1. Introduction
Rogue waves (RWs) have different names such as freak waves, giant waves and killer waves etc. They vanish as quickly as they appear with extremely steep wave peaks. Draper [1] came up with this concept in 1965. There is no formal definition of such giant waves but in general, any wave which is two to three times larger than the normal wave is regarded as RW and can be found in major deep seas as well as on shores causing accidents.
It is extremely difficult task to completely study the development and evolution process of RWs as their occurrence is uncertain and transitory. However, there are a number of hypotheses about the formation process of RWs which can be categorized in two classes. First one is due to the external environmental effects, including typhoon, wave group focus, seabed topography, water depth and external energy. The second type comprises of its intrinsic effects which include frequency, focusing, wave-wave linear superposition and nonlinear effects. RWs have found applications in various fields of Physics which include hydrodynamics [2], [3], optics [4], [5], [6], [7], plasmas [8] and superfluid helium [9].
Due to the advancement of computer algebra systems and symbolic computation tools [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], the RW solutions of lower order nonlinear evolutions equations (NLEE) have become a hot topic of research. In recent years, the extraction of soliton solutions for NLEEs have become a popular topic of research [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43]. RWs solutions of higher order NLEEs are incorporated to study complex physical problems. The spotlight of this work is to investigate a couple of higher order NLEEs for RWs of different orders.
The layout of this paper is. In Section 2, the RWs of 1st, 2nd and 3rd-order for the generalized HSI are investigated. Section 3 is devoted to extract 1st, 2nd and 3rd-order RWs of extended (3 + 1)-dimensional Jimbo-Miwa equation. After that, concluding remarks are given in Section 4.
The symbolic computation technique which is based on direct variable transformation and utilizing Hirota’s bilinear form investigates the multiple rogue wave solutions of two well known PDEs. The generalized Hirota-Satsuma-Ito (HSI) equation and the newly proposed extended (3 + 1)-dimensional Jimbo-Miwa (JM) equation are being addressed for the very first time in this article for extracting rogue waves of 1st, 2nd and 3rd-orders. The dynamics of the obtained results are also explained with the help of graphical illustrations that includes 3D plots, contour plots and density plots by assigning specific values of free parameters.
2. Generalized Hirota-Satsuma-Ito (HSI) equation
where and are arbitrary constants. Eq. (2) is called generalized Hirota-Satsuma-Itoequation (gHSI). Zhang et al. [50] reported lump solutions for Eq. (2). Ma et al. [51] studied interaction solutions for Eq. (2) while Zhou et al.[52] proposed lump and lump-soliton solutions for Eq. (2). In this work, we study multiple rational rogue wave solutions for Eq. (2) using symbolic computation technique.
2.1. 1st- order RW for generalized HSI equation
The RW of 1st-order is obtained in this section for the generalized Hirota-Satsuma-Ito equation. Applying traveling wave transformation on Eq. (2) yields
Next assume the subsequent variable transformation as
Indulging Eq. (4) into Eq. (3), the subsequent bilinear form for Eq. (2) has been obtained as
Where
Clarkson and Dowie proposed a rational solution [53] for the Boussinesq equation of a special form. The extension of such rational solutions formulation leads to the introduction of a new symbolic computation approach [54], which is capitalized to investigate the Nth-order RWs of NLEEs. The Nth-order RWs has the following form
$\begin{array}{l} \begin{aligned} G= & G_{n+1}(\xi, y ; m, n)=A_{n+1}(\xi, y)+2 m y B_{n}(\xi, y)+2 n \xi C_{n}(\xi, y) \\ & +\left(m^{2}+n^{2}\right) A_{n-1} \end{aligned}\\ \end{array}$
Without loss of generality, take and putting Eq. (10) into Eq. (5). Then equating the coefficients of all powers of ξ and y to zero. The following values of unknowns are calculated as
Eq. (10) takes the form as
Using the transformation , the 1st-order RW solutions for Eq. (2) has the form
Fig. 1. The Rogue wave of 1st-order for Eq. (13) with $m=n=0,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
Fig. 2. The rogue wave of 1st-order for Eq. (13) with $m=n=-6,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
2.2. 2nd-order RWs for the generalized HSI equation
The RWs of second-order for Eq. (2) can be obtained by putting n=1 in Eq. (9). The solution of Eq. (2) is expressed as
$\begin{array}{l} \begin{aligned} G= & G_{2}(\xi, y ; m, n)=A_{2}(\xi, y)+2 m y B_{1}(\xi, y)+2 n \xi C_{1}(\xi, y) \\ & +\left(m^{2}+n^{2}\right) A_{0} \\ = & \left(q+g y^{2}+h y^{4}+j y^{6}\right)+\left(c+d y^{2}+e y^{4}\right) \xi^{2}+\left(a+b y^{2}\right) \xi^{4}+\xi^{6} \\ & +2 m y\left(s+u y^{2}+t \xi^{2}\right)+2 n \xi\left(q+r y^{2}+i \xi^{2}\right)+\left(m^{2}+n^{2}\right) \end{aligned}\\ \text { (1) } \end{array}$
Inserting Eq. (14) in Eq. (5) and comparing the coefficients of all powers of y and ξ to zero. We obtain the following values of unknowns as
SET 1
SET 2
$\begin{aligned} & a=-\frac{25}{d_2+d_4}, \quad b=\frac{3\left(d_2+d_4\right)}{d_5}, \quad c=-\frac{125}{\left(d_2+d_4\right)^2}, \\ & d=-\frac{90}{d_5} \\ & e=\frac{3\left(d_2+d_4\right)^2}{d_5^2}, \quad j=\frac{\left(d_2+d_4\right)^3}{d_5^3}, \quad h=-\frac{17\left(d_2+d_4\right)}{d_5^2}, \\ & g=\frac{475}{\left(d_2+d_4\right) d_5} \\ & q=\frac{\sqrt{d_2+d_4} \pm \sqrt{d_2+d_4+7500 n^2+4\left(d_2+d_4\right)^3 n^4}}{2 \sqrt{\left(d_2+d_4\right)^5} n^2}, \\ & i=\frac{\sqrt{d_2+d_4} \pm \sqrt{d_2+d_4+7500 n^2+4\left(d_2+d_4\right)^3 n^4}}{2 \sqrt{\left(d_2+d_4\right)^3} n^2} \\ & r=-\frac{3\left(\sqrt{d_2+d_4} \pm \sqrt{d_2+d_4+7500 n^2+4\left(d_2+d_4\right)^3 n^4}\right)}{2 d_5 \sqrt{d_2+d_4} n^2}, \\ & m=0 \text {. } \\ & \end{aligned}$
And SET 3
$\begin{aligned} & a=-\frac{25}{d_2+d_4}, \quad b=\frac{3\left(d_2+d_4\right)}{d_5}, \quad c=-\frac{125}{\left(d_2+d_4\right)^2}, \\ & d=-\frac{90}{d_5}, \\ & e=\frac{3\left(d_2+d_4\right)^2}{d_5^2}, \quad j=\frac{\left(d_2+d_4\right)^3}{d_5^3}, \quad h=-\frac{17\left(d_2+d_4\right)}{d_5^2}, \\ & g=\frac{475}{\left(d_2+d_4\right) d_5}, \quad \\ & s=-\frac{5 t}{3\left(d_2+d_4\right)}, \quad u=-\frac{\left(d_2+d_4\right) t}{3 d_5}, \quad q=\frac{3\left(d_2+d_4\right)^3+\sqrt{\Delta}}{2 \sqrt{6\left(d_2+d_4\right)^5} n^2}, \\ & i=\frac{3\left(d_2+d_4\right)^3+\sqrt{\Delta}}{2 \sqrt{6\left(d_2+d_4\right)^4} n^2}, \quad r=-\frac{3\left(d_2+d_4\right)^3+\sqrt{\Delta}}{2 d_5 \sqrt{6\left(d_2+d_4\right)^3} n^2},\end{aligned}$
where . The rogue waves of 2nd-order for Eq. (2) are obtained for SET 1, SET 2 and SET 3 as
Fig. 3. The rogue waves of 2nd-order for Eq. (15) with $m=n=0,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
Fig. 4. The rogue waves of 2nd-order for Eq. (15) with $m=n=200,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
2.3. The 3rd-order rogue waves for the generalized HSI equation
The rogue waves of 3rd-order for Eq. (2) are obtained by putting n=2 in Eq. (9), we have
$\begin{aligned} G= & G_{3}(\xi, y ; m, n)=A_{3}(\xi, y)+2 m y B_{2}(\xi, y)+2 n \xi C_{2}(\xi, y) \\ & +\left(m^{2}+n^{2}\right) A_{1}(\xi, y) \\ = & \xi^{12}+\left(a+b y^{2}\right) \xi^{10}+\left(c+d y^{2}+e y^{4}\right) \xi^{8}+\left(g+h y^{2}+i y^{4}+j y^{6}\right) \xi^{6} \\ & +\left(k+l y^{2}+m_{1} y^{4}+n_{1} y^{6}+o y^{8}\right) \xi^{4}+\left(a_{1}+a_{2} y^{2}+a_{3} y^{4}+a_{4} y^{6}\right. \\ & \left.+a_{5} y^{8}+a_{6} y^{10}\right) \xi^{2}+b_{1}+b_{2} y^{2}+b_{3} y^{4}+b_{4} y^{6}+c_{1} y^{8} \\ & +c_{2} y^{10}+c_{3} y^{12} \\ & +2 m y\left(D_{0}+\left(D_{1}+D_{2} \xi^{2}+D_{3} \xi^{4}\right) y^{2}+\left(D_{5}+D_{6} \xi^{2}\right) y^{4}+y^{6}\right. \\ & \left.+D_{7} \xi^{2}+D_{8} \xi^{4}+D_{9} \xi^{6}\right) \\ & +2 n \xi\left(e_{0}+e_{1} y^{2}+e_{2} y^{4}+e_{3} y^{6}+\left(e_{4}+e_{5} y^{2}+e_{6} y^{4}\right) \xi^{2}\right. \\ & \left.+\left(e_{7}+e_{8} y^{2}\right) \xi^{4}+\xi^{6}\right) \\ & +\left(m^{2}+n^{2}\right)\left(\xi^{2}-\frac{3}{d_{2}+d_{4}}+\frac{y^{2}\left(d_{2}+d_{4}\right)}{d_{5}}\right) \end{aligned}$
Substituting Eq. (16) in Eq. (5). Then comparing the coefficients ξ and y to zero. We retrieve the following values of unknowns as
SET 1
$\begin{array}{l} a=98, \quad b=2, \quad c=735, \quad d=230, \quad e=\frac{5}{3} \\ g=\frac{75460}{3}, \quad j=\frac{20}{27}, \quad h=\frac{18620}{3}, \quad i=\frac{1540}{9} \\ k=-\frac{5187875}{3}, \quad l=73500, \quad m_{1}=\frac{37450}{9}, \\ n_{1}=\frac{1460}{27}, \quad \quad=\frac{5}{27}, \\ b_{1}=\frac{878826025}{9}, \quad b_{2}=\frac{300896750}{9}, \quad b_{3}=\frac{16391725}{27}, \\ b_{4}=\frac{798980}{81}, \\ a_{1}=\frac{159786550}{3}, \quad a_{2}=18865, \quad a_{3}=-\frac{4900}{3} \\ a_{4}=\frac{35420}{27}, \quad a_{5}=\frac{190}{27}, \\ a_{6}=\frac{2}{81}, \quad c_{1}=\frac{1445}{27}, \quad c_{2}=\frac{58}{243}, \quad c_{3}=\frac{1}{729} \\ m=n=0 . \end{array}$
And SET 2
$\begin{array}{l} a=98, \quad b=2, \quad c=735, \quad d=230, \quad e=\frac{5}{3}, \quad g=\frac{75460}{3}, \quad j=\frac{20}{27} \\ h=\frac{18620}{3}, \quad i=\frac{1540}{9}, \quad k=-\frac{5187875}{3}, \quad l=73500, \quad m_{1}=\frac{37450}{9}, \\ n_{1}=\frac{1460}{27}, \quad \quad \quad=\frac{5}{27}, \quad e_{0}=\frac{12005}{3}, \quad e_{1}=\frac{535}{3}, \quad e_{2}=5, \quad e_{3}=\frac{5}{27}, \\ e_{4}=-245, \quad e_{5}=-\frac{230}{3}, \quad e_{6}=-\frac{5}{9}, \quad e_{7}=13, \quad e_{8}=-3, \\ D_{0}=169785, \quad D_{1}=-2205, \quad D_{2}=-1710, \quad D_{3}=-45, \quad D_{5}=-21, \\ D_{6}=-27, \quad D_{7}=-17955, \quad D_{8}=2835, \quad D_{9}=135, \\ b_{1}=\frac{878826025+59022 m^{2}}{9}, \\ b_{2}=\frac{300896750+6558 m^{2}}{9}, \quad b_{3}=\frac{16391725}{27}, \quad b_{4}=\frac{798980}{81}, \\ a_{1}=\frac{159786550+6558 m^{2}}{3}, \quad a_{2}=18865, \quad a_{3}=-\frac{4900}{3}, \quad a_{4}=\frac{35420}{27}, \\ a_{5}=\frac{190}{27}, a_{6}=\frac{2}{81}, \quad c_{1}=\frac{1445}{27}, \quad c_{2}=\frac{58}{243}, \quad c_{3}=\frac{1}{729}. \end{array}$
The rogue waves of 3rd-order for of Eq. (1) are obtained for SET 1 and SET 2 as
where is given in Eq. (16). The graphical representations of the 3rd-order rogue waves are displayed in Figs. 5 and 6. In Fig. 5, the concentration of 3rd-order RWs at (0,0) shows the consistent behavior of these waves for m=n=0. However, for sufficiently large values of m and n, Fig. 6portrays central peak alongside five other peaks.
Fig. 5. The 3rd-order rogue wave Eq. (17) with $m=n=0,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
Fig. 6. The rogue waves of 3rd-order for Eq. (17) with $m=n=4000,{{s}_{0}}=1,{{d}_{2}}=1,{{d}_{4}}=-2,{{d}_{5}}=-3$(a) 3D plot, (b) the contour plot, (c) the respective density plot. |
3. Extended (3 + 1)-dimensional Jimbo-Miwa (JM) equation
The JM equation in (3 + 1)-dimensions
is second equation in integrable systems of KP hierarchy [55]. In [56], Cao proposed the exact solutions for Eq. (18). In [57], Xu studied soliton, dormions for model (18). The model explains intriguing (3 + 1)-dimensional waves without passing any integrability criteria. Eq. (18) gives a number of interesting physical structures. In [58], A.M. Wazwaz extended Eq. (18) with same order and dimensions of Eq. (18). The new extended (3 + 1)-dimensional JM equation is considered as
In [59], Sun et al. studied lump and lump-kink solutions of Eq. (19). In 2018, Ali et al. reported soliton solutions for extended (3 + 1)-dimensional JM equation [60]. Interaction solutions for extended (3 + 1)-dimensional JM equation were studied by Wang et al. [61]. This work will explore the dynamics of 1st and higher order rogue waves profiles for Eq. (19).
3.1. The 1st-order rogue waves for JM equation
The 1st-order RWs are obtained in this section for the extended (3 + 1)-dimensional JM equation. Applying transformation on Eq. (19) yields
Consider the subsequent traveling wave transformation as
Indulging Eq. (21) into Eq. (20), the following bilinear form for Eq. (19) has been retrieved as
Upon choosing n=0, Eq. (9) takes the following form
Without loss of generality, take and putting Eq. (23) into Eq. (22). Then comparing the coefficients of ξ and y to zero. We obtain the following values of unknowns as
The solution of Eq. (22) takes the following form
Using the transformation the 1st-order rogue wave solutions for Eq. (19) has the form
3.2. The 2nd-order RWs for JM equation
The RWs of 2nd-order are retrieved by taking n=1 in Eq. (9). The corresponding solution expression takes the form
$\begin{array}{l} \begin{aligned} G= & G_{2}(\xi, y ; m, n)=A_{2}(\xi, y)+2 m y B_{1}(\xi, y)+2 n \xi C_{1}(\xi, y) \\ & +\left(m^{2}+n^{2}\right) A_{0} \\ = & \left(q+g y^{2}+h y^{4}+j y^{6}\right)+\left(c+d y^{2}+e y^{4}\right) \xi^{2}+\left(a+b y^{2}\right) \xi^{4}+\xi^{6} \\ & +2 m y\left(s+u y^{2}+t \xi^{2}\right)+2 n \xi\left(q+r y^{2}+i \xi^{2}\right)+\left(m^{2}+n^{2}\right) .(27) \end{aligned}\\ \end{array}$
Substituting Eq. (27) in Eq. (22). By comparing the coefficients of ξ and y, a system of algebraic equations is obtained and solved for the following set of parameters:
SET 1
And SET 2
The rogue waves of 2nd-order for Eq. (19) are obtained for SET 1 and SET 2 as
where is given in Eq. (27).
3.3. The 3rd-order RWs for JM equation
The 3rd-order RWs of Eq. (19) are obtained by putting n=2 in Eq. (9), we have
$\begin{aligned} G= & G_{3}(\xi, y ; m, n)=A_{3}(\xi, y)+2 m y B_{2}(\xi, y)+2 n \xi C_{2}(\xi, y)+\left(m^{2}+n^{2}\right) A_{1}(\xi, y) \\ = & \xi^{12}+\left(a+b y^{2}\right) \xi^{10}+\left(c+d y^{2}+e y^{4}\right) \xi^{8}+\left(g+h y^{2}+i y^{4}+j y^{6}\right) \xi^{6} \\ & +\left(k+l y^{2}+m_{1} y^{4}+n_{1} y^{6}+o y^{8}\right) \xi^{4} \\ & +\left(a_{1}+a_{2} y^{2}+a_{3} y^{4}+a_{4} y^{6}+a_{5} y^{8}+a_{6} y^{10}\right) \xi^{2} \\ & +b_{1}+b_{2} y^{2}+b_{3} y^{4}+b_{4} y^{6}+c_{1} y^{8}+c_{2} y^{10}+c_{3} y^{12} \\ & +2 m y\left(D_{0}+\left(D_{1}+D_{2} \xi^{2}+D_{3} \xi^{4}\right) y^{2}+\left(D_{5}+D_{6} \xi^{2}\right) y^{4}+y^{6}+D_{7} \xi^{2}+D_{8} \xi^{4}\right. \\ & \left.+D_{9} \xi^{6}\right) \\ & +2 n \xi\left(e_{0}+e_{1} y^{2}+e_{2} y^{4}+e_{3} y^{6}+\left(e_{4}+e_{5} y^{2}+e_{6} y^{4}\right) \xi^{2}+\left(e_{7}+e_{8} y^{2}\right) \xi^{4}+\xi^{6}\right) \\ & +\left(m^{2}+n^{2}\right)\left(\xi^{2}-\frac{2}{3} y^{2}-\frac{3}{2}\right). \end{aligned}$
Substituting Eq. (29) in Eq. (22) and repeating the same procedure, the following values of unknowns as listed below
SET 1
$\begin{array}{l} a=-49, \quad b=-4, \quad c=\frac{735}{4}, \quad d=230, \quad e=\frac{20}{3}, \\ g=-\frac{18865}{6}, \quad j=-\frac{160}{27}, \quad h=-\frac{9310}{3}, \quad i=-\frac{3080}{9}, \\ k=-\frac{5187875}{48}, \quad l=18375, \quad m_{1}=\frac{37450}{9}, \quad n_{1}=\frac{5840}{27}, \quad o=\frac{80}{27}, \\ b_{1}=\frac{878826025}{576}, \quad b_{2}=\frac{150448375}{72}, \quad b_{3}=\frac{16391725}{108}, \quad b_{4}=\frac{798980}{81}, \\ a_{1}=-\frac{79893275}{48}, \quad a_{2}=-\frac{94325}{4}, \quad a_{3}=\frac{2450}{3}, \quad a_{4}=-\frac{70840}{27}, \\ a_{5}=-\frac{1520}{27}, \\ a_{6}=-\frac{64}{81}, \quad c_{1}=\frac{5780}{27}, \quad c_{2}=\frac{928}{243}, \quad c_{3}=\frac{64}{729}, \quad m=n=0. \end{array}$
And SET 2
$\begin{array}{l} a=-49, \quad b=-4, \quad c=\frac{735}{4}, \quad d=230, \quad e=\frac{20}{3}, \\ g=-\frac{18865}{6}, \quad j=-\frac{160}{27}, \quad h=-\frac{9310}{3}, \quad i=-\frac{3080}{9}, \\ k=-\frac{5187875}{48}, \quad l=18375, \quad m_{1}=\frac{37450}{9}, \quad n_{1}=\frac{5840}{27}, \quad o=\frac{80}{27}, \\ e_{0}=-\frac{12005}{24} \quad e_{1}=-\frac{535}{6}, \quad e_{2}=-10, \quad e_{3}=-\frac{40}{27}, \quad e_{4}=-\frac{245}{4}, \quad e_{5}=-\frac{230}{3}, \\ e_{6}=-\frac{20}{9}, \quad e_{7}=-\frac{13}{2}, \quad e_{8}=6, \quad D_{0}=\frac{169789}{64}, \quad D_{1}=-\frac{2205}{16}, \quad D_{2}=\frac{855}{4} \\ D_{3}=-\frac{45}{4}, \quad D_{5}=-\frac{21}{4}, \quad D_{6}=\frac{27}{2}, \quad D_{7}=\frac{17955}{32}, \quad D_{8}=\frac{2835}{16}, \quad D_{9}=-\frac{135}{8} \\ b_{1}=\frac{5\left(703060820+12501 m^{2}\right)}{2304}, \quad b_{2}=\frac{5\left(240717400+1389 m^{2}\right)}{576}, \\ b_{3}=\frac{16391725}{108}, \quad b_{4}=\frac{798980}{81}, \quad a_{2}, \quad a_{2}=-\frac{94325}{4}, \quad a_{3}=\frac{2450}{3}, \quad a_{4}=-\frac{70840}{27} \\ a_{1}=-\frac{5\left(127829240+1389 m^{2}\right)}{384}, \quad a_{6}=-\frac{64}{81}, \quad c_{1}=\frac{5780}{27}, \quad c_{2}=\frac{928}{243}, \quad c_{3}=\frac{64}{729} . \\ a_{5}=-\frac{1520}{27}, \quad a_{0} \end{array}$
The 3rd-order RWs of Eq. (19) are obtained for SET 1 and SET 2 as
where is given in Eq. (29).
4. Conclusion
This paper investigates (2 + 1)-dimensional genralized HSI equation and (3 + 1)-dimensional JM equation by a symbolic computation approach. By incorporating a direct variable transformation and utilizing Hirota’s bilinear form, multiple rogue wave structures of different orders are obtained for both generalized HSI and JM equation. The solutions are verified by inserting them into original equations. To discuss and emphasize our findings, three dimensional graphs of rogue wave solutions of generalized HSI equation alongside contour and density plots are portrayed. In Figs. 1 and 2, 3D plots of rogue wave of 1st-order are displayed along side their density and contour maps. For m=n=0, it has been observed that the rogue waves of 1st-order fuse together at center (Fig. 1) and for nonzero values of m,n, the peaks move away from center. Similarly, 2nd (Figs. 3 and 4)and 3rd-order rogue waves (Figs. 5 and 6) are also observed to be following the same pattern of waves fusion. The graphs for (3 + 1)-dimensional JM equation are not included in the manuscript but can be provided on request.
The results presented in this research are novel and can be a valuable addition in the literature for such rich models.
CRediT authorship contribution statement
Saima Arshed: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Nauman Raza: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Asma Rashid Butt: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Ahmad Javid: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. J.F. Gómez-Aguilar: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation.
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
CRediT authorship contribution statement
Saima Arshed: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Nauman Raza: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Asma Rashid Butt: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. Ahmad Javid: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation. J.F. Gómez-Aguilar: Conceptualization, Data curation, Formal analysis, Writing - original draft, Writing - review & editing, Validation.
Acknowledgments
José Francisco Gómez Aguilar acknowledges the support provided by CONACyT: cátedras CONACyT para jóvenes investigadores 2014 and SNI-CONACyT.
