Multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation on different backgrounds

In this work, we analyze the multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation (NLSE) in order to understand more thoroughly the origin and appearance of optical rogue waves in this system. Such structures are obtained on uniform backgrounds by using the Darboux transformation scheme for finding analytical solutions of the NLSE under various conditions. In particular, we solve the eigenvalue problem of the Lax pair of order n in which the first m evolution shifts are equal, nonzero, and eigenvalue dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of order n-2m\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n-2m$$\end{document} appears at the origin of the (x, t) plane and can be considered as the central rogue wave of the so-formed cluster. We show that the high-intensity narrow peak, with the characteristic intensity distribution in its vicinity, is enclosed by m ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a wavy background, we utilize the modified Darboux transformation scheme to build such solutions on a Jacobi elliptic dnoidal background. We analyze the vertical semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the (x, t) plane is broken when the shift value is increased above a threshold. We apply the same analysis on the Hirota equation, to examine the influence of a real parameter and Hirota’s operator on the cluster appearance. The same analysis can be applied to the infinite hierarchy of extended NLSEs. The main outcomes of this paper are the new multi-rogue wave solutions of the nonlinear Schrödinger equation and its extended family on uniform and elliptic backgrounds.

ious conditions. In particular, we solve the eigenvalue problem of the Lax pair of order n in which the first m evolution shifts are equal, nonzero, and eigenvalue dependent, while the imaginary parts of all eigenvalues tend to one. We show that an Akhmediev breather of order n − 2m appears at the origin of the (x, t) plane and can be considered as the central rogue wave of the so-formed cluster. We show that the high-intensity narrow peak, with the characteristic intensity distribution in its vicinity, is enclosed by m ellipses consisting of the first-order Akhmediev breathers. The number of maxima on each ellipse is determined by its index and the solution order. Since rogue waves in nature usually appear on a wavy background, we utilize the modified Darboux transformation scheme to build such solutions on a Jacobi elliptic dnoidal background. We analyze the vertical semi-axis of all ellipses in a cluster as a function of an absolute evolution shift. We show that the cluster radial symmetry in the (x, t) plane is broken when the shift value is increased above a threshold. We apply the same analysis on the Hirota equation, to examine the influence of a real parameter and Hirota's operator on the cluster appearance. The same analysis can be applied to the infinite hierarchy of extended NLSEs. The main outcomes of this paper are the new multi-rogue wave solutions of the nonlinear Schrödinger equation and its extended family on uniform and elliptic backgrounds.
Keywords Nonlinear Schrödinger equation · Rogue waves · Circular and triangular rogue wave clusters · Darboux transformation

Introduction
The cubic nonlinear Schrödinger equation (NLSE) [1][2][3] is one of the most studied partial differential equations in nonlinear sciences since it was first introduced in 1960s. The extensive research on NLSE solutions and their dynamical stability is still being conducted, due to its huge importance in various fields of physics, such as nonlinear optics [4][5][6][7][8], Bose−Einstein condensates [9,10], oceanography [11,12], and plasmas [13]. The NLSE is a general equation for the nonlinear wave propagation that can describe a variety of phenomena in nonlinear regimes of different physical systems. However, due to its simple form, it cannot be used for more accurate explanation of some higher-order effects in nonlinear optics. To this end, in recent times an extended family of nonlinear Schrödinger equations (ENLSEs) that may include an arbitrary number of higher-order dispersion terms with additional nonlinearities, has been proposed and investigated in [14,15]. The extension of NLSE to the hierarchy of higherorder equations originated from the need to explain the propagation of ultrashort pulses through optical fibers [16,17]. So far, attention was mostly focused on the Hirota [18,19] (with the third-order dispersion) and quintic equations (containing the dispersions up to the fifth-order) [20][21][22][23].
Although NLSE is a well-known equation, it remains a subject of broad interest for several reasons. First, both NLSE and its extended variants are completely integrable in one dimension. The possibility of deriving exact analytical NLSE solutions has motivated a number of experimental studies in many branches of physics where NLSE appears. Second, the same mathematical procedure used for deriving NLSE solutions can be applied to the ENLSE as well. Finally, the characteristics of NLSE solutions are similar to those of the more complicated equations emanating from the ENLSE. Therefore, one can analyze the simpler NLSE solutions and still predict the properties of the same solution class of the extended family.
The one-dimensional NLSE that will be mostly considered in this work has the form The transverse spatial variable is denoted by t, the retarded time in the moving frame by x, while the slowly varying envelope of the electric field corresponds to the wave function ψ ≡ ψ(x, t). This form of NLSE is appropriate for the propagation of light pulses in fibers. However, if pulses are very short, additional operators have to be introduced that add finer details to the basic NLSE solutions. Thus, in this work we will also deal with the Hirota equation, comprising of a new operator (with a third-order dispersion along taxis and additional nonlinearities) added to the basic NLSE, multiplied by a real parameter α: Both the NLSE and its extended variants exhibit similar classes of localized solutions, among which the most important seem to be Akhmediev breathers (ABs) [24,25] and different solitons [26]. An AB consists of a series of intensity maxima on a finite background that are localized in time and periodic in space. The term soliton in general describes a solitary wave packet that propagates in some direction in the (x, t) plane on a zero background, without any distortion in its shape. The technique that is often used to derive exact analytical solutions is the Darboux transformation (DT) [27]. It utilizes the Lax pair formalism and recursive relations to calculate higher-order solutions of the NLSE, starting from the trivial zeroth-order seed function which satisfies Eq. (1.1).
The importance of DT for this work is its ability to provide higher-order Akhmediev breathers on uniform [28] and periodic backgrounds [29,30]. The breather emerges as a high-intensity narrow peak with a complex intensity distribution at its base. Such structures can be considered as rogue waves (RWs), which "appear from nowhere and disappear without a trace." The RW is localized both in space and time and is defined by one dominant peak. The simplest example of a RW is the Peregrine soliton [31]. The notion of rogue waves is now widely spread around, in studies of deep ocean waves [12,32], nonlinear optics [33,34], superfluidity [35], Bose-Einstein condensates [36], and others. The current hot topic in the nonlinear science is to investigate the cause and nature of optical rogue waves [37].
In this paper, we add new results to the field of RWs by investigating the special multi-elliptic rogue wave clusters of the NLSE. These solutions are also periodic along t-axis and throughout the paper we consider intensity distributions within a single transverse period. They consist of a rogue wave peak (ABs of the second order or higher), surrounded by the first-order ABs positioned on a number of concentric ellipses centered on the peak (see Fig. 1). We obtain these structures on uniform and Jacobi elliptic dnoidal backgrounds, by using the DT of order n, having the first m evolution shifts equal, nonzero, and eigenvalue dependent. We show that the order of the central rogue wave and the number of ellipses are determined by the two mode numbers, n and m.
Various multi-rogue wave solutions have been previously analyzed as triplets [50], triangular cascades [51][52][53], and circular clusters [28,[54][55][56]. The classification of various hierarchies of multi-RW structures into different families was presented in [57,58]. Although our results are similar to those in [56], where the authors used the determinant representation of DT, we believe that the first important contribution of our study is the simple method for generating elliptical clusters by using the evolution shifts in the Darboux transformation scheme. The second novelty in our results is the analysis of semi-axes of ellipses as functions of the evolution shifts and an estimate when the radial symmetry of the cluster will break up. Our third contribution to the field is the generation of elliptic RW clusters on a dn background, which was not presented before, to the best of our knowledge. Finally, the significance of our work is also the determination of new solutions of the Hirota equation in the form of elliptic RW clusters. We also point to ways how to generalize this analysis to the infinite hierarchy of NLSEs.
We stress out that the solutions presented in this paper are in the form of two-dimensional (2D) or threedimensional (3D) color plots. The clusters on the uniform background are calculated by using the exact analytical DT procedure. However, the mathematical expressions of higher-order DT solutions are extremely lengthy and complicated, and would require many journal pages to be written down. We therefore omit the derivation of such expressions. In turn, we pick the convenient grid and calculate the numerical values at each point with infinite accuracy (that is, up to the machine precision limit).
To ensure the correctness of our calculations, our DT method has been extensively validated by: (1) comparing it to directly solved NLSE, verified by the conservation of energy [38], (2) showing that our DT algorithm exactly satisfied the Peak-height formula [30], and (3) always halving the grid-size when doing the DT iterations and confirming that the results are stable and unchanged.
The paper is organized as follows. In Sect. 2, we briefly discuss the main properties of higher-order Akhmediev breathers. In Sect. 3, we present various NLSE solutions in the form of multi-elliptic rogue wave clusters on a uniform background. In Sect. 4, we analyze the properties of such clusters, in particular the lengths of semi-axes of elliptical rings, by going up to the four ellipses in a cluster. In Sect. 5, we exhibit the NLSE cluster solutions on a Jacobi elliptic dnoidal background. In Sect. 6, we generalize our findings to the Hirota equation that includes the third-order dispersion and additional nonlinearities. In Sect. Conclusion, we summarize our results.

Higher-order Akhmediev breathers
Here, we briefly describe Akhmediev breathers of the NLSE and how to use DT scheme to generate higherorder RW solutions. The first-order AB is a singleperiodic function along the t-axis [24,25]: that rides on a finite background and is localized along the x-axis. The period L and the angular frequency ω of an AB of any order are determined by a single parameter a, with 0 < a < 0.5 [38]: AB turns into the Peregrine RW at a = 0.5 and becomes the Kuznetsov-Ma soliton when a > 0.5. An arbitrary AB can be derived using DT, starting from the seed solution ψ 0 = e i x . The n-th-order AB (its wave function ψ n (x, t)) turns out to be a nonlinear superposition of n first-order ABs, each characterized by the complex eigenvalue λ j = r j +iν j and the evolution x j , and spatial shifts t j ( j = 1, . . . , n). The existence of such abundance of relevant parameters offers an incredible variety of possible RW solutions. The Lax pair procedure and the recursive relations in the DT scheme that are used to calculate ψ n (x, t) from ψ 0 are described in details in [28]. Here, we briefly mention that computational complexity for calculating the n-th-order DT solution exhibits a quadratic growth (≈ O(n 2 )). Therefore, by increasing n, the number of iterations and the complexity of analytical expressions rise significantly. For this reason, we do not write down these expressions explicitly here, but only represent them graphically.
It is important to note that the imaginary part ν of an AB is simply related to the parameter a: ν = √ 2a. By taking into account relation (2.3), one can see that the imaginary part of AB's eigenvalue is completely determined by its angular frequency: ν = 1 − ω 2 /4.

Multi-elliptic rogue wave clusters on uniform background
To generate multi-elliptic RW clusters, we require that the frequencies of constituent single-order breathers are all different, but close to zero. This goal can be achieved by defining them as harmonics of ω 1 = ω → 0, so that ω j = jω, where j ≥ 2 [28]. In this work, we take the simplest possibility that all real parts are zero: r j = 0. The n ABs are thus formed by using their imaginary parts calculated from the corresponding frequencies: It is easy to see that all ν j tend to 1. Having set the eigenfrequencies, it remains to choose the evolution and spatial shifts. Different choices lead to very different solutions. We introduce a slight modification with respect to [28]: The first m evolution shifts x j are set to be equal, nonzero, and eigenvalue dependent. We assume them to be given via an expansion for j ≤ m, and x j = 0 for j > m. In addition, we simply set all t j shifts to be zero. We also assume that all X jl = 0 except one particular value that is explicitly stated in the text. Although seemingly an oversimplification, this choice of parameters nonetheless leads to an interesting family of new RW clusters. And, as mentioned, all this is provided for by an incredible richness in the choice of four sets of parameters. It turns out that such an n-th-order Darboux solution with m nonzero shifts x j is characterized by a single Akhmediev breather of order n − 2m placed at the origin (0, 0) (a central rogue wave, labeled as RW n−2m ) and m ellipses (rings) around the RW. The outer ellipse contains 2n − 1 ABs of order 1 (AB1), and each following ring toward the center has four AB1s less, as analyzed in [56,57]. We term this Darboux solution as the multi-RW cluster.
This remarkable intensity pattern can be described as follows. If all x j shifts are zero, the nonlinear superposition of all n DT components will arise at the origin of (x, t) plane, forming an Akhmediev breather of order n. If only one shift is applied, say x 1 = 0, the central AB of lower order and intensity will remain, but it will partially break up and a ring structure of 2n − 1 rational solutions centered at (0, 0) will be displayed [28]. The minimal n value for this picture is n = 3. As mentioned above, if one chooses m > 1 components shifts to be nonzero, they will split and decrease the intensity of the central structure even more and produce more rings. The exact splitting mechanism could be understood if one applies the mathematical analysis of exact analytical DT expression. However, deriving and analyzing such complex expressions for big n is a very tedious job that offers little insight and was not performed anywhere before; hence, it will not be performed here either.
In Fig. 1, we present the multi-rogue wave cluster on uniform background having 2 ellipses. Hence, we set m = 2 and vary the value of n in order to change the order of the central RW. The main frequency ω is set to 10 −1 so that the imaginary part ν is getting close to one, according to Eq. (3.1). Also, we choose X j4 = 10 6 , to set evolution shift x j in the order of 1 (Eq. 3.2). In Fig.  1a, we set n = 6 and obtain the second-order RW at the center. The outer and inner ellipses consist of c 1 = 11 and c 2 = 7 AB1s, respectively. In Fig. 1b, we set n = 7 to get a third-order RW. The number of AB1s on two  (x, t) plane. The shifts are calculated for X j4 = 10 6 . The orders of Darboux transformation and the Akhmediev breather representing the high-intensity central peak are: a n = 6 with the second-order rogue wave, b n = 7 with the third-order rogue wave, c n = 8 with the fourthorder rogue wave, and d n = 9 with the fifth-order rogue wave ellipses is c 1 = 13 and c 2 = 9. One can further increase n to get higher-order RWs that are rarely or never seen before. For n = 8, the RW4 is obtained with c 1 = 15 and c 2 = 11 (Fig. 1c). For n = 9, the RW5 is formed with c 1 = 17 and c 2 = 13 (Fig. 1d). It is seen that higher the order of the central RW, the narrower and stronger the RW peak at (0, 0). The highest intensities in Fig. 1a-d are, respectively: 22.98, 44.45, 77.26, and 105.81. We have also computed solutions with other frequencies, for instance ω = 0.05. The appearance of this RW cluster was very similar to the 10 −1 case (not shown), so we proceeded with the 0.1 value.
We last show the results for m = 4. The analysis is analogous to the previous two cases. When n = 10, we get RW2 and 4 rings surrounding the central peak. The number of AB1 on four ellipses, from outer to the inner, is c 1 = 19, c 2 = 15, c 3 = 11, and c 4 = 7, respectively (Fig. 3a). Next, we take n = 11 and obtain RW3 with c 1 = 21, c 2 = 17, c 3 = 13, and c 4 = 9 (Fig. 3b).
In general, under the conditions for DT computation presented in this section, our conjecture is that the RW of n − 2m order is obtained at (0, 0) with m ellipses around the peak for n ≥ 2m + 2. If we index the rings from 1 to m, going from the outer to the inner one, then the number of AB1 on each ring is c i = 2n − 4i + 3.

The semi-axes of ellipses in clusters
In paper [28], dealing with a single circular rogue wave cluster, the authors proposed a formula for the radius of the ring depending on Darboux shifts along the xand t-axes. Having ellipses at hand, we present how the length of the vertical semi-axis depends on an absolute  origin (0, 0) of the (x, t) plane. The orders of Darboux transformation and the Akhmediev breather representing the high-intensity central peak are: a n = 8 with the second-order rogue wave, b n = 9 with the third-order rogue wave, and c n = 10 with the fourth-order rogue wave Fig. 3 2D color plots of rogue wave clusters on the uniform background having four ellipses (m = 4) around n − 2m order rogue wave, formed at the origin (0, 0) of the (x, t) plane. Shifts are obtained for X j4 = 10 6 . The orders of Darboux transforma-tion and the Akhmediev breather representing the high-intensity central peak are: a n = 10 with the second-order rogue wave, and b n = 11 with the third-order rogue wave evolution shift for all ellipses, up to four rings (m = 4) in the cluster. In Fig. 4, we show the n = 10 and m = 4 case and indicate the AB1s on an i-th ellipse with numbers i = 1 to i = 4 (from the inner-most toward the outer-most ring). It turns out that all rings, for any m, have AB1s at t = 0 with alternating positions along this vertical line: the inner-most AB1 is positioned above the central RW. The next AB1 with index 2 is below the maximum at (0,0), the AB1 marked 3 is in the upper half of the (x, t) plane, and so on. We therefore define the length of the vertical semi-axis R xi as the distance between the central RW at the origin and AB1 indexed with i. Since the higher-order DT solution is expressed by a very complicated and cumber-some analytical expression, which is difficult to write and analyze, we applied the numerical calculation of AB1 positions along the t = 0 line.
In Fig. 5a, we show the R x dependence on x shift = x 1 = ... = x m for two rings surrounding RW2 at the center (n = 6, m = 2) at two main frequencies: ω = 0.1 and ω = 0.05. We see that the position of AB1 at the first ring is increasing as the evolution shift becomes larger, in contrast to AB1's x-coordinate on the outer ring, which first increases but then saturates and finally starts to decline slowly. Therefore, one can differ two regions in the (x, t) plane: the first one (I) is roughly estimated as the half-plane before the interception of R x1 and R x2 curves. In this region, the cluster has a Fig. 4 2D color plots of the quadruple-elliptic rogue wave cluster, obtained for n = 10 and m = 4, with numbers 1, 2, 3, and 4 indicating the single-order Akhmediev breathers located at t = 0 on each ellipse. The distance along x-axis between the RW center at (0, 0) and the maximum of the breather labeled with i corresponds to the vertical semi-axis R xi of the i-th ellipse regular "concentric ellipses" -like shape. For ω = 0.1, the region I is determined by x shift < 11.7. When ω = 0.05, the region I is given by x shift < 23. The example of a RW cluster in the second region (II) is shown in Fig. 5b. It is clearly seen that the two rings are deformed and thus no longer elliptical in the shape. In Fig. 5c and 5d, the R x1 and R x2 dependence is shown for the case of RW2 and RW3 at the center, respectively, only in the region I, where the radial symmetry is preserved.
In Fig. 6a, we plot the graph of R x as a function of x shift in the case of three rings around a RW2 cluster (n = 8 and m = 3). We see that the vertical semi-axis of the first and third ellipses is an increasing function of the evolution shift, in contrast to R x2 . In Fig. 6b, the RW2 with four rings is analyzed (n = 10 and m = 4). Graphs in both figures are computed in region I. Our conclusion is that the x position of AB1 (with t = 0) on odd-labeled ellipses (1, 3) grows constantly with the increasing shift, while R x of even indexed rings (2, 4) first increases, then saturates and finally slowly declines until the symmetry is broken.

Multi-elliptic rogue wave clusters on Jacobi elliptic dnoidal background
In this section, we demonstrate that multi-elliptic RW clusters can be obtained on a periodic background defined by Jacobi elliptic dnoidal function dn, by using the modified DT scheme for the NLSE [29]. The seed where g is the elliptic modulus and m dn = g 2 is the elliptic modulus squared. The choice of eigenvalues and shifts is the same as described in previous sections, but the procedure for calculating ψ(x, t) of order n is different. As explained in [29], the exact analytical values of wave function can be obtained only when t = 0. In order to compute ψ(x, t = 0) over the entire (x, t) grid, the numerical calculation is required. In this work, we use the fourth-order Runge-Kutta algorithm. We manage to obtain a two-ring cluster around RW2 (n = 6 and m = 2) on an elliptic background (m dn = 0.4 2 ) using this numerical procedure. The result is shown in Fig. 7a. We also present the 2-elliptic cluster around RW3 (n = 7, m = 2, m dn = 0.4 2 ) in Fig. 7b. By a careful look at both 3D plots, one can observe the low-amplitude background waves on which the AB1 structures and high-intensity AB2/AB3 peaks are generated.

Multi-elliptic rogue wave clusters for extended NLSE family
Finally, we generalize our results to Hirota equation, which is a first member of the extended family of nonlinear Schrödinger equations [14,15]. It is important to note that the DT technique retains the same recursive relations for the Lax pair and higher-order ψ functions as before. We therefore can generate solitons and The R x1 and R x2 as functions of x shift for n = 9 in the region of undeformed elliptic cluster breathers of any order using the sets of eigenvalues and transverse/evolution shifts as explained above. The intensity distribution of such solutions will differ from the simple cubic NLSE, due to free parameters in the extended families and a bunch of additional dispersion and nonlinear terms, but the procedure for their analytical buildup remains the same. In other words, one can utilize the same algorithm and take identical sets of shifts to compute multi-elliptic RW clusters for any equation from the extended NLSE hierarchy. The only difference in the DT scheme between NLSE and Hirota equation is in analytical expressions for the Lax pair functions r and s, but all recursive relations remain the same, as stated above. The Hirota DT scheme is presented in detail in [19]. Here, we generate the multi-elliptic RW cluster on a uniform background with ψ 0 = e i x as the seed. Our main goal is to investi-  are calculated for X j4 = 10 6 . The free parameter is: a α = −0.07 and b α = 0.07 gate the cluster appearance when Hirota operator (the term in Eq. 1.2 multiplied by α) sets in. For this purpose, we take α = ±0.07 and build two clusters having RW2 at the (x, t) center, surrounded by two rings (n = 6 and m = 2). When α is negative, the entire cluster is tilted toward the positive direction of t-axis. In addition, the radial symmetry of the central RW2 is broken, since the local maxima in the vicinity of RW2 are more pronounced in the tilted direction.
The intensity distribution for α = −0.07 is shown in Fig. 8a. If one changes the sign of α then the skew angle (between the vertical axis of the cluster and x-axis) will just change the sign. The results for α = 0.07 are shown in Fig. 8b. The measured skew angle for α = ±0.07 is θ ≈ ∓18.6 • . In addition, both ellipses are stretched for nonzero α, since the distances between AB1s on inner and outer rings (marked in Fig. 4 with 1 and 2) and the central RW2 are bigger than in the NLSE case. For the cubic NLSE (α = 0: Fig. 1a) the semi-axis lengths are R x1 = 10.1 and R x2 = 22. In the Hirota case, shown in Fig. 8, the semi-axis lengths are R 1 = 10.76 and R 2 = 23.7. Here we claim that larger the α, the bigger the skew angle and the amount of cluster stretching (results not shown).
The analysis can be further applied on even higherorder equations of the extended NLSE hierarchy using the same procedure-with similar results. The subject of ongoing research is the influence of higher-order dispersions, additional nonlinearities, and real parameters on the overall shape of multi-elliptic RW clusters for this highly nonlinear systems. The limitation of this study is the complication for producing the rogue wave clusters dynamically. Even using the single period box and appropriate initial conditions from DT, the modulation instability sets in during numerical integration, decreasing the AB1 intensities on the rings and introducing additional peaks. The challenge for obtaining RW clusters numerically for NLSE and ENLSE remains the topic for the next research. In addition, the verification and analysis of the conservation laws of such numerical solutions using different methods, such as structure-preserving method [59][60][61][62][63][64] could be considered.

Conclusion
In this paper, we have presented the multi-elliptic rogue wave clusters of the nonlinear Schrödinger equation on uniform and nonuniform (Jacobi elliptic dnoidal) backgrounds. We showed that the Darboux transformation scheme of an arbitrary order n (up to n = 11) with m equal and nonzero evolution shifts, can produce Akhmediev breathers of n − 2m order positioned at the origin of the (x, t) plane. We showed that this high-intensity narrow peak, a central rogue wave, is encircled by m concentric elliptical rings of the firstorder Akhmediev breathers. The number of AB1s on each ring depends on the ring index and the solution order.
In order to better understand the cluster geometry, we have numerically investigated the lengths of semiaxes of all rings around the central RW. We provided the graphs where distances from RW at (0, 0) to AB1 on the vertical t = 0 line at each ellipse were plotted as functions of the absolute evolution shift. Our results suggest that the radial symmetry is destroyed for large evolution shifts.
We next used the modified Darboux transformation scheme to numerically build RW clusters on a periodic background. Although the intensity of higherorder Akhmediev breather at the center significantly surpasses the amplitude of elliptic waves, we were able to observe the weak background oscillations on which the cluster is constructed.
We concluded our analysis by applying the Darboux transformation scheme on the Hirota equation. We exhibited that the Hirota operator introduces the titling and stretching of the entire cluster in a direction determined by the sign of the single real Hirota parameter.
We believe that further research of multi-rogue wave clusters on the cubic NLSEs is warranted in the future, owing to many degrees of freedom offered by the Darboux transformation scheme (the choice of eigenvalues and the spatial and temporal shifts). Research possibilities grow even more if one considers the cluster solutions for the infinite hierarchy of extended nonlinear Schrödinger equations. The analysis and results presented in this paper could be used to understand better the origin and generation of rogue waves in physical systems governed by cubic and extended nonlinear Schrödinger equations. Data availability All data generated or analyzed during this study are included in the published article.