2 = 0.
This condition is satisfied if
q4 =
q32
4q2
(26)
(the other solution will be considered after).
Then Eq. (10) reduces to
(16)
(17)
8ρp 2 D 2 − 80η
(18)
and Eq. (9) to
Equations (16) and (17) reduce to
q=−
(21)
Equation (12) can be solved to yield
ϕ 1 : g1 + ηp 4 q22 g1 − Dg1 p 2 q2
− 3β f 2 g1 + αg1 = 0,
(10)
15
q2 q3 g1 + 16q22 g2
ϕ 2 : g2 + ηp 4
2
− 3β f 2 g2 + f g12
3
q3 g1 + 4q2 g2 + αg2 = 0,
(11)
−Dp 2
2
15
ϕ 3 : ηp 4 20q2 q4 g1 + 65q2 q3 g2 + q32 g1
2
− Dp 2 (2q4 g1 + 5q3 g2 ) − β g13 + 6 f g1 g2 = 0,
q3 2
ρp .
2
d(ρp 2 )
+ βηq22 ρ 3 p 4 − 80αηρp 2 = 0
dt
(27)
24ρ D 3 − 360ηq2 ρp2 D 2 + 15βηq22 ρ 3 p4 D − 720αηρ D
From Eq. (15), we see that
g2 = ρq4 p 2 ,
+ 3600η2 q2
(19)
d(ρp2 )
−25βη2 q23 ρ 3 p6 + 3600αη2 q2 ρp2 = 0.
dt
(28)
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2118
H. Triki et al.
Eliminating the derivative between (27) and (28) yields
a polynomial equation for ρp 2 . Thus, the differential
equations (27) and (28) can be solved when this polynomial identically vanishes. It is easily seen that this
is not possible if both the equation and the solution do
not reduce to a trivial case. Hence,
d(ρp 2 )
= 0,
dt
(29)
D − 5ηp 2 q2
If we set the second factor to zero, compute α, and
report it into Eq. (27), we see that the equation has no
real solution. But if we set
(31)
and report it into Eq. (27), we see that it is solved under
the condition
α=
8D 2
.
50η
(32)
This first solution (“solution A”) exists under condition
(32), q2 is given by (31), q3 remains free, q4 is given
by (26), g1 and g2 are related to p, which remains free
provided that condition (29) is fulfilled, according to
(21) and (19), respectively. It is easily checked that f
vanishes.
Let us now return to (25), and assume that it is zero.
After some computation, it is seen that q4 must be zero.
It is more clear and safe to seek from the beginning a
solution with q4 = 0. It is seen from Eq. (15) that
g2 = 0. Then Eqs. (14) and (13) are automatically
satisfied. From Eq. (12), we get
g1 = Rp ,
2
(33)
with
R2 =
15ηq32
.
2β
Equation (11) yields
123
D − 5ηp 2 q2
D 2 − 10ηp 2 q2 D + 25η2 p 4 q22 − 30αη = 0.
(36)
D 2 − 10ηp 2 q2 D + 25η2 p 4 q22 − 30αη = 0.
D
,
5ηp 2
(35)
(30)
q2 =
−(D − 5ηp 2 q2 )R
.
15ηq3
Comparing Eqs. (9) and (10), it is seen in the same
way as in the previous case that they cannot coincide,
and consequently, we restrict us to the solution with
constant g1 and f . Equation (28) factorizes as above
into
and Eq. (28) can be factorized into
f =
(34)
Using the same procedure as in the case with nonzero
q4 , we find that setting the second factor to zero does
not yield a real solution. Hence, we set q2 as in (31).
Then Eq. (10) reduces to
α = 4ηp 4 q22 ,
(37)
which yields a second solution (“solution B”), with g1
given by (33), g2 = 0, f = 0, p free, q2 given by (31),
q3 free, q4 = 0.
Let us look in some detail to the auxiliary ordinary differential equation (8). We have found some new
soliton-like solutions to Eq. (8) as follows:
(i) When 4q2 q4 − q32 > 0, q2 > 0,
ϕ (ξ ) =
q3 +
−2q2
4q2 q4 − q32 sinh
√ ,
q2 ξ
(38)
(ii) When q32 − 4q2 q4 > 0, q2 > 0,
√ q
−2q2 sech2 2 2 ξ
ϕ (ξ ) = √ .
q
2 q32 − 4q2 q4 −
q32 − 4q2 q4 − q3 sech2 2 2 ξ
(39)
Furthermore, it is known that Eq. (8) possesses the
following exact solutions [46]:
(iii) When q2 > 0,
√ q
−q2 q3 sech2 ± 2 2 ξ
ϕ (ξ ) =
√ 2 ,
q
q32 − q2 q4 1 − tanh ± 2 2 ξ
(iv) When q32 − 4q2 q4 > 0, q2 > 0,
(40)
Soliton solutions of nonlinear diffusion–reaction-type equations
√ q2 ξ
ϕ (ξ ) = √ .
q32 − 4q2 q4 − q3 sech q2 ξ
2q2 sech
2119
(41)
Substituting the solutions (38)–(41) into (6), we
obtain the following new soliton-like solutions of Eq.
(4):
Type 1. If 4q2 q4 − q32 > 0 and q2 > 0,
2q2 g1 (t)
u 1 (x, t) = f (t) −
√ q3 + 4q2 q4 − q32 sinh q2 ξ
+
q3 +
4q22 g2 (t)
4q2 q4 − q32 sinh
,
√ 2
q2 ξ
(42)
Type 2. If
q32
− 4q2 q4 > 0 and q2 > 0,
u 2 (x, t) = f (t)
√ q
2q2 g1 (t)sech2 2 2 ξ
− √ q
2 q32 − 4q2 q4 −
q32 − 4q2 q4 − q3 sech2 2 2 ξ
√ q
4q22 g2 (t)sech4 2 2 ξ
+
√ 2 ,
q2
2
2
2
q3 − 4q2 q4 − q3 sech
2 q3 − 4q2 q4 −
2 ξ
(43)
Type 3. If q2 > 0,
√ q
q2 q3 g1 (t)sech2 ± 2 2 ξ
u 3 (x, t) = f (t) −
√ 2
q
q32 − q2 q4 1 − tanh ± 2 2 ξ
√ q
g2 (t)q22 q32 sech4 ± 2 2 ξ
+
√ 2 2 ,
q
2
q3 − q2 q4 1 − tanh ± 2 2 ξ
(44)
− 4q2 q4 > 0 and q2 > 0,
√ 2g1 (t)q2 sech q2 ξ
u 4 (x, t) = f (t) + √ q32 − 4q2 q4 − q3 sech q2 ξ
√ 4g2 (t)q22 sech2 q2 ξ
+ .
√ 2
2
q3 − 4q2 q4 − q3 sech q2 ξ
Type 4. If
q32
(45)
In each case, ξ = p (t) x +q (t), and the coefficients
are given by one of the two solutions A and B of system
(9)–(17), namely:
Fig. 1 Solution (46) to Eq. (1), for parameters η = 0.5, β = 0.3,
D = 1, δ = 0.5, and p = 1. Blue solid line and red dashed line
are the real and imaginary parts of u, respectively. (Color figure
online)
– In the case A, q, q2 , q4 , g1 , and g2 are given (18),
(31), (26), (21), and (19), respectively, with ρ given
by (20), q3 remains free, f = 0, α satisfies (37).
– In the case B, q, q2 , g1 , given by (18), (31), (33),
respectively, with R given by (34), q4 = 0, g2 = 0,
f = 0, q3 free, α satisfies (37).
The Type 1 solution, in the case A, reduces to zero.
In the case B, we obtain
√
1
− 30D
√ ,
u 1 (x, t) = √
5 ηβ) 1 + i ± sinh q2 ξ
(46)
where
q2 =
D
.
5ηp 2
(47)
u 1 (x, t) is not real. Indeed, the condition 4q2 q4 − q32 >
0 is never satisfied since q4 = 0, and q3 = 0. An
example of this solution is shown on Fig. 1, it is a
localized bright soliton.
The condition g1 = 0, which results from Eq. (92),
does not induce any restriction on the t-dependency of
the coefficients, since q3 can be chosen as a function of
t in such a way that g1 is a constant (cf. Eq. (77)) and
then vanishes from the final expression of u(x, t).
The Type 2 solution also vanishes in case A. In case
B, it reduces to
√
sech2 q2 ξ/2
30 D
√
u(x, t) = −
ηβ 5 2 − (1 − )sech2 q2 ξ/2
(48)
123
2120
H. Triki et al.
with
q2 =
D
,
5ηp02
(49)
and = sgn(q3 ) = ±1. For positive q3 , it reduces to
a sech-square solution. For negative q3 , solution (48)
remains valid and real, but is singular on the line ξ = 0
in the (x, t) plane. It reduces to a hyperbolic cosecant
square, as
√
− 30D
.
√
u= √
10 ηβ sinh2 q2 ξ/2
(50)
The Type 3 solution in case B is also nontrivial but of
less interest since it is the known sech-square solution:
The Type 3 solution in case A is
√
√
q
sech2 2 2 ξ
−4 30D
u(x, t) =
√
2
√
q
5 ηβ
tanh 2 2 ξ − 3
Fig. 2 Solution (51) to Eq. (1), for parameters η = 0.50, β =
0.3, D = 1, δ = 0.5, and p0 = 1
(51)
√
√
q2
−D 15
ξ ,
sech2
u(x, t) = √
2
5 2βη
(56)
with
with
D
q2 =
,
5ηp(t)2
q2 =
(52)
and ξ given by Eq. (7), with q given by (18). The condition that g1 must be constant can be canceled by an
adequate choice for q3 , but the condition that g2 must
be constant remains. It can be reduced to
√
g2 =
√
30 ηβ
,
48D
(53)
√
ηβ
D
= 0.
(54)
Solution (51) is real provided that ηβ > 0 and ηD > 0.
An example of it is shown in Fig. 2.
However, it can be proved that
(tanh X − 3)2 cosh2 X = 8 cosh2 (X − atanh(1/3))
(55)
from which we deduce that solution (51) is nothing but
a shifted sech-square.
123
(57)
It also requires that βη > 0 and ηD > 0 to be real.
The solution of Type 4 is zero in the case A. In case
B, it can be written as
√ √
sech q2 ξ
−2 15D
√ ,
√
5 2βη 1 + sech q2 ξ
(58)
with q2 as in (57). Since
1
sech2X
= sech2 X,
1 + sech2X
2
and consequently, we must have
d
dt
D
.
5ηp 2
(59)
we see that solution (58) is a sech-square. Apart from
the shift in ξ , it is identical to (51).
2.2 Exact solutions of Eq. (2)
Here we are interested in finding soliton-like solutions
of the time-dependent DR equation (2):
u t + η(t)u x x x x − D(t)u x x + α(t)u − β(t)u 3
+ γ (t)u 5 + δ(t)u x = 0,
(60)
Considering homogeneous balance between u x x x x
and u 5 terms in (60), we get: M + 4 = 5M. This
Soliton solutions of nonlinear diffusion–reaction-type equations
2121
implies that M = 1. Accordingly, we adopt the ansatz
of Zhao et al. [42] with a modification for the solution
of (60) as follows
Equation (69) is identical to (16) and gives the same
expression of q as (18) above. Equation (68) gives
g1 = ρp,
u = f (t) + g1 (t)ϕ (ξ ) ,
(70)
(61)
with
where the definition of ξ stays the same as in (7).
Here f (t), g1 (t), p(t), and q(t) are time-dependent
functions that will be determined, and ϕ (ξ ) satisfies an
auxiliary equation of the form [46]
dϕ
dξ
2
= q4 ϕ 4 + q3 ϕ 3 + q2 ϕ 2 ,
(62)
where q2 , q3 , and q4 are constants. Substituting (61)
and (62) into (60) and equating the coefficients of ϕ i
(where i = 0, . . . , 5), and ϕ to zero, we, respectively,
obtain
ϕ 0 : f + α f − β f 3 + γ f 5 = 0,
(63)
ρ4 =
−24ηq42
.
γ
Then (67) yields
f =
q3
g1 .
4q4
− 2Dp 2 q4 g1 − βg13 + 10γ f 2 g13 = 0,
8q2 q42 ρ 2 Dγ − 8ηq22 q42 g12 γ + 3ηq34 g12 γ − 24βηq32 q42 = 0.
(73)
(64)
(65)
(66)
ϕ 4 : 30ηp 4 q3 q4 g1 + 5γ f g14 = 0,
(67)
ϕ : 24ηp 4 q42 g1 + γ g15 = 0,
ϕ : g1 p x + q + δg1 p =
(68)
5
0
(72)
We report (70) and (72) into Eqs. (63) and (64), then
multiply Eq. (63) by 4q4 and Eq. (64) by q3 , and subtract the latter from the former, which yields
ϕ 1 : g1 + ηp 4 q22 g1 − Dg1 p 2 q2 + αg1
−3β f 2 g1 + 5γ f 4 g1 = 0,
15 4
ϕ2 :
ηp q2 q3 g1
2
3
− Dq3 g1 p 2 − 3β f g12 + 10γ f 3 g12 = 0,
2 15
3
ϕ : ηp 4 20q2 q4 g1 + q32 g1
2
(71)
Equation (73) is a polynomial equation for g1 . However, g1 is free to vary if the polynomial is identically
zero. After some computation, it is seen that no solution of this kind exist. Hence, we restrict to a constant
g1 , which is computed from (73) so that
g12
− 24q42 8q2 q42 D + βq32 ρ 2
2 2
=
.
8q2 q4 − 3q34 ρ 2 γ
(74)
Equation (74) is reported into Eq. (66) (together with
(70) and (72) ), and Eq. (66) is reduced to
(69)
24q42 4q2 q4 − q32 72q2 q42 D − 12q32 q4 D − 4βq2 q4 ρ 2 + 9βq32 ρ 2
2 2
= 0.
8q2 q4 − 3q34
(75)
123
2122
H. Triki et al.
u 2 (x, t) = f (t)
Hence,
q4 =
q32
.
4q2
(76)
√ q
2q2 g1 (t)sech2 2 2 ξ
− √ ,
q
2 q32 − 4q2 q4 −
q32 − 4q2 q4 − q3 sech2 2 2 ξ
(82)
This condition is necessary. Indeed, it is straightforward
to see that q4 cannot be 0. Further, if we compute D
so that the last factor in (75) is zero, and insert it into
Eq. (65), we get after simplification the condition (76)
itself. Then (74) reduces to
g12 =
3q34 D + 6βq2 q32 ρ 2
√ q
q2 q3 g1 (t)sech2 ± 2 2 ξ
u 3 (x, t) = f (t) −
√ 2 ,
q
2
q3 − q2 q4 1 − tanh ± 2 2 ξ
(77)
10q23 ρ 2 γ
We report (74) and (76) into Eqs. (63)–(65). Equation (65) is automatically satisfied, and then Eq. (64) is
solved to yield
γ =
provided that q32 − 4q2 q4 > 0 and q2 > 0,
−12β 2 ηq32
.
(83)
if taking q2 > 0, and
√ q2 ξ
u 4 (x, t) = f (t) + √ ,
q32 − 4q2 q4 − q3 sech q2 ξ
2g1 (t)q2 sech
(78)
(84)
Then (78) is reported into Eq. (63), and Eq. (63) is
automatically satisfied.
However, expression (71) of ρ involves q4 , which
depends on q2 according to (76). But q2 itself depends
on ρ according to (78). We solve (78) in terms of q2 as
which exist provided that q32 − 4q2 q4 > 0 and q2 > 0.
Here g1 is given by (77), q2 from (79), q3 is free, q4
is given by (76). Then f is deduced from g1 using (72),
and p using (70), where ρ is an arbitrary parameter.
Substitution gives:
−g1 q3 12β 2 η − 50αηγ + 3D 2 γ
,
u1 = u2 = u4 =
2βρ 2 Dγ
(85)
q2 =
3q32 D 2
+ 2βq2 ρ 2 D − 50αηq32
−(3γ q32 D 2 + (12β 2 η − 50αηγ )q32 )
,
2βγρ 2 D
(79)
and report the result into (71) using (76). ρ vanishes
from the resulting equation, which reduces to
and these solutions are constant ones.
−6β 2 ηρ 4 D 2 γ = (3D 2 γ − 50αηγ + 12β 2 η)2 .
u3 =
(80)
−g1 q3 12β 2 η − 50αηγ + 3D 2 γ
2βρ 2 Dγ
+
√ q
2g1 q3 12β 2 η−50αηγ + 3D 2 γ sech2 ± 2 2 ξ
,
√ 2
q
βρ 2 Dγ 4 − 1 − tanh ± 2 2 ξ
Hence, the solution exists only if condition (80) on the
coefficients is fulfilled.
Further substitution of the expressions (38)–(41)
into (61), respectively, gives the following soliton-like
solutions to (60):
(86)
with
u 1 (x, t) = f (t) −
2q2 g1 (t)
√ ,
q3 + 4q2 q4 − q32 sinh q2 ξ
q2 =
−q3 2 12β 2 η − 50αηγ − 3q3 2 D 2 γ
.
2β Dγρ 2
(87)
(81)
provided that
123
4q2 q4 − q32
> 0 and q2 > 0.
(g1 given by (77) has not been substituted except where
g12 did appear).
Soliton solutions of nonlinear diffusion–reaction-type equations
2123
2.3 Exact solutions of Eq. (3)
From (96) and (97), we get
We now look for soliton-like solutions of the nonlinear
DR equation with time-dependent coefficients (3):
g1 =
u t + η(t)u x x x x − D(t)u x x + α(t)u
− β(t) (u x ) + δ(t)uu x = 0,
(88)
Notably, we have found that Eq. (88) does not admit
physically viable solutions using the generalized auxiliary ordinary differential equation (8) or (62). We will
use a reduced auxiliary equation of the form [45]:
dϕ
= b + ϕ2,
dξ
(89)
where b is a constant and ξ is the same as in (7).
Balancing u x x x x with (u x )2 in (88) gives
M + 4 = 2 (M + 1) so that M = 2. Accordingly, we
assume an ansatz solution of (88) of the form
u = f (t) + g1 (t)ϕ (ξ ) + g2 (t)ϕ 2 (ξ ) ,
(90)
where f (t), g1 (t), g2 (t), p(t), and q(t) are functions
of t, which are unknown and to be further determined.
Substituting (90) along with (89) into (88) and then
setting the coefficients of ϕ i (where i = 0, . . . , 6) to
zero, we obtain the following set of algebraic equations:
ϕ 0 : f + g1 b p x + q + 16ηp 4 g2 b3
− 2Db2 g2 p 2 + α f − βg12 b2 p 2 + δ f g1 pb = 0,
(91)
ϕ 1 : g1 + 2g2 b p x + q + 16ηp 4 g1 b2 − 2Dbg1 p 2
+ αg1 − 4βb g1 g2 p + 2δbg2 p f
2
ϕ : g1 p x + q +
2
+ δbg12 p
= 0,
(92)
g2
+ 136ηp g2 b
4
2
− 8Dbg2 p 2 + αg2 − 2βbg12 p 2 − 4βb2 g22 p 2
+ δg1 p f + 3δbg1 g2 p = 0,
3
ϕ : 2g2 p x + q + 40ηp 4 g1 b − 2Dg1 p 2
(93)
− 8βg1 g2 bp 2 + 2δg2 p f + δg12 p + 2δbg22 p = 0,
(94)
ϕ : 240ηp g2 b − 6Dg2 p
4
4
2
− 8βbg22 p 2
+ 3δpg1 g2 − βg12 p 2 = 0,
ϕ :
5
ϕ6 :
−4βg1 g2 p + 2δg22 p + 24ηg1 p 4
120ηp 4 g2 − 4βp 2 g22 = 0,
2
(98)
30ηp 2
β
(99)
and
2
2
75δηp
4β 2
(95)
= 0,
g2 =
The substitution of Eqs. (98) and (99) into Eq. (95)
yields an important constraint equation between the
model coefficients δ(t), η(t), D(t), and β (t) as
475δ 2 η − 64Dβ 2 = 0,
which means that the parameters δ(t), η(t), D(t), and
β (t) are not independent and the corresponding solitary wave solutions are obtained in the framework of
this relationship.
By multiplying (94) by b, subtracting the latter from
(92), and using (96), we obtain
g1 + αg1 = 0.
(101)
Evidently, the integration of (101) determines the
function g1 (t) as follows
g1 (t) = k1 e−
α(t)dt
,
(102)
where k1 is an arbitrary constant. Now using (98) and
(102), we obtain
p(t) =
4k1 β 2 − α(t)dt
e
.
75δη
(103)
Inserting (103) into (99), we have
g2 =
32k12 β 3 −2 α(t)dt
e
.
375δ 2 η
(104)
By multiplying (93) by b and subtracting the resulting
equation from (91) and using (95), we obtain
f + α f = b g2 + αg2 ,
(105)
which gives after the integration:
(96)
(97)
(100)
f − bg2 = k2 e−
α(t)dt
,
(106)
123
2124
H. Triki et al.
where k2 is an arbitrary constant. Using the expressions (104) and (106), it is now possible to calculate
the function f (t) as
f = k 2 e−
α(t)dt
+
32k12 bβ 3 −2 α(t)dt
e
.
375δ 2 η
(107)
Lastly, the t-dependence of the wave parameter q(t)
is found from integrating (94) as
375δηpδ 2
65δηp 3 b
−p x −
q (t) =
+
64β 3
2β
5δ Dp
+
− δp f dt,
(108)
8β
where b being an arbitrary nonzero constant.
Returning to the auxiliary ordinary differential equation (89), we can see that it has the following general
solutions [44,45]
√
√
−bξ ,
ϕ (ξ ) = − −b tanh
(109)
if b < 0. By inserting (109) into (90), we obtain an
exact soliton solution of the form:
√
75δηp √
u = f (t) −
−b
tanh
−bξ
4β 2
√
2
30ηp b
tanh2
−bξ ,
(110)
−
β
where ξ = p (t) x + q (t). Here the soliton parameters
p(t), f (t), and q(t) are given in (103), (107), and (108),
while b is an arbitrary nonzero constant.
There remains one equation which is not satisfied
(Eq. 91), which reduces after substitution to
− bk12 e−6αt 253125δ 2 e4αt D 4 + 2462400αβ 2 e4αt D 3
+ 324900bβ 2 δ 2 k12 e2αt D 2 − 130321b2 β 4 δ 2 k14
3888000β D 4 = 0,
(111)
(assuming α constant for simplicity). This equation
cannot be satisfied unless b = 0 or k1 = 0. In both
cases, solution (110) reduces to
u = k2 e−αt .
(112)
Indeed, if u does not depend on x, Eq. (3) reduces to
u t + α(t)u = 0,
123
(113)
which obviously admits solution (112), if we assume
α to be constant. The generalization to nonconstant α
does not modify the conclusion.
3 Conclusions
We have considered three variants of nonlinear diffusion–
reaction-type equations with time-dependent coefficients having both short-range and long-range diffusion terms. A variety of new soliton-like solutions are
obtained by means of a generalized auxiliary equation
method for solving the first two variants of equations
and a simplified auxiliary equation for solving the third
variant. Besides, conditions for the existence of soliton
solutions have also been reported. To the best of our
knowledge, the soliton-like solutions obtained by the
used method are entirely new and have not been previously presented for the focusing diffusion–reaction
equations with variable coefficients. Note that the study
of nonlinear models with variable coefficients that support soliton-type solutions is very important to understand nonlinear physical phenomena arising in inhomogeneous media.
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