The present work deals with the solutions of the Gerdjikov–Ivanov(G–I) equation with the Riesz fractional derivative by means of the time-splitting spectral approach. In this approach, the G–I equation is split into two equations and the proposed technique viz. time-splitting spectral method is employed for discretizing the equation in space and then subsequently integrating in time exactly. Furthermore, an implicit finite difference method (IMFD) is utilized here to compare the results with the above-mentioned seminumerical method viz. time-splitting spectral technique. Moreover, it has been established that the proposed method is unconditionally stable. In addition to these, the error norms have been also presented here.

## References

References
1.
Gerdjikov
,
V. S.
, and
Ivanov
,
M. I.
,
1983
, “
The Quadratic Bundle of General Form and the Nonlinear Evolution Equations—I: Expansions Over the “Squared,” Solutions are Generalized Fourier Transforms
,”
Bulg. J. Phys.
,
10
(
1
), pp.
13
26
.http://cds.cern.ch/record/140137
2.
Gerdjikov
,
V. S.
, and
Ivanov
,
M. I.
,
1983
, “
The Quadratic Bundle of General Form and the Nonlinear Evolution Equations—II: Hierarchies of Hamiltonian Structures
,”
Bulg. J. Phys.
,
10
(
2
), pp.
130
143
.
3.
Kaup
,
D. J.
, and
Newell
,
A. C.
,
1978
, “
An Exact Solution for a Derivative Nonlinear Schrödinger Equation
,”
J. Math. Phys.
,
19
(
4
), pp.
798
801
.
4.
Tavazoei
,
M. S.
, and
Haeri
,
M.
,
2009
, “
Describing Function Based Methods for Predicting Chaos in a Class of Fractional Order Differential Equations
,”
Nonlinear Dyn.
,
57
(
3
), pp.
363
373
.
5.
Zhang
,
Y.
,
2013
, “
Time-Fractional Camassa-Holm Equation: Formulation and Solution Using Variational Methods
,”
ASME J. Comput. Nonlinear Dyn.
,
2013
(
4
), p.
0410201
.
6.
Podlubny
,
I.
,
1999
,
Fractional Differential Equations
,
,
New York
.
7.
Samko
,
S. G.
,
Kilbas
,
A. A.
, and
Marichev
,
O. I.
,
1993
,
Fractional Integrals and Derivatives: Theory and Applications
,
Gordan and Breach
,
New York
.
8.
Saha Ray
,
S.
,
2016
,
Fractional Calculus With Applications for Nuclear Reactor Dynamics
,
CRC Press
,
Boca Raton, FL
.
9.
Oldham
,
K. B.
, and
Spanier
,
J.
,
1974
,
The Fractional Calculus
,
,
New York
.
10.
Miller
,
K. S.
, and
Ross
,
B.
,
1993
,
An Introduction to Fractional Calculus and Fractional Differential Equations
,
Wiley
,
New York
.
11.
Bao
,
W.
, and
Yang
,
L.
,
2007
, “
Efficient and Accurate Numerical Methods for the Klein-Gordon-Schrödinger Equations
,”
J. Comput. Phys.
,
225
(
2
), pp.
1863
93
.
12.
Bao
,
W.
,
Jin
,
S.
, and
Markowich
,
P. A.
,
2002
, “
On Time-Splitting Spectral Approximations for the Schrödinger Equations in the Semiclassical Regime
,”
J. Comput. Phys.
,
175
(
2
), pp.
487
584
.
13.
Muslu
,
G. M.
, and
Erbay
,
H. A.
,
2003
, “
A Split-Step Fourier Method for the Complex Modified Korteweg-de Vries Equation
,”
Comput. Math. Appl.
,
45
(
1–3
), pp.
503
14
.
14.
Borluk
,
H.
,
Muslu
,
G. M.
, and
Erbay
,
H. A.
,
2007
, “
A Numerical Study of the Long Wave-Short Wave Interaction Equations
,”
Math. Comput. Simul.
,
74
(
2–3
), pp.
113
25
.
15.
Fan
,
E.
,
2000
, “
Darboux Transformation and Soliton-Like Solutions for the Gerdjikov-Ivanov Equation
,”
J. Phys. A
,
33
(
39
), pp.
6925
6933
.
16.
Guo
,
L.
,
Zhang
,
Y.
,
Xu
,
S.
,
Wu
,
Z.
, and
He
,
J.
,
2014
, “
The Higher Order Rogue Wave Solutions of the Gerdjikov-Ivanov Equation
,”
Phys. Scr.
,
89
(3), p.
11
.
17.
Yilmaz
,
H.
,
2015
, “
Exact Solutions of the Gerdjikov-Ivanov Equation Using the Darboux Transformations
,”
J. Nonlinear Math. Phys.
,
22
(
1
), pp.
32
46
.
18.
,
N.
, and
Jafari
,
H.
,
2017
, “
Analytical Solutions of the Gerdjikov-Ivanov Equation by Using $exp(−ϕ(ξ))$-Expansion Method
,”
Optik.
,
139
, pp.
72
76
.
19.
Yang
,
Q.
,
Liu
,
F.
, and
Turner
,
I.
,
2010
, “
Numerical Methods for Fractional Partial Differential Equations With Riesz Space Fractional Derivatives
,”
Appl. Math. Modell.
,
34
(
1
), pp.
200
218
.
20.
Saha Ray
,
S.
,
2014
, “
Soliton Solutions of Nonlinear and Nonlocal Sine-Gordon Equation Involving Riesz Space Fractional Derivative
,”
Z. Für Naturforsch. A
,
70
(
8
), pp.
659
667
.
21.
Saha Ray
,
S.
, and
Sahoo
,
S.
,
2015
, “
Analytical Approximate Solutions of Riesz Fractional Diffusion Equation and Riesz Fractional Advection-Dispersion Equation Involving Nonlocal Space Fractional Derivatives
,”
Math. Methods Appl. Sci.
,
38
(
13
), pp.
2840
2849
.
22.
Celik
,
C.
, and
Duman
,
M.
,
2012
, “
Crank-Nicolson Method for the Fractional Diffusion Equation With the Riesz Fractional Derivative
,”
J. Comput. Phys.
,
231
(
4
), pp.
1743
1750
.
23.
Saha Ray
,
S.
,
2017
, “
A Novel Approach With Time-Splitting Spectral Technique for the Coupled Schrödinger-Boussinesq Equations Involving Riesz Fractional Derivative
,”
Commun. Theor. Phys.
,
68
(
3
), pp.
301
308
.