Soliton Molecules, Rational Positon Solution and Rogue Waves for the Extended Complex Modied KdV Equation

We consider the integrable extended complex modified Korteweg–de Vries equation, which is generalized modified KdV equation. The first part of the article considers the construction of solutions via the Darboux transformation. We obtain some exact solutions, such as soliton molecules, positon solutions, rational positon solutions, and rogue waves solutions. The second part of the article analyzes the dynamics of rogue waves. By means of the numerical analysis, under the standard decomposition, we divide the rogue waves into three patterns: fundamental pattern, triangular pattern and ring pattern. For the fundamental pattern, we define the length and width of the rogue waves and discuss the effect of different parameters on rogue waves.


Introduction
It is well-known that the integrable partial differential equations(PDEs) are integral part of modern mathematical and theoretical physics with far-reaching implications from pure mathematics to the applied sciences. Integrable equations have many useful properties: such as Lax pairs, Hamilton structure, conservation laws, exact solutions, and so on. Many phenomena in science can be described by integrable equations, so for an integrable equation, it is important to find their exact solutions. At present, there are many methods to find the exact solution of the equation, such as Lie group [4], the Darboux transformation (DT) [10,24,28], the Hirota bilinear method [11], Bäcklund transformation [23], algebraic geometry method [3], and the famous inverse scattering transformation [1,2], etc.
The Darboux transformation is a powerful method to construct solutions for integrable PDEs. In particular, the well-known soliton solution appearing in many physical motivated PDEs like the NLS equation, complex modified KdV equation can be computed thereby. In 2012, Ling, Guo et al obtain that the so-called rogue waves solutions of the nonlinear Schrödinger equation by the generalized Darboux transformation [7]. The concept of rogue waves originated from oceanography [15]. Oceanic rogue waves are surface gravity waves which height is much larger than expected for the sea state. The common operational definition requires them to be at least twice as large as the significant wave height. At first, people did not understand the mechanism of this phenomenon, but with the development of science and technology, some ocean probes observed this phenomenon. For example, the shape of large surface waves on the open sea and the Draupner new year wave [29]. Now rogue waves have been proposed in many fields, such as Nonlinear optics [27], Finance [32], Bose-Einstein condensates [12], plasmas [25], etc. For the more research on rogue waves, please refer to monograph [9] and its references.
Recently, so-called soliton molecules were obtained in optical experiments and attracted people's attention. Whereafter, the scientists have discovered soliton molecules in Bose-Einstein condensates [21], Few-cycle mode-locked laser [13], etc. In 2020, Lou presented a velocity resonance mechanism and theoretically obtained soliton molecules of integrable systems and asymmetric solitons three-dimensional fluid system [18].
The classical complex modified Korteweg-de Vries (cmKdV) equations can be written as where q = q(t, x) is a complex function. The cmKdV equation has many applications in science. For example, the cmKdV equation has been proposed as a model for nonlinear evolution of plasma waves [16], it has been derived to describe the propagation of transverse waves in a molecular chain model [8] and in a generalized elastic solid [5,6]. In [14], He et al constructed a generalized Darboux transformation for the cmKdV equation which obtain the rogue waves solution and analyze the dynamic of rogue waves. The soliton molecules for the cmKdV equation considered in [33].
In this paper, we investigate an extended complex modified Korteweg-de Vries (ecmKdV) equation, which takes the form where α ≪ 1 and β ≪ 1 stand for the third-order and fifth-order dispersion coefficients matching with the relevant nonlinear terms, respectively. If we take the β = 0, the Eq.(2) reduce to cmKdV equation (1). If we use −β instead of β and take the q(x, t) is real function in ecmKdV equation (2), the ecmKdV equation can be reduced to the following equation Wazwaz and Xu [30] have considered the Painlevé test and multi-soliton solutions via the simplified Hirota'direct method for Eq.(3). The conservation laws, Darboux transformation and periodic solutions obtained in [31]. The soliton molecules of the Eq.(3) are obtained in [26]. The long time asymptotic for the equation (3) with initial data or initial-boundary values are considered in [19,20]. In [17], the authors have obtained the explicit solitons and breather solutions for the equation (3) by the Riemann-Hilbert method. Our aim is to construct the exact solutions for Eq.(2) through the Darboux transformation technique in this paper. Our manuscript is organized as follows: In Section 2, we introduce the Lax pair and the Darboux transformation for Eq.(2). In Section 3, we obtain soliton molecules, positon solutions of Eq.(2) from seed solution q = 0. In Section 4, we obtain rational positon solutions of Eq.(2) from nonzero seed solution q = c. In particular, we construct the higher order rogue waves solutions from a periodic seed with constant amplitude and analyze their structures in detail by choosing suitable system parameters in Section 5. We give the conclusions in Section 6.

Lax pair
Introducing r = q * , Eq.(2) can be rewrite as follows, x r x − 20rq x q xx − 10q(r x q xx + q x r xx + rq xxx ) − q xxxxx ] = 0. According to the AKNS method [1], we obtain the Lax pair corresponding to Eq.(2), where Taking r = q * , if we consider the compatibility condition M t − W x + [M, W ] = 0, one can yield ecmKdV equation (2).

Zero seed solution
Taking the seed solution q = r * = 0, the spectral problem (4) reduces to By the simple calculation, the eigenfunctions corresponding to eigenvalue λ 2j−1 are given by the following, where ζ is a real constant.

Soliton molecules
In this subsection, we will consider soliton molecules of (2). Let n = 1, ζ ̸ = 0, and λ 1 = with It is easy to see that the interactions between l solitons and m molecules consisting of two same solitons when Eq.(10) satisfies the following resonance conditions from the expression of H, The collision process of two soliton molecules consisting of two same solitons are shown in Figure 1 (b). To find a molecule consisting of n solitons, the parameters in Eq.(10) are selected as follows: In addition, if we want to make the distance between two adjacent solitons in the molecule equal, then the constraint condition needs to add one more bases on Eq. (14): where v 0 and d 0 are real constants. Figure 1 (c) and (d) show the molecule consisting of 3 solitons under condition (14) and (15). If we take α = 1, β = 0, then the soliton molecules (12) reduced to the case of complex modified KdV equation as shown in [33].

Proposition 3.1
The n−fold DT in the degenerate limit λ 2j−1 → λ 1 generates (n − 1)-positon solution of the ecmKdV equation, which is given by Here denotes the floor function of x which is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x.
If we take n = 2 in Proposition 3.1, one can obtain the explicit expression of 1-positon solution,

Rational positon solutions
In this section, we consider non-zero seed solution, where a, b, c are real constants. By using the principle of superposition of the linear differential equations, the new eigenfunctions corresponding to λ j can be rewritten as where . where z 1 , z 2 are arbitrary complex constants. Proposition 4.1 Suppose that the eigenfunction obtained by the n-fold Darboux transformation degenerates at the eigenvalue λ 0 , when λ 2j−1 → λ 0 , the degenerate n-fold Darboux transformation produces new solution where Here denotes the floor function. When we set z 1 = cd 1 , z 2 = 0, the equation (17) can be rewritten as, where d 1 = e ic 1 (s 0 +s 1 ϵ+s 2 ϵ 2 +···+s n−1 ϵ n−1 ) , d 2 = e −ic 1 (s 0 +s 1 ϵ+s 2 ϵ 2 +···+s n−1 ϵ n−1 ) s i ∈ C(i = 0, 1, 2, ..., n − 1), a, b, c ∈ R. It is easy to verify that the eigenfunctions degenerate at λ 0 = c, i.e., ϕ j (λ 0 ) = 0 in the case of a = 0. Combining the Proposition 4.1, we find that the rational positon solutions when we set s i = 0, i = 0, 1, .., n − 1, The expressions of L 1 , L 2 and L 3 are given in the Appendix A. Figure 3 shows the second-order rational positon solution |q 2−r | and its density plot.
It can easily be verified that the eigenfunctions are defined in (19) degenerate at λ 0 = − i 2 a + c. Combining Proposition 4.1, we obtain the expression of the n-order RWs q [n] . Due to the length and complexity of higher order RWs, we only give the explicit expression of the first-order RWs, where A 1 is given in Appendix B.
Through simple calculations, we find that |q [1] | 2 = c 2 when x → ∞, t → ∞ and |q [1] | 2 ≤ 9c 2 . It is not difficult to find that the selection of parameters d 1 , d 2 or equivalent to s i , i = 0, 1, ..., n−1 will produce different types of RW. We assume α = β = 0.5 for the convenience of discussion. Next, Let's discuss the first-order to fifth-order RW because of the complexity of higher-order RW. Setting s i = 0, i = 0, 1, ..., n − 1, from the first-order RW |q [1] | 2 to fifth-order RW |q [5] | 2 in Figure 4, it's clearly see that the n-order RW |q [n] | 2 takes the maximum value at (x, t) = (0, 0), and there are n peaks on each side of t = 0(n > 1). We call it the fundamental pattern of RW. Next, We analyze the contour line of the |q [1]  It's a hyperbola which has two asymptotes, major axist = 0, imaginary axis: 2. At height c 2 + 1, A 2 = 0 which has two end points 3. At height c 2 2 , A 3 = 0, two centers of valleys P 3 = (0, Figure 5 (a) gives the densityplots of |q [1] | 2 . Figure 5 (b),(c) and (d) show the contour of first-order RW |q [1] | 2 with h = c, c 2 + 1, c 2 2 . Similar to the idea of length and width defined in [14], the distance between P 1 and P 2 is length of first-order RW, 25 a 8 − 570 a 6 + 3489 a 4 − 2736 a 2 + 580, the projection of line segment P 3 P 4 in width direction is width of the first-order RW, Figure 6 (b) shows the length and width of the first-order RW when c = 1. Combining Eq. (20) and Eq. (21), we can find that, 1. when 0 < a < 0.66, the length keeps decreasing and the width keeps increasing; 2. when 0.66 < a < 1.64, the length keeps increasing and the width keeps decreasing; 3. When 0.66 < a < 1.64, the length keeps decreasing and the width keeps increasing; 4. When a > 1.64, the length keeps increasing and the width keeps decreasing; 5. When a < 0, the length and width plots are symmetrical to the a > 0 plots. This is the first effect of a. In addition, we find that the RW rotates counterclockwise with the increase of a from the slope of the l 3 . This is the second effect of a. Continue the previous discussion and calculation ideas, when c ̸ = 1, the length and width of first-order RW, Taking s 1 ≫ 1, n ≥ 2, s i = 0, i = 0, 2, 3..., n − 1, the RW will appear similar to the triangle structure, we call it the triangular patterns. Figure 7 shows triangular patterns of RW from second-order to fifth-order. Setting s n−1 ≫ 1, n ≥ 3, s i = 0, i = 0, 1, 2, 3..., n − 2, the RW will have a ring-like structure, we call it the ring pattern. Figure 8 shows ring patterns of RW from third-order to fifth-order. We can also get some combinations of triangular patterns and ring patterns. For example, from Figure 9(a)(b) we can see that a ring contains a triangle pattern. In Figure 9(c), a ring contains a ring pattern. The RW formed by the combination of the above three patterns can be called the standard decomposition of RW. Due to the diversity of the value of the parameter s i , i = 0, 1, ..., n − 1, we can of course obtain more than the above three patterns. Figure 10 shows non-standard decomposition of RW.

Remark 5.1
In the process of obtaining the fundamental patterns, triangular patterns and ring patterns above, we all choose s 0 = 0. If we set s 0 ̸ = 0, We can also get similar above three patterns. Taking the third-order RW |q [3] | 2 as an example, we can get the above three patterns with s 0 = 10 in Figure 11.

Conclusions
In this paper, we have presented the soliton molecules, positon solutions, rational positon solutions and rogue waves solutions for the extended complex modified KdV equation (2), and the plots of solutions are shown in the figures which are throughout the paper. If we consider the special case of the α = 1, β = 0 or use −β instead of β and take the q(x, t) is real function, our results can be reduce to the case of complex modified KdV equation which consider in [14,31,33]. Figure 1 Please see the Manuscript PDF le for the complete gure caption   Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption    Please see the Manuscript PDF le for the complete gure caption Figure 5 Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption Please see the Manuscript PDF le for the complete gure caption  Please see the Manuscript PDF le for the complete gure caption Figure 10 Please see the Manuscript PDF le for the complete gure caption Figure 11 Please see the Manuscript PDF le for the complete gure caption