Dynamics of degenerate and nondegenerate solitons in the two-component nonlinear Schrödinger equations coupled to Boussinesq equation

The paper studies the dynamics of degenerate and nondegenerate bright solitons and their collisions in the two-component nonlinear Schrödinger equations coupled to Boussinesq equation. The degenerate solitons that have only single-hump profiles, exhibit both elastic and inelastic collisions, and their velocities are identical in the two short wave (SW) components and in the long wave (LW) component. The nondegenerate single solitons have two different forms: one of them has double- or single-hump profiles and identical velocities in all SW and LW components, and the other one only admits single-hump profiles and has unequal velocities in the two SW components. The collisions of nondegenerate solitons cannot result in the redistribution of soliton intensities. Three different types of collisions of nondegenerate two-soliton solutions are studied in detail.


Introduction
The nonlinear Schrödinger equation (NLSE) and its generalizations play a crucial role in the development of integrable systems [1][2][3][4]. The vector coupled NLSE and NLSE coupled with other integrable equations are important generalizations of the NLSE in mathematics and physics, such as the M-coupled NLSEs (M-CNLSEs) and the NLSE coupled the Maxwell-Bloch equations, and the NLSE coupled to Boussinesq equation (NLS-Boussinesq). This is due to the fact that the generalizations of the NLSE have a lot of applications in many physical settings, spanning from nonlinear optics [5] to water waves [6,7], plasma physics [8], biophysics [9], Bose-Einstein condensates [10], metamaterial technologies [11], and other research areas [12][13][14][15]. The vector generalizations of the NLSE or their variants are integrable when the corresponding nonlinear coefficients satisfy particular restrictions [16,17]. A typical feature of these integrable generalized M-CNLSEs and NLS/CNLS-type equations is that they admit multiple soliton solutions [4].
Due to the important applications in physics, the multi-hump soliton solutions were recently studied in the case of the CNLSEs and other vector systems by the Hirota's direct method [42,43,45,61,62]. Since the corresponding solutions with identical wave numbers are single-hump solitons, thus in order to construct these multi-hump solitons the authors of Refs. [42,43] improved the Hirota's direct method by choosing nonidentical wave numbers of solitons solutions. That is an essential improvement in the Hirota's direct method for deriving multi-hump soliton solutions to CNLSEs. Furthermore, by altering the nonidentical wave numbers to identical ones, the multi-hump solitons degenerate to the single-hump solitons. To distinguish the multi-hump soliton solutions and single-hump soliton solutions, the authors in Refs. [42,43] referred to the former ones as nondegenerate solitons and to the later ones as degenerate solitons. Very recently, such nondegenerate solitons with profiles of multiple humps were also investigated by the Darboux transformation [44]. From the results reported in the literature, we can figure out that the non-degenerate solitons with multiple humps are interesting from both mathematical and physical points of view, as well as the degenerate solitons with profiles of single humps. Hence, both degenerate and non-degenerate solitons are worth studying in physically relevant integrable nonlinear systems.
The NLS-Boussinesq equation was first proposed by Rao [63,64] The complex short wave (SW) component u and the long wave (LW) component v belong to the NLSE and the Boussinesq equation, respectively. Since Eq. (1) is a coupled system comprising of the NLSE and Boussinesq equation and these two equations have wide applications in diverse areas of physics, thus Eq. (1) also has many applications in nonlinear science, such as in a homogeneous magnetized plasma [63][64][65][66]. Hence it is natural that researchers focused on Eq. (1), including the study of its integrability and of the families of soliton solutions [67][68][69][70][71][72][73][74][75][76][77]. It is known that the Boussinesq equation and other nonlinear partial differential equations have important applications in fluid motion.
In the past few years, many relevant results in the area of fluid motion, including the study of nanofluid flows were reported [78,79].
The nondegenerate solitons with multi-hump profiles have not been investigated for Eq. (2) before, to the best of our knowledge. Furthermore, although some particular degenerate solitons possessing single-hump profiles have been studied in Ref. [80], but the general N th-order degenerate soliton solutions in determinant forms were not reported yet, to the best of our knowledge. This paper focuses on the following three objectives for Eq. (2): -The obtaining of general bright soliton solutions, which can be classified into degenerate solitons, nondegenerate solitons, and partial nondegenerated solitons under particular parameter restrictions related to the so-called wave numbers; see Refs. [42,43]. -The study of dynamics of degenerate solitons and their interaction scenarios. -The study of dynamics of nondegenerate solitons and the corresponding interaction scenarios.
This paper is organized as follows. In Sect. 2, the bright soliton solutions in determinant forms are given via the bilinear KP-reduction method. In Sect. 3 and Sect. 4 we systematically study the dynamics of degenerate solitons and nondegenerate solitons, respectively. The conclusions of this work are presented in Sect. 5.

The bright soliton solutions to the 2NLS-Boussinesq equation in forms of determinants
In this Section, we give the bright soliton solutions to the 2NLS-Boussinesq equation (2) that are expressed by determinants. To achieve this purpose, we convert Eq. (2) into the following bilinear equations: through the dependent variable transformations where for 1 ≤ s, j ≤ N , the superscript T represents the transpose, and p s , α ( ) s are free complex parameters, and N is a positive integer.
Proof To construct the above solutions, we start with the following set of bilinear equations in the multicomponent KP hierarchy and their tau functions [83][84][85][86][87]: with In these tau functions, the elements of matrix M are with Then we take the restrictions on the parameters q s , q j , r s , r j satisfying the following relations: Upon this parametric restriction, the tau function τ 0 meets the following constraint: Through this constraint, from the last two bilinear equations in Eq. (7) we get the following bilinear equation: This bilinear equation and the first bilinear equation in Eq. (7) reduce to the bilinear equations (3) of Eq. (2) by taking the following transformations: for = 1, 2, and Furthermore, to realize the complex conjugation constraint condition: we further take the parametric restrictions: Finally, since y 1 and y 2 are dummy variables, thus we take y 1 = y 2 = 0 for convenience. Hence, the above tau functions become to the solutions in Theorem 1 for γ s = −(4 p 3 s + p s ). Theorem 1 is then proved.

Dynamics of degenerate solitons and their collisions
This Section considers the degenerate solitons and their collisions in the 2NLS-Boussinesq equation (2), which are depicted by the solutions (4) with parametric constraint α ( ) s = 0 for s = 1, 2, · · · , N .

The dynamics of degenerate single soliton solutions
Setting N = 1 and α ( ) 1 = 0 in formulae (4)-(6), the single degenerate soliton solutions of Eq. (2) are derived, and the corresponding functions f and g ( ) of solutions (4) are expressed as: where ξ 1 = p 1 x + i p 2 1 t, γ 1 = −(4 p 3 1 + p 1 ). By carrying out some simple calculations, the single soliton solutions are explicitly written as  (1) , u (2) , v) → (0, 0, 0) as ξ 1R → ±∞, which can further verify that the solutions (18) are bright solitons. The solitons attain maximum amplitudes along their centers where ξ 1R − θ 0 = 0: The velocities of these single bright solitons are 2 p 1I , thus the maximum amplitudes of the single solitons in the SW component u ( ) are also related to their velocities while they are independent of their velocities in the LW component v. That is the essential difference between the bright solitons in the SW components and the LW component. Figure 1 shows the single degenerate soliton solutions with parameters in which the maximum amplitudes of the single degenerate solitons are Besides, we can also directly confirm that the single degenerate solitons in all components u (1) , u (2) , and v have identical velocities. (4)

The dynamics of degenerate two solitons
where m s, j = e In order to study the interaction behaviours of these two degenerate solitons, for convenience, we denote the solitons along ≈ 0 as soliton 1 and soliton 2, respectively, and assume p 1R , p 2R > 0. Through asymptotic analysis for these two solitons, one can derive their asymptotic expressions as follows: where λ 1 = 1 2 ln (ii) A f ter collision(t → +∞): where λ 1 = 1 2 ln   Figure 2 shows the elastic collisions of these degenerate two solitons with parameters The amplitudes of soliton 1 in the SW components u (1) , u (2) , and the LW component v are 2.24 units, 1.41 units, and 2 units, respectively, while the amplitudes of soliton 2 are 1.28 units, 1.80 units, and 0.21 units in the components u (1) , u (2) , v, respectively. We can confirm that the intensities of these two solitons remain unaltered after collisions. Figure 3 depicts the inelastic collisions of two degenerate solitons with the following parameters For larger N and α ( ) j = 0, the higher-order degenerate solitons can be derived from Theorem 1. Particularly, with choices of p 1I = p 2I = · · · p N I , the corresponding degenerate solitons are bound states. For instance, by taking p 1I = p 2I in formulae (22), the bounded state degenerate two-soliton solution can be obtained. Figure 4 displays a bound state degenerate two-soliton solution, which can form periodic localized waves along both x and t coordinates.

The fundamental non-degenerate solitons
The fundamental non-degenerate solitons (i.e., nondegnerate one-soliton solutions) correspond to the solutions (4) In this case, the solutions (22) are rewritten as the following hyperbolic forms where The above solutions contain four free complex parameters p 1 , p 2 , α (1) 1 , α (2) 2 . Particularly, the above solutions reduce to the fundamental degenerate soliton solutions (18) when p 2 = p 1 . Furthermore, upon the relations between parameters p 1I and p 2I with p 1 = p 2 , which are related to the velocities of the solitons, the nondegenerate solitons are classified into two different types: (i) When p 1I = p 2I , the solutions (22) in all components (u (1) , u (2) , v) only comprise nondegenerate solitons with profiles of double-or singlehump localized structures, which are designated as (1, 1, 1)-soliton [61]. Particularly, these nondegenerate solitons in u (1) , u (2) , and v components have identical velocities, which are 2 p 1I . (ii) When p 1I = p 2I , the solutions (22) are referred as (1, 1, 2)-soliton solutions, in which the SW components u (1) , u (2) are nondegenerate one-soliton solutions with single-hump profiles, while the LW component v is the two-soliton solution with two single humps profiles as for the degenerate two-soliton solutions displayed in Fig. 2. It is different from the former case (i.e., the case of p 1I = p 2I ), namely the nondegenerate one solitons in these two SW components u (1) and u (2) possess unequal velocities, which are 2 p 1I and 2 p 2I in u (1) and u (2) components, respectively.
Below, we will study these two different types of nondegenerate one-soliton solutions depending on the velocities of the solitons in the components u (1) and u (2) being either identical with each other (2 p 1I = 2 p 2I ) or not (2 p 1I = 2 p 2I ).
When p 1I = p 2I , the quantities φ 1I and φ 2I are zeros, which results the solutions (29) are becoming of the following form , These solutions correspond to the (1, 1, 1)-soliton solutions. A physical difference between the degenerate solitons and nondegenerate solitons is that the nondegenerate one-soliton solutions can form double humps, while the degenerate one-soliton solutions are single humps waveforms. Figure 5 depicts the nondegenerate one-soliton solution with profiles of double humps with parameters By altering the values of the parameters p j and α ( j) j ( j = 1, 2), the humps of the nondegenerate one-soliton solution in the SW components u (1) , u (2) can also be changed. According to the number of the humps in the SW components u (1) , u (2) , the nondegenerate solitons are classified into three kinds: a double-hump soliton in one SW component and a one-hump soliton in the other SW component; a double-hump soliton in both SW components; a one-hump soliton in both SW components. Figure 6 shows these three different kinds of intensity profiles of the nondegenerate solitons with different values of parameters p j and α ( j) j ( j = 1, 2). The symmetric and asymmetric nature of the nondegenerate soliton can be indicated from the relative separation distance Δt 12 between two components: where φ 1R and φ 2R are defined below Eq. (29). The nondegenerate one-soliton solutions have symmetric profiles for Δt 12 = 0 and asymmetric ones for Δt 12 = 0. Fig. 5 The nondegenerate one-soliton solution (30) with parameters (31), all components u (1) , u (2) , v have double-hump profiles. The lower row shows the intensity plot of the nondegenerate soliton at t = 0  (1) , u (2) , v are double-hump solitons with parameters p 1 = 1 + i, p 2 = 9 10 + i, α 1 = 0, α (2) 2 = 11 10 + i, δ 1 = 1, δ 2 = 1; The middle panel: The com-ponents u (2) and v are double-hump solitons and component u (1) is a one-hump soliton with parameters The rightmost panel: The components u (1) and u (2) are one-hump solitons and the component v is a double-hump soliton with parameters p 1 = We then study the (1, 1, 2)-soliton solutions, namely, the solutions (29) with p 1I = p 2I . In this case, the SW components u (1) and u (2) are always one-hump solitons, and their velocities are distinct. This feature is the essential difference between the nondegenerate one-hump solitons and the degenerate single solitons, which have the same velocities in the u (1) and u (2) components. However, the LW component v behaves as the degenerate two solitons displayed in Figs. 2 and 3, and one of them possesses the same velocity as the soliton in SW component u (1) and the other one has the same velocity as the soliton in the SW component u (2) . Figure 7 shows the (1, 1, 2)-soliton solutions (29) with parameters In this figure, the velocities of the single-hump soliton in the SW components u (1) and u (2) are 2 and −4, respectively, while the velocities of the two solitons in the LW component v are 2 and −4.

The dynamics of nondegenerated two-soliton solutions
The nondegenerate two-soliton solutions correspond to the solutions (4) with N = 4, α The explicit determinant forms of functions f and g ( ) of the correspond- where m s, j = e ( ps + p * The above solutions describe the collisions of two nondegenerate solitons. As discussed previously, the single nondegenerate solitons are classified into two types upon their velocities in the SW components being equal or not. Hence, the two nondegenerate solitons can be classified into three types related to the velocities of these two nondegenerate solitons: (i) the two nondegenerate solitons in the u (1) component have identical velocities as the two nondegenerate solitons in the u (2) component when p 1I = p 2I , p 3I = p 4I ; (ii) the two nondegenerate solitons in the u (1) component possess different velocities as compared to the two solitons in the u (2) component when p 1I = p 2I , p 3I = p 4I ; (iii) one nondegenerate soliton in the u (1) component has the same velocity with the one in the u (2) component while the other one in the u (1) component has a different velocity as compared the other one in the u (2) component when p 1I = p 2I , p 3I = p 4I or p 1I = p 2I , p 3I = p 4I . In these three different parametric conditions, the LW component v has two solitons, three solitons, and four solitons, respectively. Thus, these three types of two nondegenerate solitons are referred to as (2, 2, 2) solitons, (2, 2, 3) solitons, and (2, 2, 4) solitons, respectively.
In the case of (2, 2, 2) solitons, corresponding to the solutions (34) with p 1I = p 2I , p 3I = p 4I , the velocities of these two nondegenerate solitons in all components u (1) , u (2) , and v are 2 p 1I and 2 p 3I . These two nondegenerate solitons can possess either symmetric or asymmetric double humps, and their collisions are elastic. Figure 8 depicts the collision scenarios of two nondegenerate solitons with profiles of symmetric double humps with p 1 = 1 2 + 1 2 i, p 2 = 501 1000 + 1 2 i, p 3 = 2 5 − 2 5 i, p 4 = 201 500 − 2 5 i. We can confirm that the intensities and shapes of these two nondegenerate solitons remains unaltered before and after collision.
In the case of (2, 2, 3) solitons, i.e., the solutions (34) with p 1I = p 2I , p 3I = p 4I , one nondegenerate soliton has equal velocities 2 p 1I in both u (1) and u (2) components. The other one has different velocities in the u (1) and u (2) components, namely 2 p 3I in the u (1) component and 2 p 4I in the u (2) component. The nondegenerate soliton possessing identical velocities in the u (1) and u (2) components can form double humps, and the other one having unequal velocities in the u (1) and u (2) components only display a single hump. The LW component v features three solitons, whose velocities are 2 p 1I , 2 p 3I , and 2 p 4I . The one, whose velocity is 2 p 1I , can possess double humps, while the other two solitons only have single humps. These interesting soliton shapes can be viewed in Fig. 9 for the set of param- (2,2,4) solitons, illustrated by the solutions (34) with p 1I = p 2I , p 3I = p 4I , the nondegenerate solitons in all components u (1) , u (2) and v are single humps and cannot form double-hump structures. The velocities of the two solitons in u (1) component are 2 p 1I , 2 p 3I , and they are 2 p 2I , 2 p 4I in the u (2) component. The LW component v has four intersecting solitons whose velocities are 2 p 1I , 2 p 2I , 2 p 3I , and 2 p 4I . The (2,2,4) solitons are shown in Fig. 10 when

Conclusion
In this paper we have studied the interesting dynamics of degenerate and nondegenerate solitons and their collision scenarios in the 2NLS-Boussinesq equation (2). The main results of this work are as follows: (1) The general bright soliton solutions in forms of determinants to the 2NLS-Boussinesq equation (2) were derived via the bilinear KP-reduction method, which is a powerful method in constructing solitary wave solutions for integrable systems [88][89][90]. By modifying the input parameters, these soliton solutions can be classified into degenerate solitons, nondegenerate solitons, and partially nondegenerate solitons. (2) The degenerate solitons and their collisions were systematically investigated for the 2NLS-Boussinesq equation (2). The amplitudes of the single degenerate solitons are related to their velocities in the SW components u (1) and u (2) , but they are independent in the LW component v. Based on the asymptotic analysis for the two-soliton solutions, we found that these solitons can exhibit either elastic collisions (see Fig. 2) or inelastic ones (see Fig. 3). The degenerate soliton only possess single-hump localized waves. (3) The nondegenerate solitons have been studied in detail. It has been shown that the fundamental nondegenerate solitons have two different forms upon their velocities in the u (1) and u (2) components being identical or not: namely (1, 1, 1) solitons and (1, 1, 2) solitons. In the case of (1, 1, 1) solitons, the nondegenerate solitons possess single-hump or double-hump wave profiles, and their velocities are identical in all three components u (1) , u (2) , and v (see Fig. 5). For the (1, 1, 2) solitons, the nondegenerate solitons only display single-hump localized waves, and their velocities are different in the u (1) and u (2) components (see Fig. 7). The nondegenerate two-soliton solutions have three different forms: (2, 2, 2) solitons (see Fig. 8), (2, 2, 3) solitons (see Fig. 9) and (2, 2, 4) solitons (see Fig. 10). The col-lisions of the nondegenerate two-soliton solutions are elastic ones.