DOI 10.1007/s11071-016-3020-x O R I G I NA L PA P E R
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 long- range 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 bio- logical problems. We employ the auxiliary equation method to derive a variety of new soliton-like solutions for these models. Parametric conditions for the exis- tence 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: trikihouria@gmail.com 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 soli- tons is normally confined to homogeneous nonlinear media. In such media, the soliton dynamics is described by nonlinear evolution equations (NLEEs) with con- stant 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 long- distance 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 vari- able coefficients has also been studied recently in the context of ocean waves, where the spatiotemporal vari- ability of the coefficients is due to the changes in the water depth and other physical conditions [2]. Basi- cally, the existence of the inhomogeneities in media influences the accompanied physical effects giving rise to spatial or temporal dispersion and nonlinearity vari- ations; 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 sys- tem with an infinite number of degrees of freedom
[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 conden- sates) [11–15], and nonlinear optics [16–28]. The dis- tinction 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 tech- nique [34,35], among others. Having obtained exact solutions is helpful for realizing properties of the non- linear equation and for understanding physical phe- nomena described by the model considered. In addi- tion, 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 ux 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 incorpo- ration of the termux 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 equa- tions with derivative-type and/or algebraic-type non- linearities having both long-range diffusion through ux x x x-term and short-range diffusion through ux x- term. Importantly, the simplest case when all coeffi- cients in three variants are constant was investigated in
[45] using a simplified auxiliary equation of the form dz/dξ =b+z2(ξ), wherebis a constant.
Here we go beyond the previous study on the non- linear 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:
ut+η(t)ux x x x−D(t)ux x
+α(t)u−β(t)u3+δ(t)ux=0, (1) ut+η(t)ux x x x−D(t)ux x+α(t)u−β(t)u3
+γ (t)u5+δ(t)ux =0, (2)
ut+η(t)ux x x x−D(t)ux x+α(t)u
−β(t) (ux)2+δ(t)uux =0, (3) where α (t) , β (t) ,D(t) , γ (t) , η (t), and δ (t) are arbitrary functions oft.
To the best of our knowledge, exact analytic solu- tions 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 aux- iliary 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 solu- tion 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 equa- tion with time-dependent coefficients (1):
ut+η(t)ux x x x−D(t)ux x+α(t)u−β(t)u3
+δ(t)ux =0, (4) Balancing the highest-order derivative termux x x xwith the nonlinear termu3in (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]
u= f(t)+g1(t)ϕ (ξ)+g2(t)ϕ2(ξ) , (6)
ξ = p(t)x+q(t). (7)
The functionϕ (ξ)satisfies an auxiliary ordinary dif- ferential equation of the form [46]
dϕ dξ
2
=q4ϕ4+q3ϕ3+q2ϕ2, (8) where q2,q3, and q4 are constants. Here, in (6) and (7), f(t),g1(t),g2(t),p(t), andq(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 ofϕi (where i = 0, . . . ,6), ϕ, and ϕϕ to zero, we arrive at the following set of equations:
ϕ0: f+αf −βf3=0, (9)
ϕ1:g1+ηp4q22g1−Dg1p2q2
−3βf2g1+αg1=0, (10) ϕ2:g2+ηp4
15
2q2q3g1+16q22g2
−3β
f2g2+ f g12
−Dp2 3
2q3g1+4q2g2
+αg2=0, (11) ϕ3:ηp4
20q2q4g1+65q2q3g2+15 2 q32g1
−Dp2(2q4g1+5q3g2)−β
g13+6f g1g2
=0, (12) ϕ4:ηp4
30q3q4g1+105
2 q32g2+120q2q4g2
−6Dq4p2g2−3β
g21g2+ f g22
=0, (13)
ϕ5:ηp4
24q42g1+168q3q4g2
−3βg1g22=0, (14) ϕ6:120ηp4q42g2−βg32=0, (15) ϕ:g1
px+q
+δg1p=0, (16) ϕϕ:2g2
px+q
+2δg2p =0, (17) Equations (16) and (17) reduce to
q= − t
px+δp
dt. (18)
From Eq. (15), we see that
g =ρq p2, (19)
with ρ2=120η
β . (20)
Then Eq. (14) gives g1= q3
2 ρp2. (21)
Equation (12) can be solved to yield
f =μp2+ν, (22)
with ν= −8D
4βρ (23)
and
μ= 100ηq2q4−15ηq32
4βq4ρ . (24)
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
16q4
4q2q4−q32 ρp4
×
4q4D−20ηq2q4p2+15ηq32p2
=0. (25)
This condition is satisfied if q4= q32
4q2
(26) (the other solution will be considered after).
Then Eq. (10) reduces to
8ρp2D2−80ηd(ρp2)
dt +βηq22ρ3p4−80αηρp2=0 (27) and Eq. (9) to
24ρD3−360ηq2ρp2D2+15βηq22ρ3p4D−720αηρD +3600η2q2d(ρp2)
dt −25βη2q23ρ3p6+3600αη2q2ρp2=0.
(28)
Eliminating the derivative between (27) and (28) yields a polynomial equation forρp2. Thus, the differential equations (27) and (28) can be solved when this poly- nomial 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(ρp2)
dt =0, (29)
and Eq. (28) can be factorized into
D−5ηp2q2 D2−10ηp2q2D+25η2p4q22−30αη
=0.
(30) If we set the second factor to zero, computeα, and report it into Eq. (27), we see that the equation has no real solution. But if we set
q2= D
5ηp2, (31)
and report it into Eq. (27), we see that it is solved under the condition
α= 8D2
50η. (32)
This first solution (“solution A”) exists under condition (32),q2is given by (31),q3remains free,q4is given by (26),g1andg2are related top, 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 thatq4must be zero.
It is more clear and safe to seek from the beginning a solution withq4 = 0. It is seen from Eq. (15) that g2 = 0. Then Eqs. (14) and (13) are automatically satisfied. From Eq. (12), we get
g1=R p2, (33)
with R2= 15ηq32
2β . (34)
Equation (11) yields
f =−(D−5ηp2q2)R
15ηq3 . (35)
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 constantg1 and f. Equation (28) factorizes as above into
D−5ηp2q2 D2−10ηp2q2D+25η2p4q22−30αη
=0. (36) 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 setq2as in (31).
Then Eq. (10) reduces to
α=4ηp4q22, (37)
which yields a second solution (“solution B”), withg1
given by (33),g2=0, f =0,pfree,q2given by (31), q3free,q4=0.
Let us look in some detail to the auxiliary ordi- nary differential equation (8). We have found some new soliton-like solutions to Eq. (8) as follows:
(i) When 4q2q4−q32>0,q2>0,
ϕ (ξ)= −2q2
q3+ 4q2q4−q32sinh√
q2ξ, (38) (ii) Whenq32−4q2q4>0,q2>0,
ϕ (ξ)= −2q2sech2
√q2 2 ξ 2 q23−4q2q4− q32−4q2q4−q3
sech2√q2 2 ξ.
(39) Furthermore, it is known that Eq. (8) possesses the following exact solutions [46]:
(iii) Whenq2>0,
ϕ (ξ)= −q2q3sech2
±√2q2ξ
q32−q2q4
1−tanh
±√2q2ξ2, (40)
(iv) Whenq32−4q2q4>0,q2>0,
ϕ (ξ)= 2q2sech√ q2ξ q32−4q2q4−q3sech√
q2ξ. (41) Substituting the solutions (38)–(41) into (6), we obtain the following new soliton-like solutions of Eq.
(4):
Type 1. If 4q2q4−q32>0 andq2>0, u1(x,t)= f(t)− 2q2g1(t)
q3+ 4q2q4−q32sinh√ q2ξ + 4q22g2(t)
q3+ 4q2q4−q32sinh√ q2ξ2,
(42) Type 2. Ifq32−4q2q4>0 andq2>0,
u2(x,t)= f(t)
− 2q2g1(t)sech2√q2 2 ξ 2 q32−4q2q4− q23−4q2q4−q3
sech2√q2 2 ξ
+ 4q22g2(t)sech4√q2 2 ξ
2 q32−4q2q4− q32−4q2q4−q3
sech2√q2 2 ξ2,
(43) Type 3. Ifq2>0,
u3(x,t)= f(t)− q2q3g1(t)sech2
±√2q2ξ q32−q2q4
1−tanh
±√2q2ξ2
+ g2(t)q22q32sech4
±√2q2ξ
q32−q2q4
1−tanh
±√2q2ξ22,
(44) Type 4. Ifq32−4q2q4>0 andq2>0,
u4(x,t)= f(t)+ 2g1(t)q2sech√ q2ξ q32−4q2q4−q3sech√
q2ξ + 4g2(t)q22sech2√
q2ξ q32−4q2q4−q3sech√
q2ξ2. (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 parametersη=0.5,β=0.3, D=1,δ=0.5, andp=1.Blue solid lineandred dashed line are the real and imaginary parts ofu, respectively. (Color figure online)
– In the case A,q,q2,q4,g1, andg2are given (18), (31), (26), (21), and (19), respectively, withρgiven by (20),q3remains free, f =0,αsatisfies (37).
– In the case B,q,q2,g1, given by (18), (31), (33), respectively, withRgiven by (34),q4=0,g2=0,
f =0,q3free,αsatisfies (37).
The Type 1 solution, in the case A, reduces to zero.
In the case B, we obtain u1(x,t)= −√
30D 5√
ηβ)
1 1+i±sinh√
q2ξ, (46) where
q2= D
5ηp2. (47)
u1(x,t)is not real. Indeed, the condition 4q2q4−q32>
0 is never satisfied sinceq4 = 0, and q3 = 0. An example of this solution is shown on Fig. 1, it is a localized bright soliton.
The conditiong1 =0, which results from Eq. (92), does not induce any restriction on thet-dependency of the coefficients, sinceq3can be chosen as a function of tin such a way thatg1is a constant (cf. Eq. (77)) and then vanishes from the final expression ofu(x,t).
The Type 2 solution also vanishes in case A. In case B, it reduces to
u(x,t)= −
30 ηβ
D 5
sech2√ q2ξ/2 2−(1−)sech2√
q2ξ/2 (48)
with q2= D
5ηp02, (49)
and=sgn(q3)= ±1. For positiveq3, it reduces to a sech-square solution. For negativeq3, solution (48) remains valid and real, but is singular on the lineξ =0 in the(x,t)plane. It reduces to a hyperbolic cosecant square, as
u= −√
30D 10√
ηβsinh2√
q2ξ/2. (50)
The Type 3 solution in case A is
u(x,t)=−4√ 30D 5√
ηβ
sech2
√q2
2 ξ
tanh
√q2 2 ξ−3
2 (51)
with q2= D
5ηp(t)2, (52)
andξ given by Eq. (7), withqgiven by (18). The con- dition thatg1must be constant can be canceled by an adequate choice forq3, but the condition thatg2must be constant remains. It can be reduced to
g2=
√30√ ηβ
48D , (53)
and consequently, we must have d
dt √
ηβ D
=0. (54)
Solution (51) is real provided thatηβ >0 andηD>0.
An example of it is shown in Fig.2.
However, it can be proved that
(tanhX−3)2cosh2X =8 cosh2(X−atanh(1/3)) (55) from which we deduce that solution (51) is nothing but a shifted sech-square.
Fig. 2 Solution (51) to Eq. (1), for parametersη=0.50,β= 0.3,D=1,δ=0.5, andp0=1
The Type 3 solution in case B is also nontrivial but of less interest since it is the known sech-square solution:
u(x,t)=−D√ 15 5√
2βη sech2 √
q2
2 ξ
, (56)
with q2= D
5ηp2. (57)
It also requires thatβη >0 andηD>0 to be real.
The solution of Type 4 is zero in the case A. In case B, it can be written as
−2√ 15D 5√
2βη
sech√ q2ξ 1+sech√
q2ξ, (58)
withq2as in (57). Since sech2X
1+sech2X =1
2sech2X, (59)
we see that solution (58) is a sech-square. Apart from the shift inξ, 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):
ut+η(t)ux x x x−D(t)ux x+α(t)u−β(t)u3
+γ (t)u5+δ(t)ux =0, (60)
Considering homogeneous balance betweenux x x x
and u5 terms in (60), we get: M +4 = 5M. This
implies thatM =1.Accordingly, we adopt the ansatz of Zhao et al. [42] with a modification for the solution of (60) as follows
u= f(t)+g1(t)ϕ (ξ) , (61)
where the definition ofξ stays the same as in (7).
Here f(t),g1(t),p(t), andq(t)are time-dependent functions that will be determined, andϕ (ξ)satisfies an auxiliary equation of the form [46]
dϕ dξ
2
=q4ϕ4+q3ϕ3+q2ϕ2, (62) whereq2,q3, andq4are constants. Substituting (61) and (62) into (60) and equating the coefficients ofϕi (wherei=0, . . . ,5),andϕto zero, we, respectively, obtain
ϕ0: f+αf −βf3+γf5=0, (63) ϕ1:g1 +ηp4q22g1−Dg1p2q2+αg1
−3βf2g1+5γf4g1=0, (64) ϕ2: 15
2 ηp4q2q3g1
−3
2Dq3g1p2−3βf g12+10γf3g12=0, (65) ϕ3:ηp4
20q2q4g1+15 2 q32g1
−2Dp2q4g1−βg31+10γf2g31=0, (66) ϕ4:30ηp4q3q4g1+5γf g14=0, (67) ϕ5:24ηp4q42g1+γg51=0, (68) ϕ:g1
px+q
+δg1p=0 (69)
Equation (69) is identical to (16) and gives the same expression ofq as (18) above. Equation (68) gives
g1=ρp, (70)
with
ρ4=−24ηq42
γ . (71)
Then (67) yields f = q3
4q4
g1. (72)
We report (70) and (72) into Eqs. (63) and (64), then multiply Eq. (63) by 4q4and Eq. (64) byq3, and sub- tract the latter from the former, which yields
8q2q42ρ2Dγ−8ηq22q42g21γ+3ηq34g12γ−24βηq32q42=0. (73) Equation (73) is a polynomial equation forg1. How- ever,g1is free to vary if the polynomial is identically zero. After some computation, it is seen that no solu- tion of this kind exist. Hence, we restrict to a constant g1, which is computed from (73) so that
g21=− 24q42
8q2q42D+βq32ρ2 8q22q42−3q34
ρ2γ . (74)
Equation (74) is reported into Eq. (66) (together with (70) and (72) ), and Eq. (66) is reduced to
24q42
4q2q4−q32 72q2q42D−12q32q4D−4βq2q4ρ2+9βq32ρ2
8q22q42−3q34 =0. (75)
Hence,
q4= q32
4q2. (76)
This condition is necessary. Indeed, it is straightforward to see thatq4cannot 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
g21=3q34D+6βq2q32ρ2
10q23ρ2γ (77)
We report (74) and (76) into Eqs. (63)–(65). Equa- tion (65) is automatically satisfied, and then Eq. (64) is solved to yield
γ = −12β2ηq32
3q32D2+2βq2ρ2D−50αηq32. (78) Then (78) is reported into Eq. (63), and Eq. (63) is automatically satisfied.
However, expression (71) of ρ involves q4, which depends onq2according to (76). Butq2itself depends onρaccording to (78). We solve (78) in terms ofq2as
q2=−(3γq32D2+(12β2η−50αηγ )q32)
2βγρ2D , (79)
and report the result into (71) using (76).ρ vanishes from the resulting equation, which reduces to
−6β2ηρ4D2γ =(3D2γ−50αηγ +12β2η)2. (80) 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):
u1(x,t)= f(t)− 2q2g1(t) q3+ 4q2q4−q32sinh√
q2ξ, (81) provided that 4q2q4−q32>0 andq2>0.
u2(x,t)= f(t)
− 2q2g1(t)sech2√q2 2 ξ 2 q32−4q2q4− q32−4q2q4−q3
sech2√q2 2 ξ,
(82) provided thatq32−4q2q4>0 andq2>0,
u3(x,t)= f(t)− q2q3g1(t)sech2
±√2q2ξ q32−q2q4
1−tanh
±√2q2ξ2,
(83) if takingq2>0,and
u4(x,t)= f(t)+ 2g1(t)q2sech√ q2ξ q32−4q2q4−q3sech√
q2ξ, (84) which exist provided thatq32−4q2q4>0 andq2>0.
Hereg1is given by (77),q2from (79),q3is free,q4
is given by (76). Then f is deduced fromg1using (72), andpusing (70), whereρis an arbitrary parameter.
Substitution gives:
u1=u2=u4= −g1q3
12β2η−50αηγ +3D2γ
2βρ2Dγ ,
(85) and these solutions are constant ones.
u3= −g1q3
12β2η−50αηγ +3D2γ
2βρ2Dγ +
2g1q3
12β2η−50αηγ +3D2γ sech2
±√2q2ξ βρ2Dγ
4−
1−tanh
±√2q2ξ2 ,
(86) with
q2= −q32
12β2η−50αηγ
−3q32D2γ
2βDγρ2 . (87)
(g1given by (77) has not been substituted except where g21did appear).
2.3 Exact solutions of Eq. (3)
We now look for soliton-like solutions of the nonlinear DR equation with time-dependent coefficients (3):
ut+η(t)ux x x x−D(t)ux x+α(t)u
−β(t) (ux)2+δ(t)uux =0, (88) Notably, we have found that Eq. (88) does not admit physically viable solutions using the generalized auxil- iary ordinary differential equation (8) or (62). We will use a reduced auxiliary equation of the form [45]:
dϕ
dξ =b+ϕ2, (89)
wherebis a constant andξ is the same as in (7).
Balancing ux x x x with (ux)2 in (88) gives M+4=2(M+1)so thatM =2.Accordingly, we assume an ansatz solution of (88) of the form
u= f(t)+g1(t)ϕ (ξ)+g2(t)ϕ2(ξ) , (90) where f(t),g1(t),g2(t),p(t), andq(t)are functions oft, which are unknown and to be further determined.
Substituting (90) along with (89) into (88) and then setting the coefficients ofϕi (wherei = 0, . . . ,6) to zero, we obtain the following set of algebraic equations:
ϕ0: f+g1b
px+q
+16ηp4g2b3
−2Db2g2p2+αf −βg21b2p2+δf g1pb=0, (91) ϕ1:g1+2g2b
px+q
+16ηp4g1b2−2Dbg1p2 +αg1−4βb2g1g2p2+2δbg2p f +δbg21p=0,
(92) ϕ2:g1
px+q
+g2 +136ηp4g2b2
−8Dbg2p2+αg2−2βbg21p2−4βb2g22p2 +δg1p f +3δbg1g2p=0, (93) ϕ3:2g2
px+q
+40ηp4g1b−2Dg1p2
−8βg1g2bp2+2δg2p f +δg21p+2δbg22p=0, (94) ϕ4:240ηp4g2b−6Dg2p2−8βbg22p2
+3δpg1g2−βg12p2=0, (95) ϕ5: −4βg1g2p2+2δg22p+24ηg1p4=0, (96) ϕ6:120ηp4g −4βp2g2=0, (97)
From (96) and (97), we get g1= 75δηp
4β2 (98)
and
g2= 30ηp2
β (99)
The substitution of Eqs. (98) and (99) into Eq. (95) yields an important constraint equation between the model coefficientsδ(t), η(t),D(t), andβ (t)as
475δ2η−64Dβ2=0, (100)
which means that the parametersδ(t), η(t),D(t), and β (t)are not independent and the corresponding soli- tary wave solutions are obtained in the framework of this relationship.
By multiplying (94) byb, subtracting the latter from (92), and using (96), we obtain
g1+αg1=0. (101)
Evidently, the integration of (101) determines the functiong1(t)as follows
g1(t)=k1e−
α(t)dt, (102)
wherek1is an arbitrary constant. Now using (98) and (102), we obtain
p(t)= 4k1β2
75δη e−α(t)dt. (103)
Inserting (103) into (99), we have g2= 32k21β3
375δ2ηe−2α(t)dt. (104) By multiplying (93) byband subtracting the resulting equation from (91) and using (95), we obtain
f+αf =b
g2 +αg2
, (105)
which gives after the integration:
f −bg =k e−
α(t)dt, (106)
where k2 is an arbitrary constant. Using the expres- sions (104) and (106), it is now possible to calculate the function f(t)as
f =k2e−
α(t)dt+32k21bβ3 375δ2ηe−2
α(t)dt. (107)
Lastly, thet-dependence of the wave parameterq(t) is found from integrating (94) as
q(t)= −px−375δηpδ2
64β3 +65δηp3b 2β +5δDp
8β −δp f
dt, (108)
wherebbeing an arbitrary nonzero constant.
Returning to the auxiliary ordinary differential equa- tion (89), we can see that it has the following general solutions [44,45]
ϕ (ξ)= −√
−btanh √
−bξ
, (109)
ifb < 0. By inserting (109) into (90), we obtain an exact soliton solution of the form:
u= f(t)−75δηp 4β2
√−btanh √
−bξ
−30ηp2b β tanh2
√
−bξ
, (110)
whereξ =p(t)x+q(t). Here the soliton parameters p(t),f(t), andq(t)are given in (103), (107), and (108), whilebis an arbitrary nonzero constant.
There remains one equation which is not satisfied (Eq.91), which reduces after substitution to
−bk21e−6αt
253125δ2e4αtD4+2462400αβ2e4αtD3 +324900bβ2δ2k12e2αtD2−130321b2β4δ2k14
3888000βD4
=0, (111)
(assuming α constant for simplicity). This equation cannot be satisfied unlessb = 0 or k1 = 0. In both cases, solution (110) reduces to
u=k2e−αt. (112)
Indeed, ifudoes not depend onx, Eq. (3) reduces to
ut+α(t)u=0, (113)
which obviously admits solution (112), if we assume αto be constant. The generalization to nonconstantα does not modify the conclusion.
3 Conclusions
We have considered three variants of nonlinear diffusion–
reaction-type equations with time-dependent coeffi- cients having both short-range and long-range diffu- sion 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 pre- viously presented for the focusing diffusion–reaction equations with variable coefficients. Note that the study of nonlinear models with variable coefficients that sup- port soliton-type solutions is very important to under- stand nonlinear physical phenomena arising in inho- mogeneous media.
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