CONTROLLABLE DYNAMICAL BEHAVIORS AND THE … · controllable dynamical behaviors and the analysis...

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CONTROLLABLE DYNAMICAL BEHAVIORS AND THE ANALYSIS OFFRACTAL BURGERS HIERARCHY WITH THE FULL EFFECTS OF

INHOMOGENEITIES OF MEDIA

H.M. JARADAT1, F. AWAWDEH2, S. AL-SHARA’ 1, M. ALQURAN3, S. MOMANI4

1Department of Mathematics, Al al-Bayt University, Mafraq 25113, JordanE-mail: [email protected], [email protected]

2Department of Mathematics, Hashemite University, Zarqa 13115, JordanE-mail: [email protected]

3Department of Mathematics and Statistics, Jordan University of Scienceand Technology, Irbid 22110, Jordan

E-mail: [email protected] of Mathematics, University of Jordan, Amman 11942, Jordan

E-mail: [email protected]

Received June 17, 2014

The aim of this work is three fold. First, via a fractional complex transformationand a simplified bilinear method, N -shock-wave solutions for the variable-coefficientfractal Burgers equations are obtained, with their relevant properties and features beingillustrated. The explicit functions describing the evolution of the amplitude, phase, andvelocity are also given. Second, through the characteristic-line method and graphicalanalysis we discuss the solitonic propagation and collision, including the bidirectionalsolitons and elastic interactions. Finally, a new approach to control the effects of in-homogeneities of media and nonuniformities of boundaries on the soliton behavior issuggested.

Key words: Modified Riemann-Liouville derivative;Fractional transform; Fractional Burgers hierarchy;Wave propagation; N -shock-wave solution.

PACS: 05.45.Yv, 05.45.Df, 47.40.Nm.

1. INTRODUCTION

This paper is devoted to the study of the whole fractional Burgers hierarchy in(3+1)-dimensions of the form

u(α)t +a(t)

∂β

∂xβ

(∂β

∂βx+u

)nu+ b(t)u(β)

x + c(t)u(γ)y +d(t)u(δ)

z = 0,

n= 0,1,2, . . . (1)

Here u(α)t = ∂αu(t,x)

∂tα denotes the modified Riemann-Liouville derivative [28–30];0 < α,β,γ,δ ≤ 1 and a(t), b(t), c(t) and d(t) are time-dependent functions. The

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2 Controllable dynamical behaviors and analysis of fractal Burgers hierarchy 325

first few elements of the (3+1)-dimensional extended hierarchy (1) are given by

0 = u(α)t +a(t)u(β)

x + b(t)u(β)x + c(t)u(γ)

y +d(t)u(δ)z ,

0 = u(α)t +a(t)

(u(2β)x + 2uu(β)

x

)+ b(t)u(β)


z ,

0 = u(α)t +a(t)

(u(3β)x + 3(u(β)

x )2 + 3uu(2β)x + 3u2u(β)

x

)+ b(t)u(β)

x

+ c(t)u(γ)y +d(t)u(δ)

z ,

0 = u(α)t +a(t)

(u(4β)x + 10u(β)

x u(2β)x + 4uu(3β)

x + 12u(u(β)x )2 + 6u2u(2β)

x + 4u3u(β)x

)+ b(t)u(β)


z ,

obtained by substituting n= 0,1,2,3.The whole Burgers hierarchy of equations are significant parts of the theory of

integrable equations. They arise in various fields of fluids, plasmas, gas dynamics,traffic, hydrodynamics, population growth in the presence of disorder and convection,magnetohydrodynamic waves in a medium with finite electrical conductivity, etc.[7, 10, 34, 35]. They are also connected with applications in acoustic phenomena andhave been used to model turbulence and certain steady-state viscous flows. Moreover,Burgers equations are used to model the formation and decay of nonplanar shockwaves, where the variable x is a coordinate moving with the wave at the speed of thesound and the dependent variable u represents the velocity fluctuations [24, 47].

1.1. BACKGROUND AND RELATED WORKS

The classical Burgers equation

ut+uux = νuxx, ν = const, (2)

has been extensively used to model a variety of physical phenomena where shockcreation is an important ingredient, from the growth of molecular interfaces, throughtraffic jams to the mass distribution for the large scale structure of the Universe[32, 40]. In the application in Ref. [45], the Burgers equation is coupled with thecontinuity equation to consider the problem of passive tracer transport in Burgers ve-locity flows. Recently there appeared numerous papers on Burgers turbulence, i.e.,the theory of statistical properties of solutions to the Burgers equation with randominitial data with intriguing connections to probability theory, stochastic partial diffe-rential equations, propagation of chaos and numerical simulations [46, 54].

The fractional Burgers equation

∂f

∂X−αf ∂f

∂θ= β

∂2f

∂θ2− δ ∂

12 f

∂θ12

(3)

describes the physical processes of unidirectional propagation of weakly nonlinear

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326 H.M. Jaradat et al. 3

acoustic waves through a gas-filled pipe. Here β is the dissipation constant and αdenotes the well-known nonlinear coefficient 1

2(γ+ l), γ being the ratio of specificheats. In (3), the fractional derivative results from the cumulative (memory) effectof the wall friction through the boundary layer [50]. Generally speaking, a boundarylayer will give rise to memory effects in the form of this fractional derivative. Theremaining terms have the same physical meanings as in the usual Burgers equationexcept for the definition of the independent variables. Here X and θ denote, res-pectively, the spatial coordinate and the retarded time measured in a frame movingwith the sound speed, so that (3) describes the spatial evolution of a fluid velocityf , appropriately normalized. Note that the dissipation constant β due to the dif-fusivity of sound is far smaller than the other constant δ due to the wall friction(0 < β δ . 1). In fact, β/δ is of order R−

12D/L, where R ( 1) denotes the

acoustic Reynolds number and D and L denote, respectively, the diameter of thepipe and a characteristic wavelength (for the details, see Refs. [49–52]).

On the other hand, there is an ample physical motivation justifying considera-tion of the fractional Burgers equations

ut =−ν (−∆)α/2u−a ·∇(ur) , (4)

where α ∈ (0,2], r ≥ 1 and a is a fixed vector. One of them being the eventual goalof studying the Navier Stokes problem [6]

ut =−ν (−∆)α/2u− (u ·∇)u−∇p∇·u= 0.

Bardos et al. (1979) studied the equation

U(x,t)t =− ∂2

∂x2[U(0, t)−U(x,t)]2−ν

(− ∂2

∂x2

)α/2U(x,t) (5)

for positive definite (in x) U . It is quite different from Eq. (4) but has a similar flavorand some of its phase transition properties (as α varies from 0 to 2) are analogousto those of (4). Equation (5), arises as a modified equation for covariance functionU in the Markov random coupling model for the Burgers homogeneous turbulence.A large variety of physically motivated fractal differential equation can be found inShlesinger et al. [53], including applications to hydrodynamics, statistical mechan-ics, physiology, and molecular biology. Fractal relaxation models are described inSaichev and Woyczynski [45]. Models of several other hydrodynamical phenomena(including hereditary and viscoelastic behavior and propagation of nonlinear acous-tic waves in a tunnel with an array of Helmholtz resonators) employing the Burgersequation involving the fractional Laplacian have also been developed [49–52]. Forapplications in the theory of nonlinear Markov processes and propagation of chaosassociated with fractal Burgers equation, see Funaki and Woyczynski [13].

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Biler et al. [8] studied local and global solutions to a class of multidimen-sional generalized Burgers-type equations with a fractal power of the Laplacian inthe principal part and with general algebraic nonlinearity. Biler et al. [9] presentedthe existence and uniqueness of the source solution to the fractal Burgers equationin the critical case. Mann and Woyczynski [36] presented asymptotic properties ofsolutions and the Monte Carlo type approximation algorithms of the fractal Burgers-KPZ equations. Recently, Stanescu et al. [56] worked a numerical method based onthe interacting particles approximation for the solution of a large class of evolutionproblems involving the fractional Laplacian operator and a nonlocal quadratic-typenonlinearity. Mustafa Inc [23] and Doha et al. [11] have extended the variationaliteration method to obtain some exact solutions for a generalized fractal Burgersequation and Momani [41] has presented nonperturbative analytical solutions of thespace- and time-fractional Burgers equations by Adomian decomposition method andDoha et al. [11] used a Jacobi collocation method for solving nonlinear Burgers-typeequations. Finally, other methods such as homotopy analysis method, the first in-tegral method, solitary ansatz methods, Sinc Galerkin method and G′/G-expansionmethod have been conducted for similar studies [1–3, 25, 26, 33, 44].

1.2. CONTRIBUTIONS

A great part of the analysis of the classical Burgers Eq. (2) and its multidimen-sional counterparts is based on the intriguing connection, via the global functionalHopf-Cole formula, between the nonlinear Burgers equation and the linear heat equa-tion. This crucial simplification is no longer available in the fractional case. On theother hand, there are few studies to find soliton solutions for Burgers equations withvariable coefficients, since they are essentially complicated and their theory is still inits early stages.

In the last 50 years there have been important developments in the soliton the-ory. Solitons have been studied by mathematicians, physicists, and engineers fortheir applicability in physical applications (including plasmas, Josephson junctions,polyacetylene molecules etc.). Hirota’s bilinear method [18–21, 23] is known asone of the most effective tools to explicitly construct multi-soliton solutions of com-pletely integrable evolution equations. The first step of this method is to make suit-able transformations of nonlinear partial differential and difference equations whichprovide that the equations are in bilinear form. To find such a transformation is noteasy for some equations and sometimes it requires the introduction of new depen-dent and sometimes even independent variables. However, Hereman et al. [15–17],introduced a simplified version of Hirota’s method, and used it to construct solitaryand soliton solutions. In this method, without bilinear forms, exact solutions canstill be constructed straightforwardly by solving a perturbation scheme on the com-

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puter, using a symbolic manipulation package. The method was applied to nonlinearevolution equations with constant coefficients.

To our knowledge, multi-soliton solutions for fractional Burgers hierarchy (1)have not been explicity computed before. The reason for that is the complicated,lengthy, and nearly impossible computation when using Hirota’s direct method. Thiswork contribute to the soliton theory by developing tools allowing to analyze andcontrol the dynamical behaviors of multi-soliton soltions for equations with fractalderivatives. We will first introduce a new simplified bilinear method for a reliabletreatment of the fractional Burgers equations. The method is an efficient transfor-mation method combined with Hereman’s simplified method [15–17]. The main ad-vantage of the introduced method is that the bilinear representation for the equationbecomes superfluous. Therefore, the method can be applied to equations for whichthe bilinear form is not known or does not exist [4, 5, 27].

Through the analytical analysis of the soliton solutions, we will discuss theevolution characteristics of solitons in inhomogeneous media. We will reveal howinhomogeneous parameters affect the dynamical properties of solitons. This makesit possible to develop the mathematical theory to control the evolution of solitons.Furthermore, by performing the asymptotic analysis, we will investigate the colli-sion process of front waves. Through graphical analysis, we will detail the propaga-tion features and collision behaviors of front waves with or without inhomogeneouseffects.

1.3. ORGANIZING OF THE PAPER

The rest of the paper is organized as follows. For the convenience of the reader,a brief background on the definition of the modified Riemann–Liouville derivativeis given in Section 1. In Section 2, a new simplified bilinear method for a reliabletreatment of the fractional Burgers equations is presented. The closed form analyticalN -shock-wave-like solution is obtained. Based on the obtained N -shock-wave-likesolution, the propagation properties and collision behavior are discussed in Section3. Furthermore, the effects of the inhomogeneities, namely, variable coefficients arediscussed. Section 4 is devoted to the conclusions.

2. MODIFIED RIEMANN–LIOUVILLE DERIVATIVE

There are different definitions for fractional derivatives; for more details seeRef. [42]. In our paper we use the modified Riemann–Liouville derivative whichwas defined by Jumarie.

In order to circumvent some drawbacks involved in the classical Riemann–Liouville definition, Jumarie proposed the following alternative referred to as themodified Riemann–Liouville derivative [28–30].

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Definition 1 (Modified Riemann–Liouville derivative) Let f : R→ R, x→ f(x),denote a continuous (but not necessarily differentiable) function.

(i) Assume that f(x) is a constant K. Then its fractional derivative of order α is

DαxK =

KΓ−1 (1−α)x−α, α≤ 0,

0, α > 0.

(ii) When f(x) is not a constant, then one will set

f(x) = f(0) + (f(x)−f(0)) ,

and its fractional derivative will be defined by the expression

f (α)(x) =Dαxf(0) +Dα

x (f(x)−f(0))

in which, for negative α, one has

Dαx (f(x)−f(0)) =

1

Γ(−α)

x∫0

(x−y)−1−α (f(y)−f(0))dy, α < 0

whilst for positive α, one will set

Dαx (f(x)−f(0)) =Dα

xf(x) =Dx

(f (α−1)(x)

).

When 1≤ n≤ α < n+ 1, one will set

f (α)(x) =(f (α−n)(x)

)(n).

The main properties of the modified Riemann–Liouville derivative were sum-marized in Ref. [28] and three useful formulas of them are given as follows:

Dαxx

β = Γ(1+β)Γ(1+β−α)x

β−α, β > 0,

Dαx [u(x)v(x)] = u(x)Dα

xv(x) +v(x)Dαxu(x),

Dαx [f(v(x))] = df

duDαxu(x) = (dudx)αDα

uf(u).

3. FRONT WAVES OF FRACTIONAL BURGERS HIERARCHY (1)

In this paper, Hereman’s bilinear method [17] will be modified to obtain multiple-front solutions for the variable-coefficient fractional Burgers hierarchy (1). The ex-plicit construction of the multi-front-wave solutions involved a large amount of te-dious algebra and calculus. The use of computer algebra is necessary to carry out thelengthy but straightforward computations.

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3.1. SINGLE-FRONT WAVES

Using the fractional transforms

T =p1t

α

Γ(1 +α),

X =p2x

β

Γ(1 +β),

Y =p3y

γ

Γ(1 +γ),

Z =p4z

δ

Γ(1 + δ),

where p1,p2,p3 and p4 are constants, we can convert fractional derivatives into clas-sical derivatives as follows

∂αu

∂tα= p1

∂u

∂T,

∂βu

∂xβ= p2

∂u

∂X,

∂γu

∂yγ= p3

∂u

∂Y,

∂δu

∂zδ= p4

∂u

∂Z.

(6)

See [55]. Taking into account (6), we can write (1) in the form

uT +aA(T )∂

∂X

(∂

∂X+u

)nu+ b(T )uX + c(T )uY +d(T )uZ = 0,

n= 0,1,2, . . . (7)

We begin our analysis by working on the one soliton solutions to the equations of theBurgers hierarchy given by (7). Substituting

u(X,Y,Z,T ) = ekiX+riY+siZ−Ωi(T )

into the linear terms of (7) gives the dispersion relation

Ωi(T ) =

∫ [kn+1i a(T ) +kib(T ) + ric(T ) +sid(T )

]dT.

It is noticed that the dispersion relations will play an important role in dominating thesolutions’ dynamics behaviors. Equations of this form can usually be bilinearized byintroducing a new dependent variable whose natural degree would be zero, e.g., lnfor f/g. In this case the first one works, so let us define

u(X,Y,Z,T ) =R (lnf)X . (8)

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To find the one-soliton solution, take

f(X,Y,Z,T ) = 1 +C1ek1X+r1Y+s1Z−Ω1(T ), (9)

where C1 =±1. Substituting (8) and (9) into Eq. (7) and solving for R we find, that

R= 1.

This enable us to define θi, i= 1,2, . . ., as

θi(X,Y,Z,T )=kiX+riY+siZ−∫ [kn+1i a(T )+kib(T )+ric(T )+Sid(T )

]dT. (10)

As a result we obtain the relation

θi(x,y,z, t) =ki

Γ(1 +β)xβ +

riΓ(1 +γ)

yγ +si

Γ(1 + δ)zδ−ωi(t), i= 1,2, . . . , (11)

where

ωi(t) =α

Γ(1 +α)

∫ [kn+1i a(t) +kib(t) + ric(t) +sid(t)

]tα−1dt.

Therefore, the single-soliton solution u(X,Y,Z,T ) can be written as

u(X,Y,Z,T ) = C1k1eθ1(X,Y,Z,T )

1 +C1eθ1(X,Y,Z,T ).

For C1 = 1, we obtain the single-wave solution

u(x,y,z, t) =k1

2

[1 + tanh(

θ1(x,y,z, t)

2)

], (12)

which can be seen as the (3+1)-dimensional Taylor shock profile [13], which wasobserved on the vortex cores produced by an oscillating grid in a rotating fluid [33].For C1 =−1, we obtain the single-singular-wave solution

u(x,y,z, t) =k1

2

[1 + coth(

θ1(x,y,z, t)

2)

]which gives a profile of an explosive pulse.

3.2. MULTIPLE-FRONT WAVES

For the two-front solutions, one starts with the auxiliary function

f(X,Y,Z,T ) = 1 +C1eθ1(X,Y,Z,T ) +C2e

θ2(X,Y,Z,T )

+C1C2a12eθ1(X,Y,Z,T )+θ2(X,Y,Z,T ), (13)

where Ci =±1, i= 1,2 and θi are given by (10).

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Upon substituting (13) and (8), into Eq. (7) and solving for the phase shift a12,we obtain

a12 = 0 (14)and hence we can generalize for other phase shifts by

aij = 0, 1≤ i < j ≤ 3. (15)

Using (14) and (13) in (8), the following two-front solutions can be obtained

u(X,Y,Z,T ) =C1k1e

θ1(X,Y,Z,T ) +C2k2eθ2(X,Y,Z,T )

1 +C1eθ1(X,Y,Z,T ) +C2eθ2(X,Y,Z,T ). (16)

As a result we obtain

u(x,y,z, t) =C1k1e

θ1(x,y,z,t) +C2k2eθ2(x,y,z,t)

1 +C1eθ1(x,y,z,t) +C2eθ2(x,y,z,t),

where θ1 and θ2 are defined in (10).By substituting C1 = C2 = 1 in (16), we obtain the two-front solutions

u(x,y,z, t) =k1e

θ1(x,y,z,t) +k2eθ2(x,y,z,t)

1 +eθ1(x,y,z,t) +eθ2(x,y,z,t). (17)

However, the two-singular-front solutions are obtained by substitutingC1 =C2 =−1in (16),

u(x,y,z, t) =k1e

θ1(x,y,z,t) +k2eθ2(x,y,z,t)

eθ1(x,y,z,t) +eθ2(x,y,z,t)−1.

For the three-front solutions, we set the auxiliary function

f(X,Y,Z,T ) = 1 +C1eθ1(X,Y,Z,T ) +C2e

θ2(X,Y,Z,T ) +C3eθ3(X,Y,Z,T ), (18)

where Ci =±1, i= 1,2,3 and θi are given by (10). Proceeding as before, the three-front solutions are given by

u(x,y,z, t) =k1e

θ1(x,y,z,t) +k2eθ2(x,y,z,t) +k3e

θ3(x,y,z,t)

1 +eθ1(x,y,z,t) +eθ2(x,y,z,t) +eθ3(x,y,z,t)(19)

and the singular three-front solutions are given by

u(x,y,z, t) =k1e

θ1(x,y,z,t) +k2eθ2(x,y,z,t) +k3e

θ3(x,y,z,t)

eθ1(x,y,z,t) +eθ2(x,y,z,t) +eθ3(x,y,z,t)−1. (20)

3.3. N -SHOCK-WAVE-LIKE SOLUTION

It is known from soliton theory that every solitonic equation that has genericN = 3 soliton solution has also soliton solutions for N ≥ 4 [18, 19]. This shows thatEq. (7) has N -soliton solutions which can be obtained for finite N , where N > 1.

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Generally, we can conjecture the N -front solutions for the (3+1)-dimensionalBurgers hierarchy (7), for any n= 0,1,2, . . ., as

u(x,y,z, t) =

∑Ni=1kie

θi(x,y,z,t)

1 +∑N

i=1eθi(x,y,z,t)

(21)

and the singular N -front solutions are

u(x,y,z, t) =

∑Ni=1kie

θi(x,y,z,t)

−1 +∑N

i=1eθi(x,y,z,t)

, (22)

where the dispersion law is given by

ωi(t) =α

Γ(1 +α)

∫ [kn+1i a(t) +kib(t) + ric(t) +sid(t)

]tα−1dt (23)

and

θi(x,y,z, t) = kixβ

Γ(1 +β)+ ri

yγ

Γ(1 +γ)+si

zδ

Γ(1 + δ)−ωi(t), i= 1,2, . . . .

It is important to notice that the multiple-front solutions (21) and (22) are justrational functions of exponentials, with the wave parameters ki and ri for each expo-nential wave. Accordingly, we could name solution (22) as the N -shock-wave-likesolution of Eq. (7). Some previously published solutions turn out to be the specialcases of solution (22), such as the shock-wave-like solutions in [5].

4. STABILITIES AND PROPAGATION CHARACTERISTICS OF SOLITARY WAVES

By virtue of the multiple soliton solutions derived, we can discuss the effects ofthe inhomogeneities, namely, variable coefficients. Furthermore, we will analyze theinteractions of the solitons through changing the wave numbers and phase factors.We will see that the dispersion relation plays a crucial role in the analysis, since itcan be used to obtain the characteristic line and velocity v for each soliton.

From solution (12), the soliton amplitude amp can be expressed as

amp= |k1| (24)

With the characteristic-line method [12, 56], the characteristic wedge for each soli-tary wave can be defined by

xβ + riyγ

kiΓ(1 +γ)Γ(1 +β)+si

zδ

kiΓ(1 + δ)Γ(1 +β)=

α

kiΓ(1 +α)Γ(1 +β)

×∫ [

kn+1i a(t) +kib(t) + ric(t) +Sid(t)

]tα−1dt, i= 1,2, . . . (25)

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Since there are three arbitrary parameters, ki, ri, si, in expression (25), it is conve-nient to control the solitonic velocity in the profile at z = y = 0 (or z,y are constants)by choosing the appropriate parameters. Correspondingly, the velocity v of eachsolitary wave along the x-axis can be expressed as

vi =1

β

(α

kiΓ(1 +α)Γ(1 +β)

∫ [kn+1i a(t) +kib(t) + ric(t) +Sid(t)

]tα−1dt

) 1β−1

×α[kn+1i a(t) +kib(t) + ric(t) +Sid(t)

]tα−1

kiΓ(1 +α)Γ(1 +β)(26)

Further, the absolute value of velocity v determines the speed, namely, velocity inmagnitude, and propagation direction of soliton is decided by the sign of v.

The analysis is illustrated by studying the time fractional Burgers equation

u(α)t +a(t)(uxx+ 2uux) + b(t)ux+ c(t)uy +d(t)uz = 0, (27)

From (21) we obtain the single shock-wave solution

u(x,y,z, t) = kieθi

1 +eθi=ki2

[1 + tanh(

θi2

)

], (28)

where

θi = kix+riy+siz−α

Γ(1 +α)

∫ [k2i a(t) +kib(t) + ric(t) +Sid(t)

]tα−1dt. (29)

For the single-front solution (28), it is obvious that the amplitude of the frontis ki, which keeps invariant during the propagation. Following the characteristic-linemethod, the characteristic face for each solitary wave can be defined by

x+rikiy+

sikiz =

α

Γ(1 +α)

∫ [kia(t) + b(t) +

rikic(t) +

sikid(t)

]tα−1dt,

i= 1,2, . . . , (30)

which can be derived from relation (29). Correspondingly, the velocity of the waveat time t can be expressed as

vx =α

Γ(1 +α)

[kit

α−1a(t) + tα−1b(t) +rikitα−1c(t) +

sikitα−1d(t)

], (31)

vy =α

Γ(1 +α)

[k2i

ritα−1a(t) +

kiritα−1b(t) + tα−1c(t) +

siritα−1d(t)

], (32)

vz =α

Γ(1 +α)

[k2i

sitα−1a(t) +

kisitα−1b(t) +

risitα−1c(t) + tα−1d(t)

]. (33)

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4.1. EFFECTS OF THE VARIABLE COEFFICIENTS

The soliton amplitude amp is independent of the variable coefficients a(t),b(t), c(t) and d(t). Figure 1 shows that the solitonic amplitude is constant ki; Fig.2 shows that the solitonic amplitude increases with increasing ki that can be alsoexplained by expression (24). However, expression (31) indicates that the propaga-tion velocity of the solitary wave is influenced by the coefficient functions a(t), b(t),c(t), and d(t); it can be observed that the initial superposed solitons travel differentdistances over a period of time for the different choices of a(t), b(t), c(t), and d(t),and it can be concluded that the larger the absolute value of v is the greater is thespeed. Moreover, if the sign of v reverses, the solitonic direction will be changedaccordingly.

−20 0 20 40 60

0

0.2

0.4

0.6

0.8

1

x

u

Fig. 1 – Evolution plots of one-soliton solution given by (12) at t = 3, y = z = 1 (a) k1 = 1, r1 = 3,s1 = 5, a(t) = t3, b(t) = 1, c(t) = t, d(t) = 2; (b) k1 = 1, r1 = 2, s1 = 4, a(t) = sin

√t, b(t) = t,

c(t) = 1, d(t) =√t; (c) k1 = 1, r1 = 2, s1 = 4, a(t) = t

910 , b(t) = 3, c(t) = t

910 +1, d(t) = 1.

When a(t), b(t), c(t), and d(t) are not constants, the characteristic lines

x=α

Γ(1 +α)

∫ [kia(t) + b(t) +

rikic(t) +

sikid(t)

]tα−1dt.

are no longer lines but curves. As special cases, some types of functions will betaken as examples to illustrate how the variable coefficients affect the evolution of

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−10 0 10 20 30 40

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

x

u

Fig. 2 – Evolution plots of one-soliton solution given by (12) at t = 3, y = z = 1,α = 0.5, r1 = 3,s1 = 5, a(t) = t3, b(t) = 1, c(t) = t, d(t) =

√t (a) k1 = 1; (b) k1 = 1.2; (c) k1 = 1.5.

the soliton. According to (31) the soliton travels along the positive x-axis directionwhen a(t), b(t), c(t), and d(t) satisfy the following inequality

a(t)>−Γ(1 +α)

α(b(t)

ki+ ric(t) +sid(t)). (34)

As special cases, some types of functions will be taken as examples to illustrate howthe variable coefficients affect the evolution of the soliton. When a(t), b(t), c(t), andd(t) are chosen as positive functions and ki ≥ 0, ri ≥ 0, and si ≥ 0, from expression(34), we concude that the soliton moves along the positive x-axis direction.

In Fig. 3, the variable coefficients are chosen as a(t) = t1.5 and b(t) = c(t) =d(t) = 0; the characteristic curve is obtained by (30) as follows

x− ki

4Γ(32)t2 +η = 0.

Then the soliton reveals the parabolic type propagation trajectory with the unalterableamplitude but continuously changeable velocity, which can be deduced from (31)-(33) for a velocity varying with the evolution of time t.

The propagation trajectory of the soliton presents the periodicity oscillationwhen a(t) = −t0.5 cos t and b(t) = c(t) = d(t) = 0 as shown in Fig. 4. Further-more, Fig. 5 describes the soliton with the continuously changeable velocity throughchoosing a(t) =−t0.5 sin5t,b(t) = t1.5 and c(t) = d(t) = 0. Since the characteristiccurve in Fig. 5 can be expressed as

x− ki

10Γ(32)

cos5t+1

2t2 +η = 0, (35)

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−6−4

−20

24

6

−6

−4

−2

0

2

4

6

0

0.5

1

1.5

t

x

Fig. 3 – The profile of the solution (17) in an inhomogeneous medium in the x− t plane with α= 0.5,a(t) = t1.5, b(t) = c(t) = d(t) = 0, k1 = 1, k2 = 1.5, r1 = 1, r2 = 2, s1 = s2 = 1.

when t approaches zero, i.e., cos(t) t2, the trajectory is snake-like type with peri-odic oscillation. Otherwise, when t is far from the origin, the trajectory is parabolic-like type one. We conclude that, besides the periodic oscillation of the solitons inthe local region, the large scale propagation trajectories for such a structure are theparabolic-typed curves. In Fig. 6, the characteristic curve is obtained by (30) asfollows

x− 9

10Γ(1910)

[−ki4t4 + (

−1

3+ri3si

)t3 + (1

2+ki2

+ri2si− si

2ki)t2

+ (risi− siki

)t+η = 0. (36)

Likewise, if the variable coefficients are taken as the other forms, the correspondingcharacteristic curves will present different characters.

4.2. BIDIRECTIONAL SOLITONS

Since Eq. (27) is a variable-coefficient one, there are choices for the parametersthat allow the existence of the bidirectional solitons. Among the above solutions toEq. (27), we would like to concentrate ourselves on the two-front wave

u(x,y,z, t) =k1e

θ1 +k2eθ2

1 +eθ1 +eθ2, (37)

where θi are defined in (29).To understand the interaction between two solitonic waves in inhomogeneous

situations, we first consider the two-front wave, given by (37). In Fig. 7 the two-front

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−10

−5

0

5

10

−10−5

05

10

0

0.5

1

1.5

tx

Fig. 4 – The profile of the solution (17) in an inhomogeneous medium in the x− t plane with α= 0.5,a(t) =−t0.5 cos t, b(t) = c(t) = d(t) = 0, k1 = 1, k2 = 1.5, r1 = 1, r2 = 2, s1 = s2 = 1.

−6−4

−20

24

6

−6

−4

−2

0

2

4

6

0

0.5

1

1.5

t

x

Fig. 5 – The profile of the solution (17) in an inhomogeneous medium in the x− t plane with α= 0.5,a(t) =−t0.5 sin5t, b(t) = t0.5, c(t) = d(t) = 0, k1 = 1, k2 = 1.5, r1 = 1, r2 = 2, s1 = s2 = 1.

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−6−4

−20

24

6

−6

−4

−2

0

2

4

6

0

1

2

t

x

Fig. 6 – The profile of the solution (17) in an inhomogeneous medium in the x− t plane with α= 0.9,a(t) =−t3.1+ t1.1, b(t) =−t2.1+ t1.1, c(t) = t2.1+ t1.1+ t0.1, d(t) =−t1.1− t0.1, k1 = 1, k2 =1.5, r1 = 1, r2 = 2, s1 = s2 = 1.

−3

−2

−1

0

1

2

3

4

−6

−4

−2

0

2

4

6

00.5

11.5

tx

Fig. 7 – The propagation of the two-soliton wave via solution (37) with α = 0.8 k1 = 0.8, r1 = 1,s1 = 1, k2 = 1.5, r2 = 2, s2 = 1, a(t) = t2.2, b(t) = 0, c(t) = 0, d(t) = 0.

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−3−2

−10

12

34

−6−4

−20

24

6

−2

−1

0

1

tx

Fig. 8 – The propagation of the two-soliton wave via solution (37) with α = 0.8 k1 = 0.8, r1 = 1,s1 = 1, k2 =−1.5, r2 = 2, s2 = 1, a(t) = t2.2, b(t) = 0, c(t) = 0, d(t) = 0.

wave behaves in a basic interaction mode, which is called the overtaking coalescence.It can be found in quadratically inhomogeneous medium, with coefficients chosen asa(t) = t2.2 and b(t) = c(t) = d(t) = 0, the two fronts with the same propagationdirections in x-axis coalescing into one large front in their interaction region of the(x,t)-plane, of which the amplitude amounts to two initial amplitudes. The front withfaster velocity overtakes the slow-velocity one, while Fig. 8 represents the head-oncollision between one right-going soliton and one left-going soliton. It can be shownthat the sign of velocity controls the directions in which the solitary waves move.It is clear that after the collision the velocity and amplitude of each soliton remainunchanged since the phase shift a12 = 0.

5. CONCLUSIONS

In summary, via a simplified bilinear method, a fractional complex transform,and symbolic computation, we have formally derived the N -shock-waves describedby the nonautonomous (3+1)-dimensional fractional Burgers models (1). The ex-plicit functions, which describe the evolution of the amplitude, phase and velocity,are obtained. The parameters have no effect on the motion of the waves’ amplitudeand phases, and they affect just the evolution of their velocities. The front wave ap-proaches asymptotically the non-zero boundary value ki, which does not depend onthe variable parameters. We have discussed the properties of the N -shock-waves,especially on the interaction processes and the full effects of inhomogeneities of me-dia. Graphical analysis has been carried out to investigate the interactions of thetwo fronts through changing the wave numbers and variable parameters. It has be

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concluded that fractional Burgers models (1) admit bidirectional wave interactionsincluding head-on and overtaking collisions. Since the problem of bidirectional soli-tary waves has been reported in shallow-water waves [12] and waves in bubbly liq-uids [37, 38], it is expected that the bidirectional soliton-like solutions to (1) maybe used to describe such interesting physical phenomena with the consideration ofthe inhomogeneities of media and nonuniformities of boundaries in nonlinear lattice,plasma physics, and ocean dynamics.

Acknowledgements. The authors of this paper would like to thank the anonymous referees fortheir in-depth reading, criticism of, and insightful comments on an earlier version of this paper.

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