We discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations. We show that the Hirota method is particular case of SEsM for specific form of the function from the Step. 2 of SEsM and for simple equations from the kind of differential equation for exponential function. We illustrate the methodology by obtaining the three-soliton solution of the Korteweg - de Vries equation, two soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials.

1.
M.
Bahrami
,
N.
Chinichian
,
A.
Hosseiny
,
G.
Jafari
,
M.
Ausloos
.
Physica A
540
,
123203
(
2020
).
2.
R.
Axelrod
,
M.
Cohen
.
Harnessing Complexity
(
Basic Books
,
New York
,
2001
)
3.
L.
Cameron
,
D.
Larsen-Freeman
.
International Journal of Applied Linguistics
17
,
226
239
(
2007
).
4.
R. M.
May
,
S. A.
Levin
,
G.
Sugihara
.
Nature
451
,
893
895
(
2008
).
5.
K
Ivanova
,
M.
Ausloos
.
Physica A
274
,
349
354
(
1999
).
6.
B.
Brehmer
.
Acta Psychologica
81
,
211
241
(
1992
).
7.
R.
Lambiotte
,
M.
Ausloos
.
Journal of Statistical Mechanics: Theory and Experiment P08026
(
2007
).
8.
9.
R.
Kutner
,
M.
Ausloos
,
D.
Grech
,
T.
Di Matteo
,
C.
Schinckus
,
H. E.
Stanley
.
Physica A
516
,
240
253
(
2019
).
10.
N. K.
Vitanov
.
Science Dynamics and Research Production. Indicators, Indexes, Statistical Laws and Mathematical Models
(
Springer
,
Cham
,
2016
).
11.
M.
Mitchell
.
Artificial Intelligence
170
,
1194
1212
(
2006
).
12.
R.
Mahon
,
P.
McConney
,
R. N.
Roy
.
Marine Policy
32
,
104
112
(
2008
).
13.
N. K.
Vitanov
,
M.
Ausloos
,
G.
Rotundo
.
Advances in Complex Systems
15, Supp. 01
,
1250049
(
2012
)
14.
N. K.
Vitanov
,
K. N.
Vitanov
.
Mathematical Social Sciences
,
80
,
108
114
(
2016
).
15.
L. A. N.
Amaral
,
A.
Scala
,
M.
Barthelemy
,
H. E.
Stanley
.
Proceedings of the National Academy of Sciences
,
97
,
11149
11152
(
2000
).
16.
D. A.
Hall
.
Journal of Materials Science
36
,
4575
4601
(
2001
).
17.
A.
Beyerchen
.
International Security
17
, No.
3
,
59
90
, Winteν=ae1+be2r
1992
/93.
18.
I.
Jordanov
,
E.
Nikolova
.
Journal of Theoretical and Applied Mechanics
43
, No.
2
,
69
76
(
2013
).
19.
A. S.
Pikovsky
,
D. L.
Shepelyansky
.
Phys. Rev. Lett.
100
,
094101
(
2008
).
20.
Z. I.
Dimitrova
.
Journal of Theoretical and Applied Mechanics
45
, No.
4
,
79
92
(
2015
).
21.
I. P.
Jordanov
.
Comp. rend. Acad. Sci. Bulg
,
61
,
307
314
(
2008
).
22.
P. G.
Drazin
.
Nonlinear Systems
(
Cambridge University Press
,
Cambridge, UK
,
1992
).
23.
T.
Boeck
,
N. K.
Vitanov
.
Physical Review E
,
65
,
037203
(
2002
).
24.
Z.
Dimitrova
.
Journal of Theoretical and Applied Mechanics
42
, No.
3
,
3
22
(
2012
).
25.
Y.
Niu
,
S.
Gong
.
Phys. Rev. A
73
,
053811
(
2006
).
26.
E.
Nikolova
,
E.
Goranova
,
Z.
Dimitrova
.
Comptes rendus de l'Academie Bulgare des sciences
69
, No.
9
,
1213
1222
(
2016
).
27.
K.
Kawasaki
,
T.
Ohta
.
Kink dynamics in one-dimensional nonlinear systems
.
Physica A
,
116
,
573
593
(
1982
).
28.
H.
Kantz
,
T.
Schreiber
.
Nonlinear Time Series Analysis
(
Cambridge University Press
,
Cambridge, UK
,
2004
).
29.
P. J.
Brockwell
,
R. A.
Davis
,
M. V.
Calder
.
Introduction to Time Series and Forecasting
. (
Springer
,
New York
,
2002
).
30.
K. T.
Ashenfelter
,
S. M.
Boker
,
J. R
Waddell
,
N.
Vitanov
.
Journal of Experimental Psychology: Human Perception and Performance
35
,
1072
1091
(
2009
).
31.
N. K.
Vitanov
,
M.
Ausloos
. Knowledge epidemics and populationdynamics models for describing idea diffusion, in
Models of Science Dynamics, edited by
A.
Scharnhorst
,
K
Boerner
,
P.
van den Besselaar
, (
Springer
,
Berin
,
2012) p.p
.
65
129
.
32.
F.
Verhulst
.
Nonlinear Differential Equations and Dynamical Systems
(
Springer
,
Berlin
,
2006
).
33.
H.
Kantz
,
D.
Holstein
,
M.
Ragwitz
,
N. K.
Vitanov
.
Physica A
342
,
315
321
(
2004
).
34.
M. H.
Ernst
.
Physics Reports
78
,
1
171
(
1981
).
35.
T. B.
Benjamin
,
J. L.
Bona
,
J. J.
Mahony
, J. J. (1972).
Philosophical Transactions of the Royal Society of London. Series A
,
272 47 – 78
(
1972
).
36.
N. K.
Vitanov
.
Physica D
136
,
322
339
(
2000
)
37.
Z. I.
Dimitrova
,
N. K.
Vitanov
.
Phys. Lett A
272
,
368
380
(
2000
)
38.
Z. I.
Dimitrova
,
N. K.
Vitanov
.
Physica A
300
,
91
115
(
2001
)
39.
R.
Struble
.
Nonlinear Differential Equations
. (
Dover
,
New York
,
2018
).
40.
Z. I.
Dimitrova
,
N. K.
Vitanov
.
Theoretical Population Biology
66
,
1
12
(
2004
).
41.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
M.
Ausloos
.
Physica A
,
389
,
4970
4980
(
2010
).
42.
D. B.
Taulbee
.
Physics of Fluids A
4
2555
2561
(
1992
).
43.
R.
Borisov
,
Z. I.
Dimitrova
,
N. K.
Vitanov
.
Entropy
,
22
,
553
(
2020
).
44.
N. K.
Vitanov
,
K. N.
Vitanov
.
Physica A
,
527
,
121174
(
2019
).
45.
S.
Grossberg
.
Neural Networks
1
,
17
61
(
1981
).
46.
E.
Hopf
.
Communications on Pure and Applied Mathematics
,
3
,
201
230
(
1950
).
47.
J. D.
Cole
.
Quarterly of Applied Mathematics
9
,
225
236
(
1951
).
48.
M. J.
Ablowitz
,
D. J.
Kaup
,
A. C.
Newell
,
H.
Segur
.
Studies in Applied Mathematics
,
53
,
249
315
(
1974
).
49.
M. J.
Ablowitz
,
P. A.
Clarkson
.
Solitons, Nonlinear Evolution Equations and Inverse Scattering
. (
Cambridge University Press
,
Cambridge, UK
,
1991
).
50.
C. S.
Gardner
,
J. M.
Greene
,
M. D.
Kruskal
,
R. R.
Miura
.
Phys. Rev. Lett.
19
,
1095
1097
(
1967
).
51.
R.
Hirota
.
Phys. Rev. Lett.
27
,
1192
1194
(
1971
).
52.
R.
Hirota
.
The Direct Method in Soliton Theory
. (
Cambridge University Press
,
Cambridge, UK
,
2004
).
53.
M.
Tabor
.
Chaos and Integrability in Dynamical Systems
(
Wiley
,
New York
,
1989
).
54.
F.
Carrielo
,
M.
Tabor
.
Physica D
,
39
,
77
94
(
1989
).
55.
F.
Carrielo
,
M.
Tabor
.
Physica D
,
53
,
59
70
(
1991
).
56.
J.
Weiss
,
M.
Tabor
,
G.
Carnevalle
.
Journal of Mathematical Physics
,
24
,
522
526
(
1983
).
57.
N. A.
Kudryashov
.
Physics Letters A
,
155
,
269
275
(
1991
).
58.
N. A. Kudryshov.
Chaos
,
Solitons & Fractals
24
,
1217
1231
(
2005
).
59.
N. A.
Kudryashov
,
N. B.
Loguinova
.
Applied Mathematics and Computation
205
,
361
365
(
2008
).
60.
N. A.
Kudryashov
.
Physics Letters A
,
342
,
99
106
(
2005
).
61.
N. A.
Kudryashov
.
Communications in Nonlinear Science and Numerical Simulation
,
17
,
2248
2253
(
2012
).
62.
N. K.
Vitanov
.
Pliska Studia Mathematica Bulgarica
,
30
,
29
42
(
2019
).
63.
N. K.
Vitanov
.
Journal of Theoretical and Applied Mechanics
,
49
,
107
122
(
2019
).
64.
N. K.
Vitanov
.
AIP Conference Proceedings
2159
,
030038
(
2019
).
65.
N. K.
Vitanov
,
Z. I.
Dimitrova
.
AIP Conference Proceedings
2159
,
030039
(
2019
).
66.
N.
Martinov
,
N.
Vitanov
.
Journal of Physics A: Mathematical and General
25
,
L51
L56
(
1992
).
67.
N.
Martinov
,
N.
Vitanov
.
Journal of Physics A: Mathematical and General
25
,
L419
L426
(
1992
).
68.
N. K.
Martinov
,
N. K.
Vitanov
.
Journal of Physics A: Mathematical and General
27
,
4611
4618
(
1994
).
69.
N. K.
Martinov
,
N. K.
Vitanov
.
Canadian Journal of Physics
,
72
,
618
624
(
1994
).
70.
N. K.
Vitanov
.
Proc. Roy. Soc. London A
,
454
,
2409
2423
(
1998
)
71.
N. K.
Vitanov
.
Journal of Physics A: Mathematical and General
,
29
,
5195
5207
(
1996
).
72.
N. K.
Vitanov
,
I. P.
Joranov
,
Z. I.
Dimitrova
.
Communications in Nonlinear Science and Numerical Simulation
14
,
2379
2388
(
2009
).
73.
N. K.
Vitanov
,
I. P.
Jordanov
,
Z. I.
Dimitrova
.
Applied Mathematics and Computation
215
,
2950
2964
(
2009
).
74.
N. K.
Vitanov
.
Communications in Nonlinear Science and Numerical Simulation
15
,
2050
2060
(
2010
).
75.
N. K.
Vitanov
,
Z. I.
Dimitrova
.
Communications in Nonlinear Science and Numerical Simulation
15
,
2836
2845
(
2010
).
76.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
H.
Kantz
.
Applied Mathematics and Computation
,
216
,
2587
2595
(
2010
).
77.
N. K.
Vitanov
.
Communications in Nonlinear Science and Numerical Simulation
,
16
,
1176
1185
(
2011
).
78.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
K. N.
Vitanov
.
Communications in Nonlinear Science and Numerical Simulation
,
16
,
3033
3044
(
2011
).
79.
N. K.
Vitanov
.
Communications in Nonlinear Science and Numerical Simulation
,
16
,
4215
4231
(
2011
).
80.
N. K.
Vitanov
.
Pliska Studia Mathematica Bulgarica
21
,
257
266
(
2012
).
81.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
H.
Kantz
.
Applied Mathematics and Computation
,
219
,
7480
7492
(
2013
).
82.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
K. N.
Vitanov
.
Computers & Mathematics with Applications
,
66
,
1666
1684
(
2013
).
83.
N. K.
Vitanov
,
Z. I.
Dimitrova
.
Applied Mathematics and Computation
,
247
,
213
217
(
2014
).
84.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
T. I.
Ivanova
.
Applied Mathematics and Computation
,
315
,
372
380
(
2017
).
85.
N. K.
Vitanov
,
Z. I.
Dimitrova
,
K. N.
Vitanov
.
Applied Mathematics and Computation
,
269
,
363
378
(
2015
).
86.
N. K.
Vitanov
,
Z. I.
Dimitrova
.
Journal of Theoretical and Applied Mechanics
, Sofia,
48
, No.
1
,
59
68
(
2018
).
87.
E. V.
Nikolova
,
I. P.
Jordanov
,
Z. I.
Dimitrova
,
N. K.
Vitanov
.
AIP Conference Proceedings
,
1895
,
07002
(
2017
).
88.
I. P.
Jordanov
,
N. K.
Vitanov
.
Studies in Computational Intelligence
793
,
199
210
(
2019
).
89.
W.
Hereman
,
W.
Zhuang
. Symbolic Computation of Solitons via Hirotas Bilinear Method, preprint,
Department of Mathematical and Computer Sciences Colorado School of Mines
(
1994
).
90.
Y.
Ishimori
.
Progress in Theoretical Physics
,
72
,
33
37
(
1984
).
91.
S.
Vijayalakshmi
,
M. Lakshmanan.
Chaos
,
Solitons and Fractals
33
203
216
(
2007
).
92.
M.
Lakshmanan
,
M.
Daniel
.
Physica A
,
107
,
533
552
(
1981
).
This content is only available via PDF.
You do not currently have access to this content.