Singular soliton, shock-wave, breather-stripe soliton, hybrid solutions and numerical simulations for a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system in fluid mechanics

In this paper, a (2+1)-dimensional Caudrey–Dodd–Gibbon–Kotera–Sawada system is investigated in fluid mechanics via the symbolic computation. With the help of the Hirota method, we derive some singular soliton, shock-wave, breather-stripe soliton and hybrid solutions. Based on the finite difference method, we get some numerical one-soliton solutions. We graphically show the singular and shock-wave solutions, and observe that the singular one-soliton solutions are explosive and unstable, but the shock-wave solutions are non-singular and stable. We observe that the breather-stripe soliton moves along the negative direction of the y axis, where y is a variable, and the amplitude and shape of the breather-stripe soliton remain invariant during the propagation. We graphically demonstrate the interaction among a rogue wave, a periodic wave and a pair of the stripe solitons: the rogue wave arises from the one stripe soliton; the rogue wave interacts with the periodic wave, the rogue wave splits into two waves and then the two waves merge into a wave; the rogue wave fuses with the other stripe soliton. We graphically present the numerical one-soliton solutions which agree with the analytic one-soliton solutions.


Introduction
Fluid mechanics has been applied in a variety of disciplines such as meteorology, geophysics, biomedical engineering, oceanography and astrophysics [1][2][3][4]. In order to gain insight into certain fluid mechanical problems, researchers have focused their attention on the solutions for some nonlinear evolution equations, such as the soliton, breather-wave, periodic-wave and roguewave solutions .
References [31][32][33][34][35][36][37][38][39][40][41][42][43] have considered the following (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada system in fluid mechanics: where u and v are both the differentiable functions with respect to the variables x, y and t, and the subscripts indicate the partial derivatives. Darboux transformations and N -soliton solutions for System (1) have been derived via the Darboux matrix method, where N is a positive integer [31]. Soliton, rational, triangular periodic, Jacobi and Weierstrass doubly periodic solu-tions for System (1) have been obtained via the algebraic method [32]. Some soliton, breather and periodic solutions for System (1) have been derived via the long wave limit method [33]. Via the Hirota-Riemann function method [34], quasi-periodic solutions for System (1) have been derived. Symmetry reductions and group-invariant solutions for System (1) have been obtained [35]. Lump, mixed rogue wave-stripe soliton and mixed lump-stripe soliton for System (1) have been obtained via the Hirota bilinear method [36]. The higher-order breather, lump and hybrid solutions for System (1) have been obtained via the long wave limit method [37]. The periodic soliton solutions for System (1) have been derived via the Hirota bilinear method [38]. Reference [39] has derived some nontraveling wave solutions for System (1) via the Lie group analysis and exp-function method. Hybrid solutions among some lumps, breathers and solitons for System (1) have been obtained via the long wave limit method [40]. Lump solutions have been obtained via the direct method [41]. A symmetry group theorem, some analytic solutions and infinite conservation laws for System (1) have been derived via the improved CK's method [42]. Hybrid solutions comprising the lumps and solitons have been obtained [43].
Under the transformations, System (1) has been converted into the bilinear form as [40] (5D 3 where f is a differentiable function about x, y and t, and D is the Hirota bilinear differentiable operator, which is defined as [44] with F(x, y, t) being a differentiable function with respect to x, y and t, G(x , y , t ) being a differentiable function about the formal variables x , y and t , and ι 1 , ι 2 and ι 3 being three nonnegative integers. However, to our knowledge, singular N -soliton solutions, shock-wave solutions, breather-stripe soliton solutions, hybrid solutions among a rogue wave, a periodic wave and a pair of the stripe solitons, and numerical solutions for System (1) have not been investigated. In Sect. 2, we will construct some singular N -soliton and shock-wave solutions for System (1). In Sect. 3, breather-stripe soliton solutions for System (1) will be established. In Sect. 4, we will discuss the hybrid solutions among a rogue wave, a periodic wave and a pair of the solitons for System (1). In Sect. 5, we will obtain certain numerical one-soliton solutions for System (1). In Sect. 6, conclusions will be drawn.

Singular soliton and shock-wave solutions for
System (1)

Singular soliton solutions for System (1)
In this part, we will construct some singular Nsoliton solutions for System (1) via the Hirota bilinear method [44]. We use the perturbation method to expand f with respect to a formal parameter as where f i 's (i = 1, 2, · · ·, N ) are the real functions of x, y and t. Substituting Expression (4) into Bilinear Form (3) and equating the coefficients of the same power of to zero, we obtain some soliton solutions of System (1). Thus, when = −1, singular N -soliton solutions can be written as where κ ı , ρ ı , ω ı and η ı0 are the real constants and ı < j < ( = 3, 4, · · ·, N ). Assuming N =1 in Solutions (5), singular onesoliton solutions can be derived as with The physical structure of Solutions (6) is shown in Fig. 1, which shows that Solutions (6) are explosive and unstable. (1) According to Ref. [45], to calculate some shock-wave solutions, we suppose that

Shock-wave solutions for System
where ζ = ax + by + ct, M is the wave amplitude, while a, b, c and λ are all the real constants. Substituting Expression (7) into u x x x x x and u 2 u x , we derive that From Expressions (8), (1), we obtain that . Therefore, we obtain the shock-wave solutions for System (1) as (9) Figure 2 shows the propagation of the shock wave. We observe that the amplitude and the shape of the shock wave keep unchanged during the propagation. Therefore, Solutions (9) are the non-singular and stable solutions.

Breather-stripe soliton solutions for System (1)
To construct the breather-stripe soliton solutions for System (1), inspired by Ref. [46], we assume that where ξ = k x + r y + p t + q ( = 1, 2, 3), and k 's, r 's, p 's, q 's, s 's and m 1 are the real constants. Substituting Expression (10) into Bilinear Form (3), we derive that , According to Expressions (10), (11) and Transformations (2), we get the breather-stripe soliton solutions for System (1) as  Figure 3 displays the propagation of the breatherstripe soliton. We observe that the breather-stripe soliton moves along the negative direction of the y axis. Amplitude and shape of the breather-stripe soliton remain invariant during the propagation.

Numerical simulations for System (1)
In the section, for simplicity, we consider the dimensional reduction ∂ x = ∂ y , then System (1) is reduced to with w being the differentiable functions about x and t. Next, we will construct the numerical one-soliton solutions for Eq. (16) via a finite difference method [48]. Choosing a finite interval = [L 1 , L 2 ], which is large enough, Eq. (16) with the vanishing boundary condition lim |x|→+∞ |w| = 0 can be approximated by According to Ref. [48], let h = L 2 −L 1 J and τ be the x-direction step size and t-direction step size, respectively, then we get the mesh points x j = L 1 + jh ( j = 0, 1, · · ·, J ) and t n = nτ , with J as a positive integer and n as a nonnegative positive integer. For simplicity, we introduce some notations, as follows [48]: where w n j denotes the approximate value of w(x j , t n ). Equation (16) at the point x j , t n+ 1 2 can be written as 36 ∂w ∂t Based on the Taylor expansion, we have Substituting Expressions (20) into Eq. (19), we derive that 36δ t w where R , we obtain that 36δ t w Equation (22) is collapsed to obtain that By the iterative method [48], the iterative scheme for Finite Difference Scheme (23) can be written as where j+1 + w n j+1 , γ = n or n + 1 or n + 1, l + 1, and l = 0, 1, 2, · · · denotes the iteration time.
From t n to t n+1 , the iteration stops when |w n+1,l+1 j − w n+1,l j | < ε, with ε being a given error bond. We let the error between the analytic solutions w(x j , t n ) and the numerical solutions w n j as and define as an approximation of the rate of convergence when both h and τ are sufficiently small. As such, if the scheme is the second-order accurate in x direction and the second-order accurate in t direction, the obtained rate should be 2.
We choose the analytic one-soliton solution for Eq. (16), Then, we simulate the one-soliton evolution by choosing Solution (27) at t = 0 as the initial data. In our simulations, we take the interval = [− 30,30], h = 0.2, τ = 0.05 and error bond ε = 10 −8 for the iterative computation of Scheme (24). Figure 5(a) shows the propagation of the one soliton via Solution (27). Figure 5(b) displays the numerical one-soliton solution. Figure 5(c) depicts a comparison of the analytic and numerical one-soliton solutions with t. Table 1 shows the errors and the convergence rates of the analytic and numerical one-soliton solutions under different h and τ at t = 1. From Table 1, we find that the errors between the analytic and numerical one-soliton solutions are small and the convergence rates are close to 2, which is the result as expected.

Conclusions
In this paper, symbolic computation has been conducted on a (2+1)-dimensional Caudrey-Dodd-Gibbon-Kotera-Sawada system in fluid mechanics, i.e., System (1). Via the Hirota method, Singular N -Soliton Solutions (5), Shock-Wave Solutions (9), Breather-Stripe Soliton Solutions (12) and Hybrid Solutions (15) have been derived. Via Finite Difference Scheme (23), we have simulated the one-soliton propagation for Eq. (16). Figures 1 and 2 have shown the propagations of the singular one soliton and the shock wave, respectively. We have observed that the singular one-soliton solutions are explosive and unstable, but the shock-wave solutions are non-singular and stable. Figure 3 has displayed the propagation of the breather-stripe soliton. We have observed that the breather-stripe soliton moves along the negative direction of the y axis. We have found that the amplitude and shape of the breather-stripe soliton remain invariant during the propagation, as shown in Fig. 3.
Based on Solutions (15), we have investigated the interaction among a rogue wave, a periodic wave and a pair of the stripe solitons, as shown in Fig. 4. It has been seen that there is a periodic wave between a pair of the stripe solitons, as shown in Figs. 4(a 1 ) and (b 1 ). As t goes on, we have observed that the rogue wave arises from the one stripe soliton and interacts with the periodic wave, next the rogue wave splits into two waves, then the two waves merge into one wave, and finally the rogue wave fuses with the other stripe soliton, as seen in Figs. 4(a 2 )-(a 5 ) and (b 2 )-(b 5 ).
Figures 5(a) and (b) have displayed the analytic onesoliton and numerical one-soliton solutions, respectively. Figure 5(c) has depicted a comparison of the analytic and numerical solutions with t, which indicates that the numerical one-soliton solution which agrees with the analytic one-soliton solution. Table 1 has shown the errors and the convergence rates of the analytic and numerical one-soliton solutions under different h and τ at t = 1. From Table 1, we have found that the errors between the analytic and numerical onesoliton solutions are small and the convergence rates are close to 2, which is the result as expected.
Hirota method has been used to find certain analytic solutions for the NLEEs. We expect that the results of this paper will be helpful to the propagation of waves in fluid mechanics.