A Unified Analysis of Exact Traveling Wave Solutions for the Fractional-Order and Integer-Order Biswas–Milovic Equation: Via Bifurcation Theory of Dynamical System

This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential equations. We use the conformable fractional derivatives. The method is based on the bifurcation theory of planar dynamical systems. To show the...

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Published in:Qualitative theory of dynamical systems Vol. 19; no. 1
Main Authors: Zhang, Bei, Zhu, Wenjing, Xia, Yonghui, Bai, Yuzhen
Format: Journal Article
Language:English
Published: Cham Springer International Publishing 01.04.2020
Springer Nature B.V
Subjects:
Bifurcation theory
Difference and Functional Equations
Dynamical systems
Dynamical Systems and Ergodic Theory
Integers
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Partial differential equations
System effectiveness
Traveling waves
Bifurcation
Periodic wave solution
Solitary wave solution
Traveling wave solution
Kink wave solution
ISSN:1575-5460, 1662-3592
Online Access:Get full text
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Abstract This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential equations. We use the conformable fractional derivatives. The method is based on the bifurcation theory of planar dynamical systems. To show the effectiveness of this method, we choose Biswas–Milovic (for short, BM) equation with conformable derivative as an application. Also comparison is presented for the exact traveling wave solutions between the integer-order BM equation and fractional-order BM equation. It is believed that this approach can be extended to other nonlinear fractional-order partial differential equations.
AbstractList This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential equations. We use the conformable fractional derivatives. The method is based on the bifurcation theory of planar dynamical systems. To show the effectiveness of this method, we choose Biswas–Milovic (for short, BM) equation with conformable derivative as an application. Also comparison is presented for the exact traveling wave solutions between the integer-order BM equation and fractional-order BM equation. It is believed that this approach can be extended to other nonlinear fractional-order partial differential equations.
ArticleNumber 11
Author Zhu, Wenjing
Zhang, Bei
Xia, Yonghui
Bai, Yuzhen
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  surname: Xia
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  organization: Department of Mathematics, Zhejiang Normal University
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  givenname: Yuzhen
  surname: Bai
  fullname: Bai, Yuzhen
  organization: School of Mathematical Sciences, Qufu Normal University
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Cites_doi 10.1142/S021812741950041X
10.1016/j.amc.2019.124576
10.3788/COL201917.020006
10.1364/OE.27.016440
10.1007/s12043-014-0837-z
10.1007/s11071-015-2516-0
10.1016/j.ijleo.2012.11.039
10.1016/j.ijleo.2015.12.051
10.1140/epjp/i2015-15255-5
10.1364/OE.27.006689
10.1007/s11071-018-4612-4
10.1007/s11071-016-3105-6
10.1007/s11082-018-1416-1
10.1016/j.ijleo.2018.05.129
10.1007/s11082-017-1225-y
10.1016/j.ijleo.2015.11.078
10.1111/sapm.12165
10.1016/j.cnsns.2014.07.022
10.1016/j.ijleo.2017.12.186
10.1016/j.jksus.2010.08.003
10.1016/j.ijleo.2018.09.002
10.1007/s11071-016-2621-8
10.1016/j.ijleo.2016.05.050
10.1109/JLT.2019.2910892
10.1016/j.ijleo.2014.03.039
10.1007/s11071-016-2724-2
10.1016/j.cnsns.2009.06.017
10.1016/j.cam.2014.01.002
10.1007/s12043-013-0565-9
10.1016/j.physleta.2007.07.051
10.1016/j.ijleo.2017.11.043
10.1016/j.ijleo.2018.07.086
10.1007/s11082-017-1310-2
10.1007/s40314-013-0098-3
10.1142/S0218127407019858
10.1007/s11071-016-2613-8
10.1088/0031-8949/82/06/065003
10.1007/s11071-015-2361-1
10.1007/s10092-015-0158-8
10.1016/j.ijleo.2018.08.067
10.1016/j.ijleo.2017.11.061
10.1080/09500340.2016.1184719
10.1016/j.camwa.2006.02.001
10.1016/j.ijleo.2018.08.023
10.1021/acsanm.9b00190
10.1140/epjp/i2015-15061-1
10.1016/j.ijleo.2016.11.130
10.1016/j.ijleo.2016.04.119
10.1016/j.ijleo.2017.09.023
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2020© Springer Nature Switzerland AG 2020
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Issue 1
Keywords Bifurcation
Periodic wave solution
Solitary wave solution
Traveling wave solution
Kink wave solution
Language English
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Springer Nature B.V
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References Wang, Li, Zhang (CR36) 2008; 372
Rizvi, Bashir, Ali, Ahmad (CR44) 2017; 131
Zhang, Xia, Zhu, Bai (CR52) 2019; 363
Eslami, Neirameh (CR3) 2018; 50
Liu, Eslami, Rezazadeh, Mirzazadeh (CR7) 2019; 95
Biswas, Mirzazadeh, Eslami (CR37) 2014; 125
Zhu, Xia, Zhang, Bai (CR53) 2019; 29
Eslami (CR39) 2016; 85
Zubair, Raza, Mirzazadeh, Liu, Zhou (CR9) 2018; 173
Najafi, Arbabi (CR30) 2016; 127
Manafian, Lakestani (CR33) 2016; 127
Li, Chen (CR50) 2007; 17
Biswas, Milovic (CR1) 2010; 15
Eslami, Neyrame, Ebrahimi (CR4) 2012; 24
Foroutana, Kumarb, Manafiand, Hoquee (CR25) 2018; 170
Nazarzadeh, Eslami, Mirzazadeh (CR8) 2013; 81
Liu, Liu, Lin, Liu, Lei, Wu, Dai, Wei (CR10) 2019; 27
Li (CR49) 2013
Wu, Huang (CR16) 2015
Liu, Liu, Liu, Quhe, Lei, Fang, Teng, Wei (CR11) 2019; 27
Kumar (CR42) 2017; 87
Eslami (CR18) 2016; 285
Kara, Biswas, Zhou, Moraru, Moshokoa, Belic (CR5) 2018; 174
Yu, Sun (CR47) 2017; 149
Jumarie (CR23) 2006; 51
Li, Zhao, Triki, Ekici, Mirzazadeh, Zhou, Liu (CR6) 2018; 175
Raza, Javid (CR48) 2018; 158
Zhou, Ekici, Sonmezoglu, Mirzazadeh (CR35) 2016; 127
Khalil, Horani, Yousef, Sababheh (CR24) 2014; 264
Rezazadeh, Korkmaz, Eslami, Vahidi, Asghari (CR21) 2018; 50
Jafari, Sooraki, Khalique (CR26) 2013; 124
Liu, Ouyang, Hou, Liu, Wei (CR14) 2019; 17
Liu, Liu, Wei (CR13) 2019; 37
Inc, Aliyu, Yusuf, Baleanu (CR43) 2018; 157
Khodadad, Nazzari, Eslami, Rezazadeh (CR20) 2017; 49
Manafian (CR31) 2015; 130
Song, Tang (CR51) 2017; 139
Podlubny (CR15) 1999
Zhou, Ekici, Sonmezoglu, Mirzazadeh, Eslami (CR34) 2016; 63
Liu (CR22) 2015; 22
Ma, Huang, Zhang (CR28) 2010; 82
Raza, Abdullah, Butt (CR45) 2018; 157
Mirzazadeh, Arnous (CR27) 2015; 3
Zhou, Ekici, Sonmezoglu, Mirzazadeh, Eslami (CR40) 2016; 84
Eslami, Rezazadeh (CR19) 2016; 53
Arshed, Biswas, Zhou, Moshokoa, Belic (CR46) 2018; 172
Ahmadiana, Darvishi (CR17) 2016; 127
Liu, Liu, Wang, Shen, Chang, Lei, Deng, Wei, Wei (CR12) 2019; 2
Manafian, Lakestani (CR32) 2015; 130
Zhou (CR41) 2016; 84
Eslami, Mirzazadeh, Neirameh (CR2) 2015; 84
Zhu, Li (CR54) 2016; 84
Mirzazadeh, Eslami, Biswas (CR38) 2014; 33
Eslami, Mirzazadeh (CR29) 2016; 83
AH Kara (352_CR5) 2018; 174
WJ Liu (352_CR11) 2019; 27
J Yu (352_CR47) 2017; 149
A Biswas (352_CR1) 2010; 15
Q Zhou (352_CR41) 2016; 84
M Inc (352_CR43) 2018; 157
STR Rizvi (352_CR44) 2017; 131
J Manafian (352_CR31) 2015; 130
ML Wang (352_CR36) 2008; 372
M Eslami (352_CR3) 2018; 50
M Najafi (352_CR30) 2016; 127
J Li (352_CR49) 2013
W Liu (352_CR12) 2019; 2
Q Zhou (352_CR40) 2016; 84
M Eslami (352_CR4) 2012; 24
A Biswas (352_CR37) 2014; 125
M Liu (352_CR13) 2019; 37
R Khalil (352_CR24) 2014; 264
Q Wu (352_CR16) 2015
G Jumarie (352_CR23) 2006; 51
A Zubair (352_CR9) 2018; 173
Y Song (352_CR51) 2017; 139
H Jafari (352_CR26) 2013; 124
S Kumar (352_CR42) 2017; 87
M Mirzazadeh (352_CR27) 2015; 3
J Manafian (352_CR32) 2015; 130
Q Zhou (352_CR35) 2016; 127
M Mirzazadeh (352_CR38) 2014; 33
FS Khodadad (352_CR20) 2017; 49
M Liu (352_CR14) 2019; 17
M Eslami (352_CR2) 2015; 84
N Raza (352_CR45) 2018; 157
J Manafian (352_CR33) 2016; 127
B Zhang (352_CR52) 2019; 363
W Zhu (352_CR54) 2016; 84
C Liu (352_CR22) 2015; 22
A Nazarzadeh (352_CR8) 2013; 81
M Eslami (352_CR39) 2016; 85
J Li (352_CR50) 2007; 17
B Li (352_CR6) 2018; 175
WJ Liu (352_CR10) 2019; 27
S Ahmadiana (352_CR17) 2016; 127
Q Zhou (352_CR34) 2016; 63
M Eslami (352_CR19) 2016; 53
M Foroutana (352_CR25) 2018; 170
M Eslami (352_CR29) 2016; 83
JG Liu (352_CR7) 2019; 95
M Eslami (352_CR18) 2016; 285
I Podlubny (352_CR15) 1999
S Arshed (352_CR46) 2018; 172
H Rezazadeh (352_CR21) 2018; 50
WX Ma (352_CR28) 2010; 82
N Raza (352_CR48) 2018; 158
W Zhu (352_CR53) 2019; 29
References_xml – volume: 22
  start-page: 92
  year: 2015
  end-page: 94
  ident: CR22
  article-title: Counterexamples on Jumarie’s two basic fractional calculus formulae
  publication-title: Commun. Nonl. Sci. Numer. Simulat.
– volume: 157
  start-page: 267
  year: 2018
  end-page: 274
  ident: CR43
  article-title: Optical solitons for Biswas–Milovic Model in nonlinear optics by Sine-Gordon equation method
  publication-title: Optik
– volume: 24
  start-page: 69
  year: 2012
  end-page: 71
  ident: CR4
  article-title: Explicit solutions of nonlinear -dimensional dispersive long wave equation
  publication-title: J. King Saud Univ. Sci.
– volume: 50
  start-page: 47
  year: 2018
  ident: CR3
  article-title: New exact solutions for higher order nonlinear Schrödinger equation in optical fibers
  publication-title: Opt. Quant. Electron.
– volume: 95
  start-page: 1027
  year: 2019
  end-page: 1033
  ident: CR7
  article-title: Rational solutions and lump dolutions to a non-isospectral and generalized variable-coefficient Kadomtsev–Petviashvili equation
  publication-title: Nonlinear Dyn.
– volume: 17
  start-page: 4049
  year: 2007
  end-page: 4065
  ident: CR50
  article-title: On a class of singular nonlinear traveling wave equations
  publication-title: Int. J. Bifurcat. Chaos
– volume: 173
  start-page: 249
  year: 2018
  end-page: 262
  ident: CR9
  article-title: Analytic study on optical solitons in parity-time-symmetric mixed linear and nonlinear modulation lattices with non-Kerr nonlinearities
  publication-title: Optik
– volume: 264
  start-page: 65
  year: 2014
  end-page: 70
  ident: CR24
  article-title: A new definition of fractional derivative
  publication-title: J. Comput. Appl. Math.
– volume: 87
  start-page: 1153
  issue: 2
  year: 2017
  end-page: 1157
  ident: CR42
  article-title: Invariant solutions of Biswas–Milovic equation
  publication-title: Nonlinear Dyn.
– volume: 84
  start-page: 1973
  year: 2016
  end-page: 1987
  ident: CR54
  article-title: Exact traveling wave solutions and birfurcations of the Biswas–Milovic equation
  publication-title: Nonlinear Dyn.
– volume: 127
  start-page: 2679
  year: 2016
  end-page: 2682
  ident: CR30
  article-title: Dark soliton and periodic wave solutions of the Biswas–Milovic equation
  publication-title: Optik
– volume: 158
  start-page: 1049
  year: 2018
  end-page: 1057
  ident: CR48
  article-title: Optical dark and singular solitons to the Biswas–Milovic equation in nonlinear optics with spatio-temporal dispersion
  publication-title: Optik
– volume: 51
  start-page: 1367
  year: 2006
  end-page: 1376
  ident: CR23
  article-title: Modified Riemann–Liouville derivative and fractional Taylor series of nondifferentiable functions further results
  publication-title: Comput. Math. Appl.
– volume: 33
  start-page: 831
  year: 2014
  end-page: 839
  ident: CR38
  article-title: Soliton solutions of the generalized Klein–Gordon equation by using -expansion method
  publication-title: Comput. Appl. Math.
– volume: 130
  start-page: 61
  year: 2015
  ident: CR32
  article-title: Optical solitons with Biswas–Milovic equation for Kerr law nonlinearity
  publication-title: Eur. Phys. J. Plus
– volume: 175
  start-page: 177
  year: 2018
  end-page: 180
  ident: CR6
  article-title: Soliton interactions for optical switching systems with symbolic computation
  publication-title: Optik
– volume: 131
  start-page: 582
  year: 2017
  end-page: 587
  ident: CR44
  article-title: Jacobian elliptic periodic traveling wave solutions for Biswas–Milovic equation
  publication-title: Optik
– year: 1999
  ident: CR15
  publication-title: Fractional Differential Equations
– volume: 285
  start-page: 141
  year: 2016
  end-page: 148
  ident: CR18
  article-title: Exact traveling wave solutions to the fractional coupled nonlinear Schrodinger equations
  publication-title: Appl. Math. Comput.
– volume: 127
  start-page: 2040
  year: 2016
  end-page: 2054
  ident: CR33
  article-title: Application of -expansion method for solving the Biswas–Milovic equation for Kerr law nonlinearity
  publication-title: Optik
– volume: 82
  start-page: 065003
  year: 2010
  ident: CR28
  article-title: A multiple exp-function method for nonlinear differential equations and its application
  publication-title: Phys. Scr.
– volume: 2
  start-page: 2697
  year: 2019
  end-page: 2705
  ident: CR12
  article-title: Thickness-dependent ultrafast photonics of nanolayers for optimizing fiber lasers
  publication-title: ACS Appl. Nano Mater.
– volume: 149
  start-page: 378
  year: 2017
  end-page: 383
  ident: CR47
  article-title: Exact traveling wave solutions to the -dimensional Biswas–Milovic equations
  publication-title: Optik
– volume: 37
  start-page: 3100
  year: 2019
  end-page: 3105
  ident: CR13
  article-title: saturable absorber with high modulation depth for erbium-doped fiber laser
  publication-title: J. Lightwave Technol.
– year: 2013
  ident: CR49
  publication-title: Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions
– volume: 174
  start-page: 195
  year: 2018
  end-page: 198
  ident: CR5
  article-title: Conservation laws for optical solitons with Chen–Lee–Liu equation
  publication-title: Optik
– volume: 27
  start-page: 6689
  year: 2019
  end-page: 6699
  ident: CR11
  article-title: Nonlinear optical properties of heterostructure in fiber lasers
  publication-title: Opt. Express.
– volume: 127
  start-page: 7694
  year: 2016
  end-page: 7703
  ident: CR17
  article-title: A new fractional Biswas–Milovic model with its periodic soliton solutions
  publication-title: Optik
– volume: 49
  start-page: 384
  year: 2017
  ident: CR20
  article-title: Soliton solutions of the conformable fractional Zakharov–Kuznetsov equation with dual-power law nonlinearity
  publication-title: Opt. Quant. Electron.
– volume: 85
  start-page: 813
  year: 2016
  end-page: 816
  ident: CR39
  article-title: Trial solution technique to chiral nonlinear Schrodinger’s equation in -dimensions
  publication-title: Nonliear Dyn.
– volume: 63
  start-page: 2131
  year: 2016
  end-page: 2137
  ident: CR34
  article-title: Analytical study of solitons to Biswas–Milovic model in nonlinear optics
  publication-title: J. Mod. Opt.
– volume: 27
  start-page: 16440
  year: 2019
  end-page: 16448
  ident: CR10
  article-title: Synthesis of high quality silver nanowires and their applications in ultrafast photonics
  publication-title: Opt. Express.
– volume: 170
  start-page: 190
  year: 2018
  end-page: 202
  ident: CR25
  article-title: New explicit soliton and other solutions for the conformable fractional Biswas–Milovic equation with Kerr and parabolic nonlinearity through an integration scheme
  publication-title: Optik
– volume: 139
  start-page: 371
  issue: 3
  year: 2017
  end-page: 404
  ident: CR51
  article-title: Stability, steady-state bifurcations, and turing patterns in a predator–prey model with herd behavior and prey-taxis
  publication-title: Stud. Appl. Math.
– volume: 17
  start-page: 020006
  year: 2019
  ident: CR14
  article-title: Q-switched fiber laser operating at 1.5 m based on
  publication-title: Chin. Opt. Lett.
– volume: 3
  start-page: 139
  year: 2015
  end-page: 146
  ident: CR27
  article-title: Exact solution of Biswas–Milovic equation using new efficient method
  publication-title: Electron. J. Math. Anal. Appl.
– volume: 124
  start-page: 3929
  year: 2013
  end-page: 3932
  ident: CR26
  article-title: Dark solitons of the Biswas–Milovic equation by the first integral method
  publication-title: Optik
– volume: 29
  start-page: 1950041
  issue: 3
  year: 2019
  ident: CR53
  article-title: Exact traveling wave solutions and bifurcations of the time fractional differential equations with applications
  publication-title: Int. J. Bifur. Chaos
  doi: 10.1142/S021812741950041X
– volume: 15
  start-page: 1473
  year: 2010
  end-page: 1484
  ident: CR1
  article-title: Bright and dark solitons of the generalized nonlinear Schröndinger equation
  publication-title: Commun. Nonlinear Sci. Numer. Simul.
– volume: 172
  start-page: 826
  year: 2018
  end-page: 831
  ident: CR46
  article-title: Optical soliton perturbation with differential group delay and parabolic law nonlinearity using -expansion method
  publication-title: Optik
– volume: 83
  start-page: 731
  year: 2016
  end-page: 738
  ident: CR29
  article-title: Optical solitons with Biswas–Milovic equation for power law and dual-power law nonlinearities
  publication-title: Nonlinear Dyn.
– volume: 127
  start-page: 6277
  year: 2016
  end-page: 6290
  ident: CR35
  article-title: Optical solitons with Biswas–Milovic equation by extended -expansion method
  publication-title: Optik
– volume: 130
  start-page: 255
  year: 2015
  ident: CR31
  article-title: On the complex structures of the Biswas–Milovic equation for power, parabolic and dual parabolic law nonlinearities
  publication-title: Eur. Phys. J. Plus
– volume: 363
  start-page: 124576
  year: 2019
  ident: CR52
  article-title: Explicit exact traveling wave solutions and bifurcations of the generalized combined double sinh–cosh-Gordon equation
  publication-title: Appl. Math. Comput.
  doi: 10.1016/j.amc.2019.124576
– volume: 50
  start-page: 150
  year: 2018
  ident: CR21
  article-title: Traveling wave solution of conformable fractional generalized reaction Duffing model by generalized projective Riccati equation method
  publication-title: Opt. Quant. Electron.
– volume: 84
  start-page: 677
  year: 2016
  end-page: 681
  ident: CR41
  article-title: Optical solitons for Biswas–Milovic model with Kerr law and parabolic law nonlinearities
  publication-title: Nonlinear Dyn.
– volume: 157
  start-page: 993
  year: 2018
  end-page: 1002
  ident: CR45
  article-title: Analytical soliton solutions of Biswas–Milovic equation in Kerr and non-Kerr law media
  publication-title: Optik
– volume: 81
  start-page: 225
  year: 2013
  end-page: 236
  ident: CR8
  article-title: Exact solutions of some nonlinear partial differential equations using functional variable method
  publication-title: Pramana
– volume: 84
  start-page: 1883
  year: 2016
  end-page: 1900
  ident: CR40
  article-title: Optical solitons with Biswas–Milovic equation by extended trial equation method
  publication-title: Nonlinear Dyn.
– year: 2015
  ident: CR16
  publication-title: Fractional Differential Equations
– volume: 84
  start-page: 3
  year: 2015
  end-page: 8
  ident: CR2
  article-title: New exact wave solutions for Hirota equation
  publication-title: Pramana
– volume: 53
  start-page: 475
  year: 2016
  end-page: 485
  ident: CR19
  article-title: The first integral method for Wu–Zhang system with conformable time-fractional derivative
  publication-title: Calcolo
– volume: 125
  start-page: 4215
  year: 2014
  end-page: 4218
  ident: CR37
  article-title: Dispersive dark optical soliton with Schödinger–Hirota equation by -expansion approach in power law medium
  publication-title: Optik
– volume: 372
  start-page: 417
  year: 2008
  end-page: 423
  ident: CR36
  article-title: The -expansion method and traveling wave solutions of nonlinear evolution equations in mathematical physics
  publication-title: Phys. Lett. A
– volume: 17
  start-page: 020006
  year: 2019
  ident: 352_CR14
  publication-title: Chin. Opt. Lett.
  doi: 10.3788/COL201917.020006
– volume: 27
  start-page: 16440
  year: 2019
  ident: 352_CR10
  publication-title: Opt. Express.
  doi: 10.1364/OE.27.016440
– volume: 285
  start-page: 141
  year: 2016
  ident: 352_CR18
  publication-title: Appl. Math. Comput.
– volume: 84
  start-page: 3
  year: 2015
  ident: 352_CR2
  publication-title: Pramana
  doi: 10.1007/s12043-014-0837-z
– volume: 84
  start-page: 677
  year: 2016
  ident: 352_CR41
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-015-2516-0
– volume: 124
  start-page: 3929
  year: 2013
  ident: 352_CR26
  publication-title: Optik
  doi: 10.1016/j.ijleo.2012.11.039
– volume: 127
  start-page: 2679
  year: 2016
  ident: 352_CR30
  publication-title: Optik
  doi: 10.1016/j.ijleo.2015.12.051
– volume: 130
  start-page: 255
  year: 2015
  ident: 352_CR31
  publication-title: Eur. Phys. J. Plus
  doi: 10.1140/epjp/i2015-15255-5
– volume: 27
  start-page: 6689
  year: 2019
  ident: 352_CR11
  publication-title: Opt. Express.
  doi: 10.1364/OE.27.006689
– volume: 95
  start-page: 1027
  year: 2019
  ident: 352_CR7
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-018-4612-4
– volume: 87
  start-page: 1153
  issue: 2
  year: 2017
  ident: 352_CR42
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-016-3105-6
– volume: 50
  start-page: 150
  year: 2018
  ident: 352_CR21
  publication-title: Opt. Quant. Electron.
  doi: 10.1007/s11082-018-1416-1
– volume: 170
  start-page: 190
  year: 2018
  ident: 352_CR25
  publication-title: Optik
  doi: 10.1016/j.ijleo.2018.05.129
– volume: 49
  start-page: 384
  year: 2017
  ident: 352_CR20
  publication-title: Opt. Quant. Electron.
  doi: 10.1007/s11082-017-1225-y
– volume: 363
  start-page: 124576
  year: 2019
  ident: 352_CR52
  publication-title: Appl. Math. Comput.
  doi: 10.1016/j.amc.2019.124576
– volume: 127
  start-page: 2040
  year: 2016
  ident: 352_CR33
  publication-title: Optik
  doi: 10.1016/j.ijleo.2015.11.078
– volume: 139
  start-page: 371
  issue: 3
  year: 2017
  ident: 352_CR51
  publication-title: Stud. Appl. Math.
  doi: 10.1111/sapm.12165
– volume: 22
  start-page: 92
  year: 2015
  ident: 352_CR22
  publication-title: Commun. Nonl. Sci. Numer. Simulat.
  doi: 10.1016/j.cnsns.2014.07.022
– volume: 158
  start-page: 1049
  year: 2018
  ident: 352_CR48
  publication-title: Optik
  doi: 10.1016/j.ijleo.2017.12.186
– volume-title: Fractional Differential Equations
  year: 2015
  ident: 352_CR16
– volume: 24
  start-page: 69
  year: 2012
  ident: 352_CR4
  publication-title: J. King Saud Univ. Sci.
  doi: 10.1016/j.jksus.2010.08.003
– volume: 175
  start-page: 177
  year: 2018
  ident: 352_CR6
  publication-title: Optik
  doi: 10.1016/j.ijleo.2018.09.002
– volume: 84
  start-page: 1973
  year: 2016
  ident: 352_CR54
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-016-2621-8
– volume-title: Singular Nonlinear Travelling Wave Equations: Bifurcations and Exact Solutions
  year: 2013
  ident: 352_CR49
– volume: 127
  start-page: 7694
  year: 2016
  ident: 352_CR17
  publication-title: Optik
  doi: 10.1016/j.ijleo.2016.05.050
– volume: 3
  start-page: 139
  year: 2015
  ident: 352_CR27
  publication-title: Electron. J. Math. Anal. Appl.
– volume: 37
  start-page: 3100
  year: 2019
  ident: 352_CR13
  publication-title: J. Lightwave Technol.
  doi: 10.1109/JLT.2019.2910892
– volume: 125
  start-page: 4215
  year: 2014
  ident: 352_CR37
  publication-title: Optik
  doi: 10.1016/j.ijleo.2014.03.039
– volume: 29
  start-page: 1950041
  issue: 3
  year: 2019
  ident: 352_CR53
  publication-title: Int. J. Bifur. Chaos
  doi: 10.1142/S021812741950041X
– volume: 85
  start-page: 813
  year: 2016
  ident: 352_CR39
  publication-title: Nonliear Dyn.
  doi: 10.1007/s11071-016-2724-2
– volume: 15
  start-page: 1473
  year: 2010
  ident: 352_CR1
  publication-title: Commun. Nonlinear Sci. Numer. Simul.
  doi: 10.1016/j.cnsns.2009.06.017
– volume: 264
  start-page: 65
  year: 2014
  ident: 352_CR24
  publication-title: J. Comput. Appl. Math.
  doi: 10.1016/j.cam.2014.01.002
– volume: 81
  start-page: 225
  year: 2013
  ident: 352_CR8
  publication-title: Pramana
  doi: 10.1007/s12043-013-0565-9
– volume: 372
  start-page: 417
  year: 2008
  ident: 352_CR36
  publication-title: Phys. Lett. A
  doi: 10.1016/j.physleta.2007.07.051
– volume: 157
  start-page: 993
  year: 2018
  ident: 352_CR45
  publication-title: Optik
  doi: 10.1016/j.ijleo.2017.11.043
– volume: 172
  start-page: 826
  year: 2018
  ident: 352_CR46
  publication-title: Optik
  doi: 10.1016/j.ijleo.2018.07.086
– volume: 50
  start-page: 47
  year: 2018
  ident: 352_CR3
  publication-title: Opt. Quant. Electron.
  doi: 10.1007/s11082-017-1310-2
– volume: 33
  start-page: 831
  year: 2014
  ident: 352_CR38
  publication-title: Comput. Appl. Math.
  doi: 10.1007/s40314-013-0098-3
– volume: 17
  start-page: 4049
  year: 2007
  ident: 352_CR50
  publication-title: Int. J. Bifurcat. Chaos
  doi: 10.1142/S0218127407019858
– volume: 84
  start-page: 1883
  year: 2016
  ident: 352_CR40
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-016-2613-8
– volume: 82
  start-page: 065003
  year: 2010
  ident: 352_CR28
  publication-title: Phys. Scr.
  doi: 10.1088/0031-8949/82/06/065003
– volume: 83
  start-page: 731
  year: 2016
  ident: 352_CR29
  publication-title: Nonlinear Dyn.
  doi: 10.1007/s11071-015-2361-1
– volume: 53
  start-page: 475
  year: 2016
  ident: 352_CR19
  publication-title: Calcolo
  doi: 10.1007/s10092-015-0158-8
– volume: 174
  start-page: 195
  year: 2018
  ident: 352_CR5
  publication-title: Optik
  doi: 10.1016/j.ijleo.2018.08.067
– volume: 157
  start-page: 267
  year: 2018
  ident: 352_CR43
  publication-title: Optik
  doi: 10.1016/j.ijleo.2017.11.061
– volume: 63
  start-page: 2131
  year: 2016
  ident: 352_CR34
  publication-title: J. Mod. Opt.
  doi: 10.1080/09500340.2016.1184719
– volume: 51
  start-page: 1367
  year: 2006
  ident: 352_CR23
  publication-title: Comput. Math. Appl.
  doi: 10.1016/j.camwa.2006.02.001
– volume: 173
  start-page: 249
  year: 2018
  ident: 352_CR9
  publication-title: Optik
  doi: 10.1016/j.ijleo.2018.08.023
– volume: 2
  start-page: 2697
  year: 2019
  ident: 352_CR12
  publication-title: ACS Appl. Nano Mater.
  doi: 10.1021/acsanm.9b00190
– volume: 130
  start-page: 61
  year: 2015
  ident: 352_CR32
  publication-title: Eur. Phys. J. Plus
  doi: 10.1140/epjp/i2015-15061-1
– volume: 131
  start-page: 582
  year: 2017
  ident: 352_CR44
  publication-title: Optik
  doi: 10.1016/j.ijleo.2016.11.130
– volume-title: Fractional Differential Equations
  year: 1999
  ident: 352_CR15
– volume: 127
  start-page: 6277
  year: 2016
  ident: 352_CR35
  publication-title: Optik
  doi: 10.1016/j.ijleo.2016.04.119
– volume: 149
  start-page: 378
  year: 2017
  ident: 352_CR47
  publication-title: Optik
  doi: 10.1016/j.ijleo.2017.09.023
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Snippet This paper presents a unified method to investigate exact traveling wave solutions of the nonlinear fractional-order and integer-order partial differential...
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SubjectTerms Bifurcation theory
Difference and Functional Equations
Dynamical systems
Dynamical Systems and Ergodic Theory
Integers
Mathematical analysis
Mathematics
Mathematics and Statistics
Nonlinear equations
Partial differential equations
System effectiveness
Traveling waves
Title A Unified Analysis of Exact Traveling Wave Solutions for the Fractional-Order and Integer-Order Biswas–Milovic Equation: Via Bifurcation Theory of Dynamical System
URI https://link.springer.com/article/10.1007/s12346-020-00352-x
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Volume 19
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