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Research ArticleFractional Soliton Dynamics and Spectral Transform ofTime-Fractional Nonlinear Systems A Concrete Example

Sheng Zhang 123 Yuanyuan Wei1 and Bo Xu4

1School of Mathematics and Physics Bohai University Jinzhou 121013 China2Department of Mathematics Hohhot Minzu College Hohhot 010051 China3School of Mathematical Sciences Qufu Normal University Qufu 273165 China4School of Education and Sports Bohai University Jinzhou 121013 China

Correspondence should be addressed to Sheng Zhang szhangchina126com

Received 25 March 2019 Accepted 19 June 2019 Published 6 August 2019

Academic Editor Mohamed Boutayeb

Copyright copy 2019 Sheng Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper the spectral transform with the reputation of nonlinear Fourier transform is extended for the first time to a localtime-fractional Korteweg-de vries (tfKdV) equation More specifically a linear spectral problem associated with the KdV equationof integer order is first equipped with local time-fractional derivative Based on the spectral problem with the equipped local time-fractional derivative the local tfKdV equation with Lax integrability is then derived and solved by extending the spectral transformAs a result a formula of exact solution with Mittag-Leffler functions is obtained Finally in the case of reflectionless potential theobtained exact solution is reduced to fractional n-soliton solution In order to gain more insights into the fractional n-solitondynamics the dynamical evolutions of the reduced fractional one- two- and three-soliton solutions are simulated It is shown thatthe velocities of the reduced fractional one- two- and three-soliton solutions change with the fractional order

1 Introduction

Since the increasing interest on fractional calculus and itsapplications dynamical processes and dynamical systemsof fractional orders have attracted much attention In 2010Fujioka et al [1] investigated soliton propagation of anextended nonlinear Schroinger equation with fractional dis-persion term and fractional nonlinearity term In 2014 Yanget al [2] used a local fractional KdV equation tomodel fractalwaves on shallow water surfaces

In the field of nonlinear mathematical physics the spec-tral transform [3] with the reputation of nonlinear Fouriertransform is a famous analytical method for constructingexact and explicit n-soliton solutions of nonlinear partialdifferential equations (PDEs) Since put forward by Gardneret al in 1967 the spectral transform method has achievedconsiderable developments [4ndash26] With the close attentionsof fractional calculus and its applications [27ndash53] some of thenatural questions are whether the existingmethods like thosein [54ndash70] in soliton theory can be extended to nonlinear

PDEs of fractional orders and what about the fractionalsoliton dynamics and integrability of fractional PDEs As faras we know there are no research reports on the spectraltransform for nonlinear PDEs of fractional ordersThis paperis motivated by the desire to extend the spectral transformto nonlinear fractional PDEs and then gain more insightsinto the fractional soliton dynamics of the obtained solutionsFor such a purpose we consider the following local tfKdVequation

119863(120572)119905 119906 + 6119906119906119909 + 119906119909119909119909 = 0 0 lt 120572 le 1 (1)Here we note that if 120572 = 1 then eq (1) becomes the celebratedKdV equation 119906119905 + 6119906119906119909 + 119906119909119909119909 = 0 In eq (1) the local time-fractional derivative119863120572119905 119906 at the point 119905 = 1199050 is defined as [30]

119863(120572)119905 119906 (119909 1199050) = 120597120572119906 (119909 119905)12059711990512057210038161003816100381610038161003816100381610038161003816119905=1199050

= lim119905997888rarr1199050

Δ120572 (119906 (119909 119905) minus 119906 (119909 1199050))(119905 minus 1199050)120572 (2)

HindawiComplexityVolume 2019 Article ID 7952871 9 pageshttpsdoiorg10115520197952871

2 Complexity

whereΔ120572(119906(119909 119905)minus119906(119909 1199050)) cong Γ(1+120572)(119906(119909 119905)minus119906(119909 1199050)) someuseful properties [33] of the local time-fractional derivativehave been used in this paper

The rest of this paper is organized as follows In Section 2we derive the local tfKdV eq (1) by introducing a linear spec-tral problem equipped with local time-fractional derivativeIn Section 3 we construct fractional n-soliton solution ofthe local tfKdV eq (1) by extending the spectral transformmethod In Section 4 we investigate the dynamical evolutionsof the obtained fractional one-soliton solution two-solitonsolution and three-soliton solution In Section 5 we concludethis paper

2 Derivation of the Local tfKdV Equation

For the local tfKdV eq (1) we have the followingTheorem 1

Theorem 1 The local tfKdV eq (1) is a Lax system which canbe derived from the linear spectral problem equipped with alocal time-fractional evolution equation

120601119909119909 = (120578 minus 119906) 120601 120578 = minus1198962 (3)

119863(120572)119905 120601 = (119906119909 + 120599) 120601 + (41198962 minus 2119906) 120601119909 (4)

where 120601 = 120601(119909 119905) and 119906 = 119906(119909 119905) are all differentiablefunctions with respect to 119909 and 119905 the spectral parameter 120578 isindependent of 119905 and 120599 is an arbitrary constant

Proof Taking the time-fractional derivative of eq (3) yields

119863(120572)119905 120601119909119909 = minus120601119863(120572)119905 119906 minus (119906 + 1198962)119863(120572)119905 120601 (5)

Substituting eq (4) into eq (5) we have

119863(120572)119905 120601119909119909 = minus [119863(120572)119905 119906 + (119906119909 + 120599) (119906 + 1198962)] 120601minus (41198962 minus 2119906) (119906 + 1198962) 120601119909 (6)

Taking the derivative of eq (4) with respect to 119909 twice gives

(119863(120572)119905 120601)119909119909

= 119906119909119909119909120601 minus (3119906119909 minus 120599) 120601119909119909 + (41198962 minus 2119906) 120601119909119909119909 (7)

With the help of eq (3) from eq (7) we have

(119863(120572)119905 120601)119909119909

= [119906119909119909119909 + (5119906 minus 1198962) 119906119909 minus 120599 (119906 + 1198962)] 120601minus (41198962 minus 2119906) (119906 + 1198962) 120601119909 (8)

On the other hand at the aribitrary point 119905 = 1199050 we have119863120572119905 120601119909119909 (119909 1199050) = lim

119905997888rarr1199050

Δ120572 (120601119909119909 (119909 119905) minus 120601119909119909 (119909 1199050))(119905 minus 1199050)120572

(119863120572119905 120601 (119909 1199050))119909119909 = lim119905997888rarr1199050

Δ120572 (120601 (119909 119905) minus 120601 (119909 1199050))119909119909(119905 minus 1199050)120572 (9)

Finally using eqs (6) (8) and (9) we arrive at eq (1)Thus we finish the proof The process of proof shows that eq(1) is a Lax integrable system

3 Local Fractional Spectral Transform

Since the local tfKdV eq (1) is a local time-fractional systemthe orders of the derivatives with respect to space variable119909 are integers So all the existing results about the spectralproblem (3) a part of the Lax pair for the classical KdVequation can be translated to the local tfKdV eq (1)

For the direct scattering problem we translate somenecessary results and definitions [9] to the local tfKdV eq(1)

Lemma 2 If the real potential 119906(119909) defined in the whole realaxis minusinfin lt 119909 lt infin and its various derivatives are differentiablefunctions which vanishes rapidly as 119909 997888rarr plusmninfin and satisfies

int+infinminusinfin

10038161003816100381610038161003816119909119895119906 (119909)10038161003816100381610038161003816 d119909 lt +infin (119895 = 0 1 2) (10)

then the linear spectral problem (3) has a set of basic solutionscalled Jost solutions120601+(119909 119896) and120601minus(119909 119896) and they are not onlybounded for all values of 119909 but also analytic for Im 119896 gt 0 andcontinuous for Im 119896 ge 0 and have the following asymptoticproperties

120601+ (119909 119896) 997888rarr ei119896119909 (119909 997888rarr +infin) (11)

120601minus (119909 119896) 997888rarr eminusi119896119909 (119909 997888rarr minusinfin) (12)

Lemma 3 Define the Wronskian

119882(120601+ (119909 119896) 120601minus (119909 119896))= 120601+119909 (119909 119896) 120601minus (119909 119896) minus 120601+ (119909 119896) 120601minus119909 (119909 119896) (13)

and let

120601minus (119909 119896) = 119886 (119896) 120601+ (119909 minus119896) + 119887 (119896) 120601+ (119909 119896) (14)

Then

119886 (119896) = 12i119896119882(120601minus (119909 119896) 120601+ (119909 119896)) 119887 (119896) = 12i119896119882(120601+ (119909 minus119896) 120601minus (119909 119896))

(15)

where 119886(119896) is analytic for Im 119896 gt 0 and continuous for Im 119896 ge0 119887(119896) is defined only on the real axis Im 119896 = 0 and theanalytic function 119886(119896) has a finite number of simple zeros 119896119898 =i120581119898 (120581119898 gt 0119898 = 1 2 119899)Lemma 4 For the linear spectral problem (3) there exists aconstant 119887119898 such that

120601minus (119909 i120581119898) = 119887119898120601+ (119909 i120581119898) (16)

int+infinminusinfin

1198882119898120601+2 (119909 i120581119898) d119909 = 1 1198882119898 = minus i119887119898119886119896 (i120581119898) (17)

Definition 5 The constant 119888119898 satisfying eq (17) is named thenormalization constant for the eigenfunction 120601+(119909 i120581119898) and119888119898120601+(119909 i120581119898) is named normalization eigenfunction

Complexity 3

Definition 6 The set

119896 (Im 119896 = 0) 119877 (119896) = 119887 (119896)119886 (119896) 119894120581119898 119888119898 119898= 1 2 119899

(18)

is named the scattering data of the linear spectral problem (3)

Lemma 7 If the eigenfunction 120601(119909 119896) satisfies the linearspectral problem (3) then

119875 = 119863(120572)119905 120601 minus (119906119909 + 120599) 120601 minus (41198962 minus 2119906) 120601119909 (19)

solves eq (3) as well

Proof A direct computation on eq (19) tells that

119875119909119909 = 119863(120572)119905 120601119909119909 minus 119906119909119909119909120601 minus (41198962 minus 2119906) 120601119909119909119909+ (3119906119909 minus 120599) 120601119909119909 (20)

With the help of eqs (3) and (20) we have

119875119909119909 + (1198962 + 119906)119875 = minus120601 (119863(120572)119905 119906 + 6119906119906119909 + 119906119909119909119909) = 0 (21)

Thus the proof is finished

For the time dependence of the scattering data we havethe followingTheorem 8

Theorem 8 If the time evolution of 119906(119909 119905) obeys the localtfKdV eq (1) then the scattering data (18) for the linear spectralproblem (3) possess the following time dependences

119863(120572)119905 120581119898 (119905) = 0119863(120572)119905 119888119898 (119905) = 41205813119898 (119905) 119888119898 (119905)

(22)

119863(120572)119905 119886 (119896) = 0119863(120572)119905 119887 (119896) = 8i1198963 (119905) 119887 (119896) (23)

Proof Substituting eqs (12) and (14) into eq (3) and using theasymptotic properties of eqs (11) and (12) as 119909 997888rarr +infin and119909 997888rarr minusinfin respectively we have

(120599 minus 4i1198963) eminusi119896119909 = 0 (24)

119863(120572)119905 119886 (119896) eminusi119896119909 + 119863(120572)119905 119887 (119896) ei119896119909= 120599 (119886 (119896) eminusi119896119909 + 119887 (119896) ei119896119909)

+ 41198962 (minusi119896119886 (119896) eminusi119896119909 + i119896119887 (119896) ei119896119909) (25)

Namely

120599 = 4i1198963 (26)

119863(120572)119905 119886 (119896) = 119886 (119896) (120599 minus 4i1198963) = 0 (27)

119863(120572)119905 119887 (119896) = 119887 (119896) (120599 + 4i1198963) = 8i1198963119887 (119896) (28)

It is easy to see that all the zeros of 119886(119896) are independent of 119905because of 119863(120572)119905 119886(119896) = 0 Therefore we arrive at 119863(120572)119905 120581119898(119905) =0

Similarly substituting eq (16) into eq (3) and using theasymptotic property of eq (11) as 119909 997888rarr +infin we have

119863(120572)119905 119887119898 (119905) = (120599 + 41205813119898 (119905)) 119887119898 (119905) = 81205813119898 (119905) 119887119898 (119905) (29)

In view of eqs (17) and (28) we obtain

119863(120572)119905 (1198882119898 (119905)) = minus119863(120572)119905 (i119887119898 (119905))119886119896 (i120581119898) = 81205813119898 (119905) 1198882119898 (119905) (30)

which can be finally reduced to the second term of eq (22)Then we finish the proof

For the inverse scattering problem we have the followingTheorem 9

Theorem 9 The local tfKdV eq (1) has an exact solution ofthe form

119906 (119909 119905) = 2 dd119909119870 (119909 119909 119905) (31)

where 119870(119909 119910 119905) satisfies the Gelrsquofand-Levitan-Marchenko(GLM) integral equation

119870(119909 119910 119905) + 119865 (119909 + 119910 119905)+ intinfin119909

119870 (119909 119911 119905) 119865 (119909 + 119911 119905) d119911 = 0 (32)

with

119865 (119909 119905) = 12120587 intinfinminusinfin

119877 (119896 119905) ei119896119909d119896 + 119899sum119898=1

1198882119898ei120581119898119909 (33)

and 119877(119896 119905) 119888119898 and 120581119898 are determined by eqs (22) and (23)

Proof Theprocess of the proof ofTheorem 9 is similar to thatof the classical KdV equation [9] with integer order and theonly differences are the scattering data To avoid unnecessaryrepetition we omit it here

For the fractional n-soliton solution we have the follow-ingTheorem 10

Theorem 10 In the case of reflectionless potential the localtfKdV eq (1) has fractional n-soliton solution of the form

119906 (119909 119905) = 2 d2

d1199092 ln (det119863(119909 119905)) (34)

where119863 (119909 119905) = (119889119895119898 (119909 119905))119899times119899

119889119895119898 (119909 119905) = 120575119895119898 + 119888119895 (119905) 119888119898 (119905)120581119895 (0) + 120581119898 (0)eminus(120581119895(0)+120581119898(0))119909(35)

119888119895 (119905) = 119888119895 (0) 119864120572 (41205813119895 (0) 119905120572) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572)

(36)

In eq (36) 119864120572(sdot) is the Mittag-Leffler function [33]

4 Complexity

Proof Firstly we further determine the scattering data Solv-ing eqs (22) and (23) yields

120581119898 (119905) = 120581119898 (0) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572) (37)

119886 (119896 119905) = 119886 (119896 0) 119887 (119896 119905) = 119887 (119896 0) 119864120572 (8i1205813 (0) 119905120572) (38)

Secondly we let 119877(119896 119905) = 0 In this case of reflectionlesseq (32) reduces to

119870(119909 119910 119905) + 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)(119909+119910)

+ 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)119910 intinfin119909

119870 (119909 119911 119905) eminus120581119898(0)119911d119911= 0

(39)

Suppose that eq (39) has a separation solution

119870(119909 119910 119905) = 119899sum119895=1

119888119895 (119905) ℎ119895 (119909) eminus120581119895(0)119910 (40)

where ℎ119895(119909) is an undetermined function which can bedetermined by the substitution of eq (40) into eq (39) Withthe determined function ℎ119895(119909) we have

119870(119909 119910 119905) = dd119909 ln (det119863(119909 119905)) (41)

Finally from eqs (31) (37) (38) and (41) we obtain eq (34)Therefore the proof is over

4 Fractional Soliton Dynamics

In order to gain more insights into the soliton dynamics ofthe obtained fractional n-soliton solution (34) we considerthe cases of 119899 = 1 2 3

When 119899 = 1 we havedet119863 (119909 119905) = 1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909 (42)

and we hence obtain from eq (34) the fractional one-soliton solution

119906 = 212058121 (0)sdot sech2 [1205811 (0) 119909 minus 12 ln

11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) ] (43)

Similarly when 119899 = 2we obtain the fractional two-solitonsolution

119906 = 2 ln[[1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909

+ 11988822 (0) 119864120572 (812058132 (0) 119905120572)21205812 (0) eminus21205812(0)119909

+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2

sdot eminus2(1205811(0)+1205812(0))119909]]119909119909

(44)

When 119899 = 3 we obtain the fractional three-solitonsolution

119906 = 2 ln[[1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909 + 11988822 (0) 119864120572 (812058132 (0) 119905120572)21205812 (0) eminus21205812(0)119909 + 11988823 (0) 119864120572 (812058133 (0) 119905120572)21205813 (0) eminus21205813(0)119909

+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2 eminus2(1205811(0)+1205812(0))119909

+ 11988821 (0) 11988823 (0) (1205811 (0) minus 1205813 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205811 (0) 1205813 (0) (1205811 (0) + 1205813 (0))2 eminus2(1205811(0)+1205813(0))119909

+ 11988822 (0) 11988823 (0) (1205812 (0) minus 1205813 (0))2 119864120572 (81205812 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205812 (0) 1205813 (0) (1205812 (0) + 1205813 (0))2 eminus2(1205812(0)+1205813(0))119909

+ 11988821 (0) 11988822 (0) 11988823 (0) (1205811 (0) minus 1205812 (0))2 (1205811 (0) minus 1205813 (0))2 (1205812 (0) minus 1205813 (0))281205811 (0) 1205812 (0) 1205813 (0) (1205811 (0) + 1205812 (0))2 (1205811 (0) + 1205813 (0))2 (1205812 (0) + 1205813 (0))2119864120572 (81205811 (0) 119905120572) 119864120572 (81205812 (0) 119905120572) 119864120572 (812058133 (0)

sdot 119905120572) eminus2(1205811(0)+1205812(0)+1205813(0))119909]]119909119909

(45)

Complexity 5

5x

02

04

06

08

10

12

u

minus5

=1

=08

=07

Figure 1 Fractional one-solitons determined by solution (43) attime 119905 = 1

0 2 4 6 8 10t

5

10

15

20

25

30v

=1

=08

=07

Figure 2 Velocity images of fractional one-solitons determined bysolution (43)

In Figure 1 we simulate the fractional one-soliton solu-tion (43) with different values of 120572 where the parameters areselected as 1205811(0) = 08 and 1198881(0) = 02 With the help ofvelocity images in Figure 2 and the formula of velocity

V = 3212057212058151 (0) 119905120572minus1 (46)

we can see that the bell-shaped solitons have differentvelocities depending on the values of 120572 At the initial stagethe smaller the value of 120572 is selected the faster the solitonpropagates But soon it was the opposite for more details seeFigures 3 and 4

For the fractional two-solitons and three-solitons deter-mined respectively by solutions (44) and (45) similarfeatures shown in Figures 5ndash7 are observed In Figures 5 and6 we select the parameters as 1205811(0) = 05 1198881(0) = 151205812(0) = 1 and 1198882(0) = minus02 While the parameters in Figure 7are selected as 1205811(0) = 1 1198881(0) = 1 1205812(0) = 11 1198882(0) = 051205813(0) = 13 1198883(0) = 2

125 130 135 140x

02

04

06

08

10

12

u

=1

=08

=07


160 170 180 190 200x

02

04

06

08

10

12

u

=1

=08

=07


5 Conclusion

In summary we have derived and solved the local tfKdVeq (1) in the fractional framework of the spectral trans-form method This is due to the linear spectral problem(3) equipped with the local time-fractional evolution (4)As for the fractional derivatives there are many defini-tions [33] except the local fractional derivative such asGrunwaldndashLetnikov fractional derivative Riemann-Liouvillefractional derivative and Caputorsquos fractional derivative Gen-erally speaking whether or not the spectral transform canbe extended to some other nonlinear evolution equationswith another type of fractional derivative depends onwhetherfractional derivative has the good properties required by thespectral transform method To the best of our knowledgecombined with the Mittag-Leffler functions the obtainedexact solution (31) the fractional n-soliton solution (34)and its special cases the fractional one- two- and three-soliton solutions (43)-(45) are all new and they have notbeen reported in literature It is graphically shown thatthe fractional order of the local tfKdV eq (1) influences

6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07

Figure 5 Dynamical evolutions of fractional two-solitons determined by solution (44) at different times

10 20 30x

05

10

15

20u

=1

=08

=07

Figure 6 Fractional two-solitons determined by solution (44) at 119905 =3

the velocity of the fractional one-soliton solution (43) withMittag-Leffler function in the process of propagations Moreimportantly the fractional scheme of the spectral transform

x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07

Figure 7 Fractional three-solitons determined by solution (45) at119905 = 1presented in this paper for constructing n-soliton solutionof the local tfKdV eq (1) can be extended to some otherintegrable local time-fractional PDEs

Complexity 7

Data Availability

The data in the paper are available from the correspondingauthor upon request

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this article

Acknowledgments

This work was supported by the Natural Science Founda-tion of China (11547005) the Natural Science Foundationof Liaoning Province of China (20170540007) the NaturalScience Foundation of Education Department of LiaoningProvince of China (LZ2017002) and Innovative TalentsSupport Program in Colleges and Universities of LiaoningProvince (LR2016021)

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[2] X-J Yang J Hristov H M Srivastava and B Ahmad ldquoMod-elling fractal waves on shallowwater surfaces via local fractionalkorteweg-de vries equationrdquo Abstract and Applied Analysis vol2014 Article ID 278672 10 pages 2014

[3] C S Gardner J M Greene M D Kruskal and R M MiuraldquoMethod for solving the Korteweg-deVries equationrdquo PhysicalReview Letters vol 19 no 19 pp 1095ndash1097 1967

[4] M J Ablowitz D J Kaup A C Newell and H Segur ldquoTheinverse scattering transform-Fourier analysis for nonlinearproblemsrdquo Studies in Applied Mathematics vol 53 no 4 pp249ndash315 1974

[5] H H Chen and C S Liu ldquoSolitons in nonuniform mediardquoPhysical Review Letters vol 37 no 11 pp 693ndash697 1976

[6] R Hirota and J Satsuma ldquoN-soliton solutions of the K-dVequation with loss and nonuniformity termsrdquo Journal of thePhysical Society of Japan vol 41 no 6 pp 2141-2142 1976

[7] F Calogero and A Degasperis ldquoCoupled nonlinear evolutionequations solvable via the inverse spectral transform andsolitons that come back the boomeronrdquo Lettere Al NuovoCimento vol 16 no 14 pp 425ndash433 1976

[8] W L Chan and K-S Li ldquoNonpropagating solitons of thevariable coefficient andnonisospectral Korteweg-deVries equa-tionrdquo Journal of Mathematical Physics vol 30 no 11 pp 2521ndash2526 1989

[9] M J Ablowitz and P A Clarkson Solitons Nonlinear EvolutionEquations and Inverse Scattering Cambridge University PressCambridge UK 1991

[10] B Z Xu and S Q Zhao ldquoInverse scattering transformationfor the variable coefficient sine-Gordon type equationrdquo AppliedMathematics-A Journal of Chinese Universities vol 9 no 4 pp331ndash337 1994

[11] Y Zeng W Ma and R Lin ldquoIntegration of the soliton hierachywith selfconsistent sourcesrdquo Journal of Mathematical Physicsvol 41 no 8 pp 5453ndash5489 2000

[12] V N Serkin and A Hasegawa ldquoNovel soliton solutions ofthe nonlinear Schrodinger equation modelrdquo Physical ReviewLetters vol 85 no 21 pp 4502ndash4505 2000

[13] V N Serkin and T L Belyaeva ldquoOptimal control of opticalsoliton parameters Part 1 The Lax representation in theproblem of soliton managementrdquo Quantum Electronics vol 31no 11 pp 1007ndash1015 2001

[14] T-k Ning D-y Chen and D-j Zhang ldquoThe exact solutionsfor the nonisospectral AKNS hierarchy through the inversescattering transformrdquo Physica A Statistical Mechanics and itsApplications vol 339 no 3-4 pp 248ndash266 2004

[15] VN Serkin AHasegawa andT L Belyaeva ldquoNonautonomoussolitons in external potentialsrdquo Physical Review Letters vol 98no 7 Article ID 074102 2007

[16] B Guo and L Ling ldquoRiemann-Hilbert approach and N-solitonformula for coupled derivative Schrodinger equationrdquo Journalof Mathematical Physics vol 53 no 7 Article ID 073506 20pages 2012

[17] S Zhang B Xu and H-Q Zhang ldquoExact solutions of a KdVequation hierarchy with variable coefficientsrdquo InternationalJournal of Computer Mathematics vol 91 no 7 pp 1601ndash16162014

[18] S Zhang and D Wang ldquoVariable-coefficient nonisospectralToda lattice hierarchy and its exact solutionsrdquoPramanamdashJournal of Physics vol 85 no 6 pp 1143ndash11562015

[19] S Zhang and X-D Gao ldquoMixed spectral AKNS hierarchy fromlinear isospectral problem and its exact solutionsrdquoOpen Physicsvol 13 no 1 pp 310ndash322 2015

[20] S Randoux P Suret and G El ldquoInverse scattering transformanalysis of rogue waves using local periodization procedurerdquoScientific Reports vol 6 Article ID 29238 10 pages 2016

[21] S Zhang and X Gao ldquoExact solutions and dynamics of ageneralized AKNS equations associated with the nonisospec-tral depending on exponential functionrdquo Journal of NonlinearSciences and Applications vol 9 no 6 pp 4529ndash4541 2016

[22] S Zhang and J Li ldquoSoliton solutions and dynamical evolutionsof a generalized AKNS system in the framework of inversescattering transformrdquoOptik - International Journal for Light andElectron Optics vol 137 pp 228ndash237 2017

[23] S Zhang and S Hong ldquoLax integrability and soliton solutionsfor a nonisospectral integro-differential systemrdquo Complexityvol 2017 Article ID 9457078 10 pages 2017

[24] S Zhang and S Hong ldquoOn a generalizedAblowitzndashKaupndashNewellndashSegur hierarchy in inhomogeneitiesof media soliton solutions and wave propagation influencedfrom coefficient functions and scattering datardquo Waves inRandom and Complex Media vol 28 no 3 pp 435ndash452 2017

[25] S Zhang and S Hong ldquoLax Integrability and Exact Solutionsof a Variable-Coefficient and Nonisospectral AKNS HierarchyrdquoInternational Journal of Nonlinear Sciences and Numerical Sim-ulation vol 19 no 3-4 pp 251ndash262 2018

[26] Z Kang T Xia and X Ma ldquoMulti-solitons for the coupledFokasndashLenells system via RiemannndashHilbert approachrdquo ChinesePhysics Letters vol 35 no 7 Article ID 070201 5 pages 2018

[27] S Zhang and H Q Zhang ldquoFractional sub-equation methodand its applications to nonlinear fractional PDEsrdquo PhysicsLetters A vol 375 no 7 pp 1069ndash1073 2011

[28] I Aslan ldquoAnalytic investigation of a reaction-diffusion brusse-lator model with the time-space fractional derivativerdquo Interna-tional Journal of Nonlinear Sciences and Numerical Simulationvol 15 no 2 pp 149ndash155 2014

[29] I Aslan ldquoAn analytic approach to a class of fractionaldifferential-difference equations of rational type via symbolic

8 Complexity

computationrdquo Mathematical Methods in the Applied Sciencesvol 38 no 1 pp 27ndash36 2015

[30] I Aslan ldquoSymbolic computation of exact solutions for fractionaldifferential-difference equation modelsrdquo Nonlinear AnalysisModelling and Control vol 20 no 1 pp 132ndash144 2015

[31] Y Wang L Liu X Zhang and Y Wu ldquoPositive solutions ofan abstract fractional semipositone differential system modelfor bioprocesses of HIV infectionrdquo Applied Mathematics andComputation vol 258 pp 312ndash324 2015

[32] X Yang andH Srivastava ldquoAn asymptotic perturbation solutionfor a linear oscillator of free damped vibrations in fractalmedium described by local fractional derivativesrdquo Communi-cations in Nonlinear Science and Numerical Simulation vol 29no 1-3 pp 499ndash504 2015

[33] X-J Yang D Baleanu and H M Srivastava Local FractionalIntegral Transforms and Their Applications Academic PressLondon UK 2015

[34] G-C Wu D Baleanu Z-G Deng and S-D Zeng ldquoLatticefractional diffusion equation in terms of a Riesz-Caputa differ-encerdquo Physica A Statistical Mechanics and its Applications vol438 pp 335ndash339 2015

[35] Y Hu and J He ldquoOn fractal space-time and fractional calculusrdquoTHERMAL SCIENCE vol 20 no 3 pp 773ndash777 2016

[36] L Guo L Liu and Y Wu ldquoExistence of positive solutionsfor singular fractional differential equations with infinite-pointboundary conditionsrdquo Nonlinear Analysis Modelling and Con-trol vol 21 no 5 pp 635ndash650 2016

[37] S Zhang M Liu and L Zhang ldquoVariable separation fortime fractional advection-dispersion equation with initial andboundary conditionsrdquo THERMAL SCIENCE vol 20 no 3 pp789ndash792 2016

[38] L L Liu X Zhang L Liu and Y Wu ldquoIterative positivesolutions for singular nonlinear fractional differential equationwith integral boundary conditionsrdquo Advances in DifferenceEquations Paper No 154 13 pages 2016

[39] B Zhu L Liu and Y Wu ldquoLocal and global existence of mildsolutions for a class of nonlinear fractional reaction-diffusionequations with delayrdquo Applied Mathematics Letters vol 61 pp73ndash79 2016

[40] H Liu and F Meng ldquoSome new generalized Volterra-Fredholmtype discrete fractional sum inequalities and their applicationsrdquoJournal of Inequalities and Applications vol 2016 no 1 articleno 213 7 pages 2016

[41] I Aslan ldquoExact Solutions for a Local Fractional DDE Associ-ated with a Nonlinear Transmission Linerdquo Communications inTheoretical Physics vol 66 no 3 pp 315ndash320 2016

[42] I Aslan ldquoExact solutions for fractional DDEs via auxiliaryequation method coupled with the fractional complex trans-formrdquoMathematicalMethods in the Applied Sciences vol 39 no18 pp 5619ndash5625 2016

[43] J Wu X Zhang L Liu and Y Wu ldquoTwin iterative solutions fora fractional differential turbulent flow modelrdquo Boundary ValueProblems Article ID 98 9 pages 2016

[44] R Xu and FMeng ldquoSome newweakly singular integral inequal-ities and their applications to fractional differential equationsrdquoJournal of Inequalities and Applications vol 2016 Article ID 7816 pages 2016

[45] X Yang J A Tenreiro Machado D Baleanu and C Cat-tani ldquoOn exact traveling-wave solutions for local fractionalKorteweg-de Vries equationrdquo Chaos An Interdisciplinary Jour-nal of Nonlinear Science vol 26 no 8 Article ID 084312 5pages 2016

[46] X Yang F Gao and H Srivastava ldquoExact travelling wavesolutions for the local fractional two-dimensional Burgers-typeequationsrdquoComputers ampMathematics with Applications vol 73no 2 pp 203ndash210 2017

[47] X-J Yang and J A Machado ldquoA new fractional operator ofvariable order application in the description of anomalousdiffusionrdquo Physica A Statistical Mechanics and its Applicationsvol 481 pp 276ndash283 2017

[48] Y Wang and J Jiang ldquoExistence and nonexistence of positivesolutions for the fractional coupled system involving general-ized p-Laplacianrdquo Advances in Difference Equations Paper No337 19 pages 2017

[49] K M Zhang ldquoOn a sign-changing solution for some fractionaldifferential equationsrdquo Boundary Value Problems vol 2017 no59 8 pages 2017

[50] X Du and A Mao ldquoExistence and multiplicity of nontrivialsolutions for a class of semilinear fractional schrodinger equa-tionsrdquo Journal of Function Spaces vol 2017 Article ID 37938727 pages 2017

[51] Y Wang and L Liu ldquoPositive solutions for a class of fractional3-point boundary value problems at resonancerdquo Advances inDifference Equations Paper No 7 13 pages 2017

[52] X Zhang L Liu Y Wu and B Wiwatanapataphee ldquoNontrivialsolutions for a fractional advection dispersion equation inanomalous diffusionrdquo Applied Mathematics Letters vol 66 pp1ndash8 2017

[53] J-H He ldquoFractal calculus and its geometrical explanationrdquoResults in Physics vol 10 pp 272ndash276 2018

[54] R Hirota ldquoExact solution of the kortewegmdashde vries equationfor multiple Collisions of solitonsrdquo Physical Review Letters vol27 no 18 pp 1192ndash1194 1971

[55] M Wang ldquoExact solutions for a compound KdV-Burgersequationrdquo Physics Letters A vol 213 no 5-6 pp 279ndash287 1996

[56] E G Fan ldquoExtended tanh-functionmethod and its applicationsto nonlinear equationsrdquo Physics Letters A vol 277 no 4-5 pp212ndash218 2000

[57] Z Yan and H Zhang ldquoNew explicit solitary wave solutions andperiodic wave solutions for Whitham-Broer-Kaup equation inshallow waterrdquo Physics Letters A vol 285 no 5-6 pp 355ndash3622001

[58] E G Fan ldquoTravelling wave solutions in terms of specialfunctions for nonlinear coupled evolution systemsrdquo PhysicsLetters A vol 300 no 2-3 pp 243ndash249 2002

[59] S Zhang and T Xia ldquoA generalized F-expansion method andnew exact solutions of Konopelchenko-Dubrovsky equationsrdquoApplied Mathematics and Computation vol 183 no 2 pp 1190ndash1200 2006

[60] J He and X Wu ldquoExp-function method for nonlinear waveequationsrdquo Chaos Solitons amp Fractals vol 30 no 3 pp 700ndash708 2006

[61] I Aslan ldquoMulti-wave and rational solutions for nonlinearevolution equationsrdquo International Journal of Nonlinear Sciencesand Numerical Simulation vol 11 no 8 pp 619ndash623 2010

[62] I Aslan ldquoAnalytic investigation of the (2 + 1)-dimensionalSchwarzian Kortewegndashde Vries equation for traveling wavesolutionsrdquo Applied Mathematics and Computation vol 217 no12 pp 6013ndash6017 2011

[63] Y Wang and Y Chen ldquoBinary Bell polynomial manipula-tions on the integrability of a generalized (2+1)-dimensionalKortewegndashde Vries equationrdquo Journal of Mathematical Analysisand Applications vol 400 no 2 pp 624ndash634 2013

Complexity 9

[64] S Y Lou ldquoConsistent Riccati expansion for integrable systemsrdquoStudies in Applied Mathematics vol 134 no 3 pp 372ndash4022015

[65] S F Tian ldquoInitial-boundary value problems for the coupledmodified Korteweg-de Vries equation on the intervalrdquoCommu-nications on Pure andAppliedAnalysis vol 17 no 3 pp 923ndash9572018

[66] P Zhao and E Fan ldquoOn quasiperiodic solutions of the mod-ified KadomtsevndashPetviashvili hierarchyrdquo Applied MathematicsLetters vol 97 pp 27ndash33 2019

[67] Z Yan ldquoThe NLS(n n) equation Multi-hump compactonsand their stability and interaction scenariosrdquo Chaos Solitons ampFractals vol 112 pp 25ndash31 2019

[68] A-M Wazwaz ldquoCompacton solutions of higher order nonlin-ear dispersive KdV-like equationsrdquo Applied Mathematics andComputation vol 147 no 2 pp 449ndash460 2004

[69] L-Q Kong J Liu D-Q Jin D-J Ding and C-Q DaildquoSoliton dynamics in the three-spine 120572-helical protein withinhomogeneous effectrdquo Nonlinear Dynamics vol 87 no 1 pp83ndash92 2017

[70] B Zhang X-L Zhang andC-QDai ldquoDiscussions on localizedstructures based on equivalent solution with different forms ofbreaking soliton modelrdquo Nonlinear Dynamics vol 87 no 4 pp2385ndash2393 2017

Hindawiwwwhindawicom Volume 2018

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2 Complexity

whereΔ120572(119906(119909 119905)minus119906(119909 1199050)) cong Γ(1+120572)(119906(119909 119905)minus119906(119909 1199050)) someuseful properties [33] of the local time-fractional derivativehave been used in this paper

The rest of this paper is organized as follows In Section 2we derive the local tfKdV eq (1) by introducing a linear spec-tral problem equipped with local time-fractional derivativeIn Section 3 we construct fractional n-soliton solution ofthe local tfKdV eq (1) by extending the spectral transformmethod In Section 4 we investigate the dynamical evolutionsof the obtained fractional one-soliton solution two-solitonsolution and three-soliton solution In Section 5 we concludethis paper

2 Derivation of the Local tfKdV Equation

For the local tfKdV eq (1) we have the followingTheorem 1

Theorem 1 The local tfKdV eq (1) is a Lax system which canbe derived from the linear spectral problem equipped with alocal time-fractional evolution equation

120601119909119909 = (120578 minus 119906) 120601 120578 = minus1198962 (3)

119863(120572)119905 120601 = (119906119909 + 120599) 120601 + (41198962 minus 2119906) 120601119909 (4)

where 120601 = 120601(119909 119905) and 119906 = 119906(119909 119905) are all differentiablefunctions with respect to 119909 and 119905 the spectral parameter 120578 isindependent of 119905 and 120599 is an arbitrary constant

Proof Taking the time-fractional derivative of eq (3) yields

119863(120572)119905 120601119909119909 = minus120601119863(120572)119905 119906 minus (119906 + 1198962)119863(120572)119905 120601 (5)

Substituting eq (4) into eq (5) we have

119863(120572)119905 120601119909119909 = minus [119863(120572)119905 119906 + (119906119909 + 120599) (119906 + 1198962)] 120601minus (41198962 minus 2119906) (119906 + 1198962) 120601119909 (6)

Taking the derivative of eq (4) with respect to 119909 twice gives

(119863(120572)119905 120601)119909119909

= 119906119909119909119909120601 minus (3119906119909 minus 120599) 120601119909119909 + (41198962 minus 2119906) 120601119909119909119909 (7)

With the help of eq (3) from eq (7) we have

(119863(120572)119905 120601)119909119909

= [119906119909119909119909 + (5119906 minus 1198962) 119906119909 minus 120599 (119906 + 1198962)] 120601minus (41198962 minus 2119906) (119906 + 1198962) 120601119909 (8)

On the other hand at the aribitrary point 119905 = 1199050 we have119863120572119905 120601119909119909 (119909 1199050) = lim

119905997888rarr1199050

Δ120572 (120601119909119909 (119909 119905) minus 120601119909119909 (119909 1199050))(119905 minus 1199050)120572

(119863120572119905 120601 (119909 1199050))119909119909 = lim119905997888rarr1199050

Δ120572 (120601 (119909 119905) minus 120601 (119909 1199050))119909119909(119905 minus 1199050)120572 (9)

Finally using eqs (6) (8) and (9) we arrive at eq (1)Thus we finish the proof The process of proof shows that eq(1) is a Lax integrable system

3 Local Fractional Spectral Transform

Since the local tfKdV eq (1) is a local time-fractional systemthe orders of the derivatives with respect to space variable119909 are integers So all the existing results about the spectralproblem (3) a part of the Lax pair for the classical KdVequation can be translated to the local tfKdV eq (1)

For the direct scattering problem we translate somenecessary results and definitions [9] to the local tfKdV eq(1)

Lemma 2 If the real potential 119906(119909) defined in the whole realaxis minusinfin lt 119909 lt infin and its various derivatives are differentiablefunctions which vanishes rapidly as 119909 997888rarr plusmninfin and satisfies

int+infinminusinfin

10038161003816100381610038161003816119909119895119906 (119909)10038161003816100381610038161003816 d119909 lt +infin (119895 = 0 1 2) (10)

then the linear spectral problem (3) has a set of basic solutionscalled Jost solutions120601+(119909 119896) and120601minus(119909 119896) and they are not onlybounded for all values of 119909 but also analytic for Im 119896 gt 0 andcontinuous for Im 119896 ge 0 and have the following asymptoticproperties

120601+ (119909 119896) 997888rarr ei119896119909 (119909 997888rarr +infin) (11)

120601minus (119909 119896) 997888rarr eminusi119896119909 (119909 997888rarr minusinfin) (12)

Lemma 3 Define the Wronskian

119882(120601+ (119909 119896) 120601minus (119909 119896))= 120601+119909 (119909 119896) 120601minus (119909 119896) minus 120601+ (119909 119896) 120601minus119909 (119909 119896) (13)

and let

120601minus (119909 119896) = 119886 (119896) 120601+ (119909 minus119896) + 119887 (119896) 120601+ (119909 119896) (14)

Then

119886 (119896) = 12i119896119882(120601minus (119909 119896) 120601+ (119909 119896)) 119887 (119896) = 12i119896119882(120601+ (119909 minus119896) 120601minus (119909 119896))

(15)

where 119886(119896) is analytic for Im 119896 gt 0 and continuous for Im 119896 ge0 119887(119896) is defined only on the real axis Im 119896 = 0 and theanalytic function 119886(119896) has a finite number of simple zeros 119896119898 =i120581119898 (120581119898 gt 0119898 = 1 2 119899)Lemma 4 For the linear spectral problem (3) there exists aconstant 119887119898 such that

120601minus (119909 i120581119898) = 119887119898120601+ (119909 i120581119898) (16)

int+infinminusinfin

1198882119898120601+2 (119909 i120581119898) d119909 = 1 1198882119898 = minus i119887119898119886119896 (i120581119898) (17)

Definition 5 The constant 119888119898 satisfying eq (17) is named thenormalization constant for the eigenfunction 120601+(119909 i120581119898) and119888119898120601+(119909 i120581119898) is named normalization eigenfunction

Complexity 3


119896 (Im 119896 = 0) 119877 (119896) = 119887 (119896)119886 (119896) 119894120581119898 119888119898 119898= 1 2 119899

(18)








119875119909119909 + (1198962 + 119906)119875 = minus120601 (119863(120572)119905 119906 + 6119906119906119909 + 119906119909119909119909) = 0 (21)




119863(120572)119905 120581119898 (119905) = 0119863(120572)119905 119888119898 (119905) = 41205813119898 (119905) 119888119898 (119905)

(22)

119863(120572)119905 119886 (119896) = 0119863(120572)119905 119887 (119896) = 8i1198963 (119905) 119887 (119896) (23)


(120599 minus 4i1198963) eminusi119896119909 = 0 (24)



Namely

120599 = 4i1198963 (26)

119863(120572)119905 119886 (119896) = 119886 (119896) (120599 minus 4i1198963) = 0 (27)

119863(120572)119905 119887 (119896) = 119887 (119896) (120599 + 4i1198963) = 8i1198963119887 (119896) (28)



119863(120572)119905 119887119898 (119905) = (120599 + 41205813119898 (119905)) 119887119898 (119905) = 81205813119898 (119905) 119887119898 (119905) (29)


119863(120572)119905 (1198882119898 (119905)) = minus119863(120572)119905 (i119887119898 (119905))119886119896 (i120581119898) = 81205813119898 (119905) 1198882119898 (119905) (30)




119906 (119909 119905) = 2 dd119909119870 (119909 119909 119905) (31)


119870(119909 119910 119905) + 119865 (119909 + 119910 119905)+ intinfin119909

119870 (119909 119911 119905) 119865 (119909 + 119911 119905) d119911 = 0 (32)

with


119877 (119896 119905) ei119896119909d119896 + 119899sum119898=1

1198882119898ei120581119898119909 (33)





119906 (119909 119905) = 2 d2

d1199092 ln (det119863(119909 119905)) (34)

where119863 (119909 119905) = (119889119895119898 (119909 119905))119899times119899

119889119895119898 (119909 119905) = 120575119895119898 + 119888119895 (119905) 119888119898 (119905)120581119895 (0) + 120581119898 (0)eminus(120581119895(0)+120581119898(0))119909(35)

119888119895 (119905) = 119888119895 (0) 119864120572 (41205813119895 (0) 119905120572) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572)

(36)


4 Complexity


120581119898 (119905) = 120581119898 (0) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572) (37)

119886 (119896 119905) = 119886 (119896 0) 119887 (119896 119905) = 119887 (119896 0) 119864120572 (8i1205813 (0) 119905120572) (38)


119870(119909 119910 119905) + 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)(119909+119910)

+ 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)119910 intinfin119909

119870 (119909 119911 119905) eminus120581119898(0)119911d119911= 0

(39)


119870(119909 119910 119905) = 119899sum119895=1

119888119895 (119905) ℎ119895 (119909) eminus120581119895(0)119910 (40)


119870(119909 119910 119905) = dd119909 ln (det119863(119909 119905)) (41)







11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) ] (43)


119906 = 2 ln[[1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909

+ 11988822 (0) 119864120572 (812058132 (0) 119905120572)21205812 (0) eminus21205812(0)119909

+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2

sdot eminus2(1205811(0)+1205812(0))119909]]119909119909

(44)



+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2 eminus2(1205811(0)+1205812(0))119909

+ 11988821 (0) 11988823 (0) (1205811 (0) minus 1205813 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205811 (0) 1205813 (0) (1205811 (0) + 1205813 (0))2 eminus2(1205811(0)+1205813(0))119909

+ 11988822 (0) 11988823 (0) (1205812 (0) minus 1205813 (0))2 119864120572 (81205812 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205812 (0) 1205813 (0) (1205812 (0) + 1205813 (0))2 eminus2(1205812(0)+1205813(0))119909


sdot 119905120572) eminus2(1205811(0)+1205812(0)+1205813(0))119909]]119909119909

(45)

Complexity 5

5x

02

04

06

08

10

12

u

minus5

=1

=08

=07


0 2 4 6 8 10t

5

10

15

20

25

30v

=1

=08

=07



V = 3212057212058151 (0) 119905120572minus1 (46)



125 130 135 140x

02

04

06

08

10

12

u

=1

=08

=07


160 170 180 190 200x

02

04

06

08

10

12

u

=1

=08

=07


5 Conclusion


6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07


10 20 30x

05

10

15

20u

=1

=08

=07



x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07


Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 3


119896 (Im 119896 = 0) 119877 (119896) = 119887 (119896)119886 (119896) 119894120581119898 119888119898 119898= 1 2 119899

(18)








119875119909119909 + (1198962 + 119906)119875 = minus120601 (119863(120572)119905 119906 + 6119906119906119909 + 119906119909119909119909) = 0 (21)




119863(120572)119905 120581119898 (119905) = 0119863(120572)119905 119888119898 (119905) = 41205813119898 (119905) 119888119898 (119905)

(22)

119863(120572)119905 119886 (119896) = 0119863(120572)119905 119887 (119896) = 8i1198963 (119905) 119887 (119896) (23)


(120599 minus 4i1198963) eminusi119896119909 = 0 (24)



Namely

120599 = 4i1198963 (26)

119863(120572)119905 119886 (119896) = 119886 (119896) (120599 minus 4i1198963) = 0 (27)

119863(120572)119905 119887 (119896) = 119887 (119896) (120599 + 4i1198963) = 8i1198963119887 (119896) (28)



119863(120572)119905 119887119898 (119905) = (120599 + 41205813119898 (119905)) 119887119898 (119905) = 81205813119898 (119905) 119887119898 (119905) (29)


119863(120572)119905 (1198882119898 (119905)) = minus119863(120572)119905 (i119887119898 (119905))119886119896 (i120581119898) = 81205813119898 (119905) 1198882119898 (119905) (30)




119906 (119909 119905) = 2 dd119909119870 (119909 119909 119905) (31)


119870(119909 119910 119905) + 119865 (119909 + 119910 119905)+ intinfin119909

119870 (119909 119911 119905) 119865 (119909 + 119911 119905) d119911 = 0 (32)

with


119877 (119896 119905) ei119896119909d119896 + 119899sum119898=1

1198882119898ei120581119898119909 (33)





119906 (119909 119905) = 2 d2

d1199092 ln (det119863(119909 119905)) (34)

where119863 (119909 119905) = (119889119895119898 (119909 119905))119899times119899

119889119895119898 (119909 119905) = 120575119895119898 + 119888119895 (119905) 119888119898 (119905)120581119895 (0) + 120581119898 (0)eminus(120581119895(0)+120581119898(0))119909(35)

119888119895 (119905) = 119888119895 (0) 119864120572 (41205813119895 (0) 119905120572) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572)

(36)


4 Complexity


120581119898 (119905) = 120581119898 (0) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572) (37)

119886 (119896 119905) = 119886 (119896 0) 119887 (119896 119905) = 119887 (119896 0) 119864120572 (8i1205813 (0) 119905120572) (38)


119870(119909 119910 119905) + 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)(119909+119910)

+ 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)119910 intinfin119909

119870 (119909 119911 119905) eminus120581119898(0)119911d119911= 0

(39)


119870(119909 119910 119905) = 119899sum119895=1

119888119895 (119905) ℎ119895 (119909) eminus120581119895(0)119910 (40)


119870(119909 119910 119905) = dd119909 ln (det119863(119909 119905)) (41)







11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) ] (43)


119906 = 2 ln[[1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909

+ 11988822 (0) 119864120572 (812058132 (0) 119905120572)21205812 (0) eminus21205812(0)119909

+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2

sdot eminus2(1205811(0)+1205812(0))119909]]119909119909

(44)



+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2 eminus2(1205811(0)+1205812(0))119909

+ 11988821 (0) 11988823 (0) (1205811 (0) minus 1205813 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205811 (0) 1205813 (0) (1205811 (0) + 1205813 (0))2 eminus2(1205811(0)+1205813(0))119909

+ 11988822 (0) 11988823 (0) (1205812 (0) minus 1205813 (0))2 119864120572 (81205812 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205812 (0) 1205813 (0) (1205812 (0) + 1205813 (0))2 eminus2(1205812(0)+1205813(0))119909


sdot 119905120572) eminus2(1205811(0)+1205812(0)+1205813(0))119909]]119909119909

(45)

Complexity 5

5x

02

04

06

08

10

12

u

minus5

=1

=08

=07


0 2 4 6 8 10t

5

10

15

20

25

30v

=1

=08

=07



V = 3212057212058151 (0) 119905120572minus1 (46)



125 130 135 140x

02

04

06

08

10

12

u

=1

=08

=07


160 170 180 190 200x

02

04

06

08

10

12

u

=1

=08

=07


5 Conclusion


6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07


10 20 30x

05

10

15

20u

=1

=08

=07



x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07


Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








4 Complexity


120581119898 (119905) = 120581119898 (0) 119888119898 (119905) = 119888119898 (0) 119864120572 (41205813119898 (0) 119905120572) (37)

119886 (119896 119905) = 119886 (119896 0) 119887 (119896 119905) = 119887 (119896 0) 119864120572 (8i1205813 (0) 119905120572) (38)


119870(119909 119910 119905) + 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)(119909+119910)

+ 119899sum119898=1

1198882119898 (119905) eminus120581119898(0)119910 intinfin119909

119870 (119909 119911 119905) eminus120581119898(0)119911d119911= 0

(39)


119870(119909 119910 119905) = 119899sum119895=1

119888119895 (119905) ℎ119895 (119909) eminus120581119895(0)119910 (40)


119870(119909 119910 119905) = dd119909 ln (det119863(119909 119905)) (41)







11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) ] (43)


119906 = 2 ln[[1 + 11988821 (0) 119864120572 (812058131 (0) 119905120572)21205811 (0) eminus21205811(0)119909

+ 11988822 (0) 119864120572 (812058132 (0) 119905120572)21205812 (0) eminus21205812(0)119909

+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2

sdot eminus2(1205811(0)+1205812(0))119909]]119909119909

(44)



+ 11988821 (0) 11988822 (0) (1205811 (0) minus 1205812 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058132 (0) 119905120572)41205811 (0) 1205812 (0) (1205811 (0) + 1205812 (0))2 eminus2(1205811(0)+1205812(0))119909

+ 11988821 (0) 11988823 (0) (1205811 (0) minus 1205813 (0))2 119864120572 (81205811 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205811 (0) 1205813 (0) (1205811 (0) + 1205813 (0))2 eminus2(1205811(0)+1205813(0))119909

+ 11988822 (0) 11988823 (0) (1205812 (0) minus 1205813 (0))2 119864120572 (81205812 (0) 119905120572) 119864120572 (812058133 (0) 119905120572)41205812 (0) 1205813 (0) (1205812 (0) + 1205813 (0))2 eminus2(1205812(0)+1205813(0))119909


sdot 119905120572) eminus2(1205811(0)+1205812(0)+1205813(0))119909]]119909119909

(45)

Complexity 5

5x

02

04

06

08

10

12

u

minus5

=1

=08

=07


0 2 4 6 8 10t

5

10

15

20

25

30v

=1

=08

=07



V = 3212057212058151 (0) 119905120572minus1 (46)



125 130 135 140x

02

04

06

08

10

12

u

=1

=08

=07


160 170 180 190 200x

02

04

06

08

10

12

u

=1

=08

=07


5 Conclusion


6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07


10 20 30x

05

10

15

20u

=1

=08

=07



x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07


Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 5

5x

02

04

06

08

10

12

u

minus5

=1

=08

=07


0 2 4 6 8 10t

5

10

15

20

25

30v

=1

=08

=07



V = 3212057212058151 (0) 119905120572minus1 (46)



125 130 135 140x

02

04

06

08

10

12

u

=1

=08

=07


160 170 180 190 200x

02

04

06

08

10

12

u

=1

=08

=07


5 Conclusion


6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07


10 20 30x

05

10

15

20u

=1

=08

=07



x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07


Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








6 Complexity

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(a) 119905 = 0

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(b) 119905 = 01

x

05

10

15

u

5 10minus5minus10

=1

=08

=07(c) 119905 = 05

x

05

10

15

20u

5 10minus5minus10

=1

=08

=07(d) 119905 = 07


10 20 30x

05

10

15

20u

=1

=08

=07



x

05

10

15

20

25

30

u

10 20 30minus10

=1

=08

=07


Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 7

Data Availability




Acknowledgments


References






























8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








8 Complexity




































Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018








Complexity 9















Journal of












Journal of







Volume 2018






Volume 2018















Journal of












Journal of







Volume 2018






Volume 2018








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