Generalized Darboux transformation and higher-order rogue wave solutions of the coupled Hirota equations
This paper is dedicated to study higher-order rogue wave solutions of the coupled Hirota equations with high-order nonlinear effects like the third dispersion, self-steepening and stimulated Raman scattering terms. By using the generalized Darboux transformation, a unified representation of th-order rogue wave solution with 3+1 free parameters is obtained. In particular, the first-order rogue wave solution containing polynomials of fourth order, and the second-order rogue wave solution consisting of polynomials of eighth order are explicitly presented. Through the numerical plots, we show that four or six fundamental rogue waves can coexist in the second-order rogue waves. By adjusting the values of some free parameters, different kinds of spatial-temporal distribution structures such as circular, quadrilateral, triangular, line and fundamental patterns are exhibited. Moreover, we see that nine or twelve fundamental rogue waves can synchronously emerge in the third-order rogue waves. The more intricate spatial-temporal distribution shapes are shown via adequate choices of the free parameters. Several wave characteristics such as the amplitudes and the coordinate positions of the highest peaks in the rogue waves are discussed.
keywords:Rogue wave; generalized Darboux transformation; coupled Hirota equations
Rogue waves (also known as freak waves and other similar names), initially termed as mysterious and exceptionally disastrous oceanic surface waves, have been the powerful and computational topic for more than a decade (1); (2); (3); (4); (5); (6). One of the key characteristic of the rogue waves is that their height or steepness can usually attain two or three times greater than the average wave state, and they appear from nowhere and disappear without a trace (7). In addition to the deep ocean (1); (4), rogue waves have also moved to other branches of physics, such as optics and lasers (3); (8), capillary waves (9), plasma physics (10), Bose-Einstein condensates (11), hydrodynamics (12), atmosphere (13), and so forth (14).
The simplest mathematical description of a rogue wave is the so-called Peregrine soliton, which is an analytical rational solution of the nonlinear Schrödinger (NLS) equation (15). In contrast to the Akhmediev breathers (ABs) and Ma solitons (16); (17), Peregrine soliton is localized in both space and time. It has been experimentally observed in the water-wave tank (12), nonlinear optics fiber (18), and even in noiselike-pulse laser emission (19). Later, in the past years, a sequence of nonlinear models have been investigated to possess lower or higher-order rogue wave solutions, the NLS equation (20); (21); (22); (23); (24); (25); (26); (28); (27); (29), derivative NLS equation (30); (31); (32), Hirota equation (33); (34), Sasa-Satsuma equation (35), Lakshmanan-Porsezian-Daniel equation (36), Gross-Pitaevskii equation (37); (38); (39), Davey-Stewartson equation (40) and so on (41); (42). However, a complete understanding of the formation mechanism for the intricate rogue wave phenomena is unclear, because of the difficult and hazardous observational conditions (7).
Recently, there has been noticeably surge of interest in studying rogue waves in coupled systems, since many complex systems must comprise several waves with different modes or frequencies instead of a single one, when considering the significant interaction processes (37); (43); (44); (45); (46); (47); (48); (49); (50); (51); (52); (53). In fact, rogue waves in coupled systems can present the more abundant dynamical structures than in the scalar ones, such as the dark rogue waves (37), the interactions between rogue waves and dark-bright solitons or breathers (43); (47), the non-symmetric doubly localized rogue waves (51), the four-petaled rogue waves (52) and the four-peaky shaped rogue waves (53). Notably, very recently, the composite rogue waves in coupled systems which can be well described by the rational fourth-order or even higher-order polynomials solutions in mathematics have attracted widespread attention (44).
In this paper, we consider the coupled Hirota (CH) equations in dimensional form, that is
where and are wave envelops, the asterisk represents complex conjugation, is the transverse variable and is the propagation distance. The real parameter represents the perturbation effects involving the third dispersion, self-steepening and stimulated Raman scattering for the Manakov system. This system was originally formulated by Tasgal and Potasek to describe the electromagnetic pulse propagation in coupled optical waveguides, and it is more accurate than the Manakov system to describe the interaction process of two surface waves in the deep ocean, and to describe the propagation of femtosecond optical pulses in the birefringent or two-mode nonlinear fibers (54); (55). Mathematically, Eqs. (1) and (2) are completely integrable in the sense of their Lax pair (54), inverse scattering transformation (56), Painlevé analysis (57), Darboux transformation (58), Hirota bilinear form and -soliton solutions (59). Recently, Chen presented the fundamental rogue wave, dark rogue wave and composite rogue wave solutions (55); (60), and we investigate the interactional localized wave and rogue-wave pair solutions of Eqs. (1) and (2) (61); (62). However, to the best of our knowledge, there are no reports on higher-order rogue wave solutions of the CH equations.
It is very known that higher-order rogue waves can be viewed as the nonlinear superposition or combination of a fixed number of fundamental rogue waves. The experimental observations in a water tank and theoretical classification of higher-order rogue waves for the scalar systems have been increasingly presented (23); (28); (34); (63). Nonetheless, as is known to us, higher-order rogue waves in coupled systems have not been widely investigated. Therefore, generalization to even higher-order rogue wave solutions of coupled systems is important and nontrivial. The objective of this paper is to explore higher-order rogue wave solutions of Eqs. (1) and (2) through the generalized Darboux transformation (DT) (26); (27); (28); (30); (53); (61); (64); (65). By means of the generalized DT, a unified formula of th-order rogue wave solution with 3+1 free parameters is derived by the direct iterative rule. Apart from the first-order rogue wave solution containing polynomials of second or fourth order , we present that the second-order rogue wave solution can be composed by the polynomials of eighth or twelfth order. Through the numerical plots, we show that four fundamental rogue waves with quadrilateral, triangular, line and fundamental patterns, as well as six fundamental rogue waves with circular, two different types of quadrilateral, triangular and fundamental patterns can coexist in the second-order rogue waves. Moreover, the third-order rogue wave solution consists of polynomials of 18th or 24th order. We exhibit that nine fundamental rogue waves with two different types of circular, two different types of quadrilateral, triangular and fundamental patterns can emerge in the third-order rogue waves. Further, twelve fundamental rogue waves with two different types of circular, two different types of pentagram, two different types of quadrilateral, triangular and fundamental patterns are explicitly shown in the third-order rogue waves. In addition, by the numerical computation, some wave characteristics such as the amplitudes and the coordinate positions of the highest peaks in the rogue waves are discussed. Also, the perturbation influences produced by the small real parameter on the higher-order rogue waves are demonstrated.
The arrangement of our paper is as follows. In section 2, we present the generalized DT and a unified th-order rogue wave solution of the CH equations. In section 3, some explicit rogue wave solutions and numerical plots are shown. The conclusion is given in the final section.
2 Generalized Darboux transformation
In this section, according to the Ablowitz-Kaup-Newell-Segur (AKNS) approach, we begin with the following linear matrix Lax pair
Here, is the vector eigenfunction and is the spectral parameter. One can directly obtain Eqs. (1) and (2) from the compatibility condition of the above linear problem.
Let be a basic solution of the Lax pair equations (3) and (4) at , and . Thus, by resorting to the standard Darboux dressing procedure of the AKNS spectral problem (55); (66); (67), the Darboux transformation of the linear system (3) and (4) reads
Here is the identity matrix, the dagger indicates complex conjugate transpose.
Hence, based on the above facts, the generalized DT can be constructed as follows. Let be a basic solution of the Lax pair equations (3) and (4) at , and . We suppose that can be expanded as Taylor series at ,
(1) The first-step generalized DT
(2) The second-step generalized DT
As the next step, we take the following limit
then the second-order generalized DT turns out that
(3) The third-step generalized DT
Similarly, considering the following limit
the third-order generalized DT yields
(4) The th-step generalized DT
Taking all the above into account, and proceeding in this way one by one, we have the general case, that is
Remark 1. It is notable to point out that the expressions (22)-(23) give rise to a unified th-order rogue wave solution of Eqs. (1) and (2). In the next section, we will present some concrete rogue wave solutions consisting of higher-order polynomials to illustrate how to employ these formulas, and exhibit a series of figures to interpret the various dynamical properties of the solutions.
3 Rogue wave solutions
As is known to all, rogue wave solutions are the limiting case of either ABs or Ma solitons which can be generated from the plane waves. Thus, in this section, we start with a plane-wave solution of Eqs. (1) and (2)
After that, in order to seek out an adequate basic solution of the Lax pair equations (3) and (4), we set , here is a small parameter. Then under this determined spectral parameter and the seed solution (24), the basic solution matrix of the Lax pair equations can be calculated as
and satisfies a cubic equation
Hereafter, let us define a basic solution of the Lax pair equations in the form of
Here, , and () are free parameters. At this time, we prove that can be expanded as the Taylor series at ,
where . Here, we give the explicit expressions of the first two terms coefficients, see appendix A.
It is straightforward to check that is a nontrivial solution of the Lax pair equations (3) and (4) at , and . So, by taking advantage of the formulas (13) and (14), the first-order rogue wave solution can be calculated with four free parameters in it, namely, , , and . We now analyze the first-order rogue wave solution into two cases based on the parameter chosen by zero or not.
Case 1. . In this case, we get the simple Peregrine soliton containing polynomials of second order. By putting , we arrive at
Case 2. . At this point, it is found that the solution is made up of polynomials of fourth order. By taking the free parameters such that or , we can work out the solution which features a composite of two well-separated fundamental rogue waves. For instance, let , then it leads to
Also, if we choose , then the solution which is characterized by a composite of two fundamental rogue waves which are closely intermingled with each other can be achieved and here we refrain from presenting the explicit expression of it.
Remark 2. Noteworthy, it should be pointed out that the validity of the above solutions (29) and (30) can be verified by putting them back into Eqs. (1) and (2), and when we take the limit , they are reduced to the solutions of the Manakov system. Here, we omit exhibiting the numerical plots of the aforementioned solutions which have been investigated by Chen and us (55); (60); (61); (62). In this paper, we focus on the higher-order rogue wave solutions of Eqs. (1) and (2), of which the dynamic distribution structures will be much richer than those for the scalar ones.
At this time, performing the following limit
and in terms of the formulas (16) and (17), one obtains the second-order rogue wave solution with seven free parameters in it, i.e. , , , , , and . Likewise, we classify the dynamic properties of the solution into two cases on basis of the the parameter chosen by zero or not.
Case 1. . In this circumstance, the solution contains polynomials of eighth order and we will show that four fundamental rogue waves can coexist in the second-order rogue waves. Here, we consider four cases of the composite structures involving four fundamental rogue waves.
When we choose , such as , then the second-order rogue wave solution takes the form given below
where and are three polynomials of eighth order in and , as well as and are two polynomials of seventh order in and , see appendix B. By aid of the symbolic computation tool, it is straightforward to check that (31) agrees with Eqs. (1) and (2), and through the numerical plots, we observe that there are four fundamental rogue waves arranging with a rhombus in the second-order rogue waves, see Figs. 1(a) and 1(b). The amplitudes of the four highest peaks in component are 2.0806, 1.8712, 2.2361 and 1.8793, and occur at (6.5036,-0.0015), (-1.6128,9.1116), (-3.0121,-8.9821) and (-10.0913,0.2432), respectively. For component, the amplitudes of the four highest peaks are 2.0806, 2.2316, 1.8661 and 1.8793, and arrive at (6.5040,0.0018), (-1.5025,8.9174), (-3.1587,-9.1815) and (-10.1295,-0.2433), respectively.
In the following, we refrain from writing down the complicated expressions of the other second-order rogue wave solutions by choosing different values of the free parameters and just show the interesting dynamic structures, although it is not difficult to check the validity of them with the aid of the symbolic computation tool. Now we set , then it is seen that in Figs. 2(a) and 2(b), two fundamental rogue waves together with a composite rogue wave which is formed by the interaction of two fundamental ones emerge with a triangular pattern on the spatial-temporal distribution; when , we see that three rogue waves including a composite one arrange with a line pattern in Figs. 3(a) and 3(b); moreover, let and the rest of the values be assumed to be zero, it is displayed that four fundamental rogue waves intermingle with each other, see Figs. 4(a) and 4(b). Particularly, it is calculated that the maximum amplitude of the peaks is achieved at the composite rogue wave which is formed by the interaction of a fixed number of fundamental ones, see table 1. We find that when the composite number is four, the maximum amplitudes of the peaks in component and component reach to 3.2697 and 3.2530, respectively, which are greater than those of the situations with number two.
Remark 3. It is worth mentioning that when by taking the limit , the second-order rogue wave solution can also be reduced to that of the Manakov system. And because of the existence of high-order nonlinear effects, the coordinate positions of some humps in the rogue waves can be changed to a certain degree, particularly in the dimension. For instance, by increasing the value of , the effects can be observed more evidently, see Figs. 5(a) and 5(b). The coordinate positions of the humps in component are (6.5029,-0.0003), (5.0775,8.8302), (-10.1055,-9.3085) and (-10.0605,-0.2430), in component are (6.5071,0.0031), (5.0440,8.6565), (-10.4401,-9.5337) and (-10.3018,-0.2437). Moreover, we would also like to note that when letting , no new composite structures can be obtained but the fundamental pattern and here we omit presenting it.