Higher-order semi-rational solutions for the coupled complex modified Korteweg-de Vries equation

We explore the Darboux-dressing transformation of the coupled complex modified Korteweg-de Vries equation. Next, with the aid of an asymptotic expansion theory, we derive the concrete forms of three types of semi-rational solutions. In particular, the seed solution is related to the normalized distance and retarded time. Interestingly, we construct a kind of novel rogue wave called as curve rogue wave. More importantly, the kinetics of semi-rational solutions are discussed in detail. We hope that these results would shed more light on comprehending of the solutions occurring in multi-component coupled systems.


Introduction
Rogue waves with high amplitudes have become a fashionable research in various realms such as science, oceanic waves, optics and even finance [1][2][3][4]. Besides, rogue waves appear and disappear both from nowhere [5]. Taking into account the adhibition of different domains needs us to study the rogue wave of nonlinear evolution equations to understand its behaviors. It should be emphasized that breather waves are divided into two kinds which tend to become rogue waves [6][7][8][9].
From in physics viewpoint, the interaction of semi-rational solutions composed of the rogue wave and the soliton or breather in integrable systems catch the attention of academics. Since Ling et al. constructed the rational solutions in vector nonlinear Schrödinger (NLS) equations [10], the localized wave solutions in other nonlinear integrable equations, for instance, coupled Hirota equations [11], coupled Lakshmanan-Porsezian-Daniel equations [12], coupled derivative NLS equation [13] and the Hirota-Maccari system [14] have been presented. Notably, Darboux transformation (DT) is an essential method to obtain soliton and localized solutions of soliton equations [15][16][17][18][19][20]. Recently, a new method is presented to obtain the localized wave solutions of integrable equations, namely, Darboux-dressing transformation (DDT) [21][22][23].
The modified Korteweg-de Vries (mKdV) equation, as everyone knows, is a fundamental completely integrable equation which plays a significant role in various physical contexts [24][25][26]. Another integrable equation is the complex mKdV equation with a large number of applications [27,28]. It can be found that results in multi-component systems are more plentiful than ones in single systems.
In this work, one aims to study the coupled complex modified Korteweg-de Vries (ccmKdV) equation where u 1 = u 1 (x, t), u 2 = u 2 (x, t) are complex functions and the symbol * denotes the complex conjugation. It is noticed that algebro-geometric solutions and N -soliton solutions have been obtained [29,30].
Beyond that, Ye et al. gave the modulation instability and vector rogue waves [31].
This study is committed to investigating semi-rational solutions. In section 2, in the light of the Lax pair, we give the DDT method. In section 3, with the aid of the asymptotic expansion method, the higher-order semi-rational solutions are presented. Peculiarly, one gains a kind of the novel rogue wave with curve shape. Moreover, the kinetics of these solutions are discussed. The conclusion is situated on the final section.

Darboux-dressing transformation
The Lax pair of Eq. (1) is written by with where Ψ indicates the eigenfunction and λ is a spectral parameter. From the zero curvature equation Then, one gives the DDT of the ccmKdV equation (1) in the following where I=diag(1,1,1) and Φ 0 (x, t) = (φ 1 ,φ 2 ,φ 3 ) T . Then the new functions read ] be a solution of (2) with λ = λ 1 (1 + ) and an initial solution with are solutions of (2) with respect to the same spectral parameter λ 1 .
Then the solution Afterwards, we begin with the initial solution u j,[0] = a j e iθ , θ = kx + t, j = 1, 2, where a j , k and are real constants. After the simple calculation, we can get the dispersion relation

Higher-order semi-rational solutions
By adopting the resulting DDT, we painlessly achieve the higher-order semi-rational solutions. Firstly, substituting the following transformation into the Lax pair (2), we can get that with and ω 2 = a 2 1 + a 2 2 , where Φ is a 3 × 3 matrix function of x and t, U 0 and V 0 are both the 3 × 3 constant matrices. Thus, the solution for system (2) can be written as and After complicated calculations, F can be written by where Likewise, G can read as where , , , , , Substituting the seed solution (9) into the DDT (4) results in explicit solutions composed of trigonometric functions and exponential functions. In what follows, the asymptotic expansion theory will be employed to construct the ration solutions. When we set λ = k ± 2iω, the exponential e i(U0x+V0t) can be regarded as consisting of two parts: exponential and polynomial functions. Setting λ = (k + 2iω)(1 + ), one can give a theorem as below.
Theorem 2. Taking into account a Taylor series expansion in (4) and (15), we have the following where F n and G n represent nth coefficient matrix of . And Z reads here Z j are constant vectors. Consequently, one gains Additionally, to divide the rational solutions, Z allows the following expression Therefore, the first-order semi-rational solutions are generated where  and (1) The first-type first-order semi-rational solutions. Fixing a 1 = a 2 = µ 1 = µ 2 = µ 3 = 1, one can arrive at the rational solution, which is degenerated from the semi-rational solution. Besides, both components are the fundamental rogue waves. The corresponding figures are neglected here. If we choose 0 < |k| < 1, the rogue wave possesses the curve shape, as shown in Fig. 1. The amplitude of the curve rogue wave is still about 3. However, different from the fundamental rogue wave, the curve rogue wave has two maximum values.
(2) The second-type first-order semi-rational solutions. If one of the parameters a 1 and a 2 is equal to zero, one can get the second-type semi-rational solutions among the bright/dark/curve rogue wave, one dark/bright soliton in Figs. 2-6. When the value of |µ 2 | increases, the rogue waves are far away from the soliton in Figs. 2-3. Interestingly, in Fig. 3(b), the peak difference between the rogue wave and the plane is almost zero. In other words, the energy to generate the rogue wave is small. For analyzing clearly the explicit collision processes in the semi-rational solutions, the plane evolution plots of the interactional processes at different times are exhibited in Fig. 4. In Fig. 4, one can find that if t < 0, the solitons in u 1 and u 2 components are a dark and bright soliton respectively. Nevertheless, when t = 0, suddenly, out of nowhere, a rogue wave appears. Then, the rogue wave disappears, and the soliton continues moving ahead whose amplitudes and velocities do not change after the collision. We call this process elastic. On the other hand, under the condition µ 2 = µ 3 , the bright rogue waves transform into the dark rogue waves in Fig. 5. More importantly, if 0 < |k| < 1, the structure of the fundamental rogue wave changes, that is to say, the curve rogue wave appears in Fig. 6.
(3) The third-type first-order semi-rational solutions. Taking a 1 a 2 = 0, one can get the third-type semi-rational solutions where the bright/dark/curve rogue wave interplays with one breather. As shown in Figs. 7-10, the first-order rogue wave and one breather separate for both components. If a 1 = a 2 = k = 1, the structures of the breathers in the u 1 and u 2 components are similar, as seen in Fig. 7. But the breathers in Fig. 8(a), (b) are very different. In Fig. 8(a), the dark breather appears. It is found that the breathers' amplitudes on the plane are lower than the ones under the plane. Nevertheless, the breathers' amplitudes are inverse in Fig. 8(b). By the same token, if 0 < |k| < 1, the fundamental rogue wave changes into the curve rogue wave in Figs. 9-10. Moreover, µ 2 = µ 3 yields that the bright rogue wave changes into the dark rogue wave in Fig. 10(a) and the amplitude of rogue wave in u 2 component becomes from almost zero to three in Fig. 10(b).
(1) The first-type second-order semi-rational solutions. Under a 1 = a 2 = l 1 = l 3 = 1, the secondorder semi-rational solutions degenerate to the second-order rogue waves which possess two forms: the fundamental and triangular pattern. Similarly, we do not show the plots here.
(2) The second-type second-order semi-rational solutions. If a 1 = 0 and a 2 = 0, one can obtain the semi-rational solutions coexisting with the second-order rogue waves, two dark and bright solitons in Figs. 11-12. If R j = 0 and S j =0, the interactional solutions are the fundamental second-order rogue waves with two dark and bright solitons in Fig. 11. Similarly, rogue waves are not apparent on the bright solitons background in Fig. 11(b). When choosing R j = 0 and S 2 = 0, the pattern of the second-order rogue waves becomes triangular.
(3) The third-type second-order semi-rational solutions. If a 1 a 2 = 0, the second-order semi-rational solutions consist of the second-order rational solutions and breathers in Figs. 13-14. Choosing R j = 0, S j = 0, the fundamental second-order rogue waves and breathers coexist in both u 1 and u 2 components.
Supposing S 2 = 0, rogue waves will become the triangular.
In the same manner, more higher-order semi-rational solutions of Eq. (1) can be generated by the DDT method. Compared with the low-order cases, we can also classify higher-order cases as three types: (1)

Conclusion
We definitely exhibit a variety of semi-rational solutions of the ccmKdV equation resorting to a novel method, namely, the DDT method. Moreover, different from other researches, we consider an initial solution with the distance and time such that curve rouge waves can be obtained. Through the adjustments of the parameters, we achieve the three categories of semi-rational solutions. Continuing the DDT process, more complicated and interesting solutions can be obtained, which possess richer dynamics.
Based on the matrix exponential function, we construct the semi-rational solutions whose propagation (a) (b) Figure 12: The second-type second-order semi-rational solutions with the same parameters as Fig. 11 except that p 1 = 20000, p 3 = 10000, S 2 = 50.
property is extremely distinct from the others. We expect that our results are advantageous to understand more complicated physical phenomena.