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BY 4.0 license Open Access Published by De Gruyter Open Access August 16, 2022

The nonlinear vibration and dispersive wave systems with extended homoclinic breather wave solutions

  • Xianqing Rao , Jalil Manafian EMAIL logo , K. H. Mahmoud , Afandiyeva Hajar , Ahmed B. Mahdi and Muhaned Zaidi
From the journal Open Physics

Abstract

This article investigates the extended homoclinic (heteroclinic) breather wave solutions and interaction periodic and dark soliton solutions to the nonlinear vibration and dispersive wave systems. The solutions include periodic, breather, and soliton solutions. The bilinear form is considered in terms of Hirota derivatives. Accordingly, we utilize the Cole–Hopf algorithm to obtain the exact solutions of the ( 2 + 1 )-dimensional modified dispersive water-wave system. The analytical treatment of extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are studied and plotted in four forms of density plots. A nonlinear vibration system will be studied. Employing appropriate mathematical assumptions, the novel kinds of the extended homoclinic breather wave solutions and interaction periodic and dark soliton solutions are derived and constructed in view of the combination of kink, periodic, and soliton for an extended homoclinic breather and also a combination of two kinks, periodic and dark soliton in terms of exponential, trigonometric, hyperbolic functions for interaction periodic and dark soliton of the governing equation. To achieve this, the illustrative example of the ( 2 + 1 )-D modified dispersive water-wave system is furnished to demonstrate the feasibility and reliability of the procedure applied in this research. The trajectory solutions of the traveling waves are offered explicitly and graphically. The effect of the free parameters on the behavior of designed figures of a few obtained solutions for two nonlinear rational exact cases was also considered. By comparing the suggested scheme with the other existing methods, the results state that the execution of this technique is succinct, extensive, and straightforward.

1 Introduction

The advent of the concept of nonlinear partial differential equations (NLPDEs) has attracted the interest of many researchers due to their importance in accurately demonstrating the dynamics of abundant real-world systems in various topics of sciences such as physics, diffusion, biology, chaos theory, chemistry, engineering, economics, commerce, and many others [1,2,3].

In particular, more and much attention has been paid to constructing exact and approximate solutions, for example, the csch-function method [4], an improved Hungarian algorithm [5], a social group optimization algorithm [6], a new solitary periodic wave solution [7], a mathematical performance framework for Hadoop data locality [8], the multiple Exp-function technique [9], the reduced chi-square and root-mean-square error [10], the Hirota’s bilinear scheme [11,12], a Pearson’s product moment correlation [13], solving absolute value problems [14], the inverse scattering transformation method [15], multiple soliton solutions and fusion interaction phenomena [16], the truncated Painlevé series [17], an optical flow approach [18], the modified Pfaffian technique [19], the conserved quantities method [20], and so forth.

In ref. [21], the N-soliton solutions, soliton molecules, and asymmetric solitons of the Korteweg–de Vries–Caudrey–Dodd–Gibbon equation were acquired by utilizing velocity resonance scheme by Ma et al. Researchers analyzed the higher-order algebraic soliton solutions of the Gerdjikov–Ivanov equation by utilizing the Darboux transformation and some limit technique, and according to the asymptotic balance between different algebraic terms, they obtained the asymptotic expressions of algebraic soliton solutions [22]. Wang studied the multi-soliton solutions of the ( 2 + 1 )-dimensional PT-symmetric couplers with varying coefficients by using homogeneous balance scheme [23]. The N-soliton solutions, M-breather solutions, and hybrid ones composed of solitons and breathers were established as a time-dependent KP equation by Wu [24].

Here, consider ( 2 + 1 )-dimensional modified dispersive water-wave system as follows:

(1.1) Σ y t + Σ x x y 2 Γ x x ( Σ 2 ) x y = 0 , Γ t Γ x x 2 ( Σ Γ ) x = 0 ,

where Σ and Γ are the unknown functions. The ( 2 + 1 )-dimensional modified dispersive water-wave system has been made in a model nonlinear and dispersive long gravity waves traveling in two horizontal directions on shallow waters of uniform depth [25]. In refs [26,27], the propagating localized excitations by using the Painlevé–Bäcklund transformation and a variable separation technique have been investigated for some ( 2 + 1 )-dimensional integrable systems. The powerful scholars have studied on nontravelling wave solutions [28]; abundant propagating, and nonpropagating solitons such as dromion, ring, peakon, and compacton [29]; solitary wave solutions, periodic wave solutions and rational function solutions [30]; the traveling wave solutions by the improved tan ( ϕ / 2 ) -expansion technique [31]; and periodic folded waves solutions [32]. In ref. [33], a pair of quartic-linear forms for the modified dispersive water-wave system was constructed by choosing an appropriate seed solution in the truncated Painlevé series. Also, the multiple soliton solutions and fusion interaction phenomena were derived with the help of the Bäcklund transformation and bilinear form technique [34]. The nonlinear vibration system behaviors was investigated by obtaining a breather type of rogue wave solution, which contains two peaks and also rogue wave by Li [35].

By reducing NLPDE with the help of Hirota’s bilinear method to nonlinear ordinary differential equation (NLODE), an algebraic equation system was obtained by Maple. By solving this system, plenty wave solutions including lump and other interactions have been found therein [36,37, 38,39]. Many researchers used various methods to study the nonlinear models by using Hirota bilinear technique (HBT), such as a ( 3 + 1 )-dimensional nonlinear evolution equation [40], generalized variable-coefficient Kadomtsev–Petviashvili equation [41], the new ( 3 + 1 )-dimensional generalized Kadomtsev–Petviashvili equation [42], and a generalized ( 3 + 1 )-D VC NLW equation [43]. Through these methods, some exact solutions of the nonlinear models of equations were obtained. To really understand these physical phenomena, it is of immense importance to solve nonlinear partial differential equations (NLPDEs), which govern these aforementioned phenomena. However, there is no general systematic theory that can be applied to NLPDEs, so that their analytic solutions can be obtained. Nevertheless, in recent times, scientists have developed effective techniques to obtain viable analytical solutions to NLPDEs, such as some nonlinear equations, for example, an analytical analysis to solve the fractional differential equations [44], an efficient alternating direction explicit method for solving a PDE [45], the semi-analytical solutions of the positive Gardner–Kadomtsev–Petviashvili equation [46], and so on. In ref. [47], M-lump solution and N-soliton solution of the ( 2 + 1 )-dimensional variable-coefficient Caudrey–Dodd–Gibbon–Kotera–Sawada equation were studied. Also, N-lump and interaction solutions of localized waves to the ( 2 + 1 )-dimensional generalized KDKK equation were investigated and obtained [48]. In particular, more and much attention has been paid to constructing exact lump solutions to the third-order evolution equation [49] and triple-soliton kind solutions to the modified HSI equation. In ref. [50], the vibration mode was acquired by finite element technology to explore the vibration response of the GIS busbar enclosure in a strong electric field. The authors of ref. [51] studied the Biswas–Arshed model in birefringent fibers for chirp-free solitons with the aid of sub-ordinary differential equations method. The new three-dimensional modified Benjamin–Bona–Mahony equations were analyzed with the introduction of the spatial and temporal fractional order derivatives using conformable fractional derivative [52]. The Hirota bilinear form has used the ( 3 + 1 )-dimensional Vakhnenko–Parkes equation to obtain the types of multiwaves, breather wave, lump-kink, lump-periodic solutions, and interaction solutions [53]. By implementing modified analytical and numerical methods, the construction of the analytical and numerical wave solutions for the Ito integro-differential dynamical equation were reached in ref. [54]. The extended form of two methods, auxiliary equation mapping and direct algebraic methods, were used to find the families of new exact travelling wave solutions of the system of equations for the ion sound and Langmuir waves [55]. N-soliton solutions have been studied recently and extensively, by the Hirota bilinear method and the Riemann–Hilbert problems, for local integrable equations in (1 + 1)-dimensions involving [56] and in ( 2 + 1 )-dimensions involving [57] and also for nonlocal integrable equations containing Riemann–Hilbert problems [58,59].

Motivated by the aforementioned studies, we apply the proposed analytical method presented by ref. [35] to solve the one problem mentioned earlier. The advantage of the suggested technique is that it can be applied to the integrable model and plenty of the cross-kink and solitary wave solutions with three kinds of plotted graphs will be obtained.

The rest of this article is organized as follows: Section 2 presents the definition of the binary Bell polynomials and their properties. Section 3 proposes the method based on the given algorithm transformation, for obtaining the extended homoclinic breather wave solutions and interaction periodic and dark soliton of system (1.1). Finally, Section 4 briefly summarizes and discusses the results in a conclusion.

2 Binary Bell polynomials

Through ref. [60], take λ = λ ( x 1 , x 2 , , x n ) is a C function with multiparameters, and the general form can be written as follows:

(2.1) ϒ n 1 x 1 , , n j x j ( λ ) ϒ n 1 , , n j ( λ d 1 x 1 , , d j x j ) = e λ x 1 n 1 x j n j e λ ,

which are named the multi-D Bell polynomials as follows:

λ d 1 x 1 , , d j x j = x 1 d 1 x j d j λ , λ 0 x i λ , d 1 = 0 , , n 1 ; ; d j = 0 , , n j ,

and we have

(2.2) ϒ 1 ( λ ) = λ x , ϒ 2 ( λ ) = λ 2 x + λ x 2 , ϒ 3 ( λ ) = λ 3 x + 3 λ x λ 2 x + λ x 3 , , λ = λ ( x , t ) , ϒ x , t ( λ ) = λ x , t + λ x λ t , ϒ 2 x , t ( λ ) = λ 2 x , t + λ 2 x λ t + 2 λ x , t λ x + λ x 2 λ t , .

The multidimensional binary Bell polynomials can be expressed as follows:

(2.3) Σ n 1 x 1 , , n j x j ( μ 1 , μ 2 ) = ϒ n 1 , , n j ( λ ) λ d 1 x 1 , , d j x j = μ 1 d 1 x 1 , , d j x j , d 1 + d 2 + + d j , is odd μ 2 d 1 x 1 , , d j x j , d 1 + d 2 + + d j , is even .

The following issues can be expressed as follows:

(2.4) Σ x ( μ 1 ) = μ 1 x , Σ 2 x ( μ 1 , μ 2 ) = μ 2 2 x + μ 1 x 2 , Σ x , t ( μ 1 , μ 2 ) = μ 2 x , t + μ 1 x μ 1 t , .

Proposition 2.1

Let μ 1 = ln ( Ω 1 / Ω 2 ) , μ 2 = ln ( Ω 1 Ω 2 ) , then the relations between binary Bell polynomials and Hirota D-operator read:

(2.5) Σ n 1 x 1 , , n j x j ( μ 1 , μ 2 ) μ 1 = ln ( Ω 1 / Ω 2 ) , μ 2 = ln ( Ω 1 Ω 2 ) = ( Ω 1 Ω 2 ) 1 D x 1 n 1 D x j n j Ω 1 Ω 2 ,

via Hirota operator

(2.6) i = 1 j D x i n i g . η = i = 1 j x i x i n i Ω 1 ( x 1 , , x j ) Ω 2 ( x 1 , , x j ) x 1 = x 1 , , x j = x j .

Proposition 2.2

Take Ξ ( γ ) = i δ i P d 1 x 1 , , d j x j = 0 and μ 1 = ln ( Ω 1 / Ω 2 ) , μ 1 = ln ( Ω 1 Ω 2 ) , we have

(2.7) i δ 1 i ϒ n 1 x 1 , , n j x j ( μ 1 , μ 2 ) = 0 , i δ 1 i ϒ d 1 x 1 , , d j x j ( μ 1 , μ 2 ) = 0 ,

which need to satisfy

(2.8) R ( γ , γ ) = R ( γ ) R ( γ ) = R ( μ 2 + μ 1 ) R ( μ 2 μ 1 ) = 0 .

The generalized Bell polynomials ϒ n 1 x 1 , , n j x j ( ξ ) is expressed as follows:

(2.9) ( Ω 1 Ω 2 ) 1 D x 1 n 1 D x j n j Ω 1 Ω 2 = Σ n 1 x 1 , , n j x j ( μ 1 , μ 2 ) μ 1 = ln ( Ω 1 / Ω 2 ) , μ 2 = ln ( Ω 1 Ω 2 ) = Σ n 1 x 1 , , n j x j ( μ 1 , μ 1 + γ ) μ 1 = ln ( Ω 1 / Ω 2 ) , γ = ln ( Ω 1 Ω 2 ) = k 1 n 1 k j n j i = 1 j n i k i P k 1 x 1 , , k j x j ( γ ) ϒ ( n 1 k 1 ) x 1 , , ( n j k j ) x j ( μ 1 ) .

The Cole–Hopf relation is expressed as follows:

(2.10) ϒ k 1 x 1 , , k j x j ( μ 1 = ln ( φ ) ) = φ n 1 x 1 , , n j x j φ ,

(2.11) ( Ω 1 Ω 2 ) 1 D x 1 n 1 D x j n j Ω 1 Ω 2 Ω 2 = exp ( γ / 2 ) , Ω 1 / Ω 2 = φ = φ 1 k 1 n 1 k j n j l = 1 j n l k l P k 1 x 1 , , k l x l ( γ ) φ ( n 1 k 1 ) x 1 , , ( n d k l ) x l ,

with

(2.12) ϒ t ( μ 1 ) = φ t φ , ϒ 2 x ( μ 1 , β ) = γ 2 x + φ 2 x φ , ϒ 2 x , y ( μ 1 , μ 2 ) = γ 2 x φ y φ + 2 γ x , y φ x φ + φ 2 x , y φ .

Theorem 2.3

According to ref. [35], the used dependent variable transformation with the logarithm of function f ( x , y , t )

(2.13) Σ = x ( ln h ) + Σ 0 , Γ = 2 x 2 ( ln h ) + Γ 0 ,

where Σ 0 = Σ 0 ( x , y , t ) and Γ 0 = Γ 0 ( x , y , t ) are known solutions of system stated by system (1.1). By substituting Σ 0 = 0 and Γ 0 = 0 in (2.13) and inserting concluded solution in (1.1), we obtain the below nonlinear PDE based on h :

(2.14) d d y 1 y 2 ( h h x t h x h t + h x h x x h h x x x ) = 0 , h = h ( x , y , t ) .

Then, one obtains

(2.15) h h x t h x h t + h x h x x h h x x x = 0 .

3 Two types of solutions for MDWW system

Lump-three soliton solutions and multiwave solutions are investigated in the following sections.

3.1 Extended homoclinic (heteroclinic) breather wave solutions

Here, we utilize to formulate the new exact solutions to the ( 2 + 1 )-dimensional MDWW system. Consider the below function for studying the extended homoclinic (heteroclinic) breather wave solutions as follows:

(3.1) h = τ 1 ( y ) e a 1 + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) + τ 4 ( y ) e a 2 ,

a l = m l x + n l ( y ) + p l ( y ) t , l = 1 , 2 .

Afterward, the values m l , n l ( y ) , p l ( y ) , and ( l = 1 : 2 ) will be found. By substituting Eq. (3.1) into (2.15) and taking the coefficients of power of exp ( a l ) , l = 1 , 2 and cos ( d ( y ) ( x + e ( y ) ) ) , sin ( d ( y ) ( x + e ( y ) ) ) to be zero, we yield a system of equations (algebraic) (these are not collected here for minimalist) for m l , n l ( y ) , p l ( y ) , ( l = 1 : 2 ) . These algebraic equations with the help of emblematic computation software like, Maple, give the following solutions by using Σ = ( ln h ) x and Γ = ( ln h ) x y :

Σ = τ 1 ( y ) m 1 e a 1 τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 3 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 4 ( y ) m 2 e a 2 τ 1 ( y ) e a 1 + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) + τ 4 ( y ) e a 2 ,

Γ = M 4 + M 1 + M 2 + M 3 + M 5 M 6 N 1 ( N 5 + N 6 + N 4 + N 3 + N 2 ) M 6 2 ,

S 1 = sin ( d ( y ) ( x + e ( y ) ) ) , S 2 = cos ( d ( y ) ( x + e ( y ) ) ) ,

M 1 = S 1 d d y τ 2 ( y ) d ( y ) + τ 2 ( y ) d d y d ( y ) + S 2 τ 3 ( y ) d d y d ( y ) + d ( y ) d d y τ 3 ( y ) ,

M 2 = τ 2 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) , M 3 = τ 3 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) , M 4 = e a 1 m 1 τ 1 ( y ) d d y p 1 ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) , M 5 = e a 2 m 2 d d y p 2 ( y ) τ 4 ( y ) t + τ 4 ( y ) d d y n 2 ( y ) + d d y q 4 ( y ) , M 6 = τ 1 ( y ) e a 1 + τ 2 ( y ) S 2 + τ 3 ( y ) S 1 + τ 4 ( y ) e a 2 , N 1 = τ 1 ( y ) m 1 e a 1 τ 2 ( y ) S 1 d ( y ) + τ 3 ( y ) S 2 d ( y ) + τ 4 ( y ) m 2 e a 2 , N 2 = e a 2 d d y p 2 ( y ) τ 4 ( y ) t + τ 4 ( y ) d d y n 2 ( y ) + d d y τ 4 ( y ) , N 3 = τ 3 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) , N 4 = τ 2 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) , N 5 = e a 1 τ 1 ( y ) d d y p 1 ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) , N 6 = d d y τ 2 ( y ) S 2 + d d y τ 3 ( y ) S 1 .

3.1.1 Set I solutions

(3.2) m l = m l , l = 1 , 2 , p 2 ( y ) = p 2 ( y ) , p 1 ( y ) = m 1 2 + d ( y ) 2 , τ 4 ( y ) = 0 , τ i ( y ) = τ i ( y ) , i = 1 , 2 , 3 ,

where m l , l = 1 , 2 , and n l ( y ) , l = 1 , 2 are free functions of y and unknown parameters. They are derived based on the bilinear frame, and the rational analytical solution can be introduced as follows:

(3.3) Σ 1 = τ 1 ( y ) m 1 e m 1 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 1 2 ) t τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 3 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) d ( y ) τ 1 ( y ) e m 1 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 1 2 ) t + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) ,

(3.4) Γ 1 = M 4 + M 1 + M 2 + M 3 M 6 N 1 ( N 5 + N 6 + N 4 + N 3 ) M 6 2 ,

S 1 = sin ( d ( y ) ( x + e ( y ) ) ) , S 2 = cos ( d ( y ) ( x + e ( y ) ) ) ,

M 1 = S 1 d d y τ 2 ( y ) d ( y ) + τ 2 ( y ) d d y d ( y ) + S 2 τ 3 ( y ) d d y d ( y ) + d ( y ) d d y τ 3 ( y ) ,

M 2 = τ 2 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) ,

M 3 = τ 3 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) ,

M 4 = e a 1 m 1 q 1 ( y ) d d y p 1 ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) ,

M 6 = τ 1 ( y ) e a 1 + τ 2 ( y ) S 2 + τ 3 ( y ) S 1 ,

N 1 = τ 1 ( y ) m 1 e a 1 τ 2 ( y ) S 1 d ( y ) + τ 3 ( y ) S 2 d ( y ) ,

N 3 = τ 3 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) ,

N 4 = τ 2 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) ,

N 5 = e a 1 q 1 ( y ) d d y p 1 ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) ,

N 6 = d d y τ 2 ( y ) S 2 + d d y τ 3 ( y ) S 1 ,

a 1 = m 1 x + n 1 ( y ) + ( m 1 2 + d ( y ) 2 ) t , a 2 = 0 .

Figures 1, 2, 3 show the analysis of the treatment of extended homoclinic (heteroclinic) breather wave solution with two-wave solution, where graphs of Γ 1 are given with the following selected parameters:

(3.5) m 1 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , n 1 ( y ) = y , d ( y ) = y , e ( y ) = y ,

(3.6) m 1 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , n 1 ( y ) = y 3 , d ( y ) = 3 , e ( y ) = y ,

(3.7) m 1 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , n 1 ( y ) = y 2 , d ( y ) = y , e ( y ) = y 3 ,

given in Eq. (3.4). By the aforementioned parameters, the structural property homoclinic breather wave solutions as presented in Figures 13 with density plot with different times. It shows a type of interaction solutions between exponential and trigonometric waves.

Figure 1 
                     Plot of the extended homoclinic breather wave solution (3.5) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       1
                                    
                                 
                              
                              {\Gamma }_{1}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 1

Plot of the extended homoclinic breather wave solution (3.5) ( Γ 1 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 2 
                     Plot of the extended homoclinic breather wave solution (3.6) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       1
                                    
                                 
                              
                              {\Gamma }_{1}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 2

Plot of the extended homoclinic breather wave solution (3.6) ( Γ 1 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 3 
                     Plot of the extended homoclinic breather wave solution (3.7) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       1
                                    
                                 
                              
                              {\Gamma }_{1}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 3

Plot of the extended homoclinic breather wave solution (3.7) ( Γ 1 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

3.1.2 Set II solutions

(3.8) m l = m l , l = 1 , 2 , p 2 ( y ) = m 2 2 + d ( y ) 2 + p 2 , p 1 ( y ) = p 1 ( y ) , τ 1 ( y ) = 0 , τ i ( y ) = τ i ( y ) , i = 2 , 3 , 4 ,

where m l , l = 1 , 2 , and n l ( y ) , l = 1 , 2 are free functions of y and unknown parameters. They are derived based on the bilinear frame, and the rational analytical solution can be introduced as follows:

(3.9) Σ 2 = τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 3 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 4 ( y ) m 2 e m 2 x + n 2 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) + τ 4 ( y ) e m 2 x + n 2 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t ,

(3.10) Γ 2 = d d y τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 3 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 4 ( y ) m 2 e m 2 x + n 2 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) + τ 4 ( y ) e m 2 x + n 2 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t .

3.1.3 Set III solutions

(3.11) m l = m l , l = 1 , 2 , p 2 ( y ) = p 2 ( y ) , p 1 ( y ) = m 2 2 + d ( y ) 2 , τ 4 ( y ) = 0 , τ i ( y ) = τ i ( y ) , i = 1 , 2 , τ 3 ( y ) = m 2 τ 2 ( y ) d ( y ) ,

where m l , l = 1 , 2 , and n l ( y ) , l = 1 , 2 , are free functions of y and unknown parameters. They are derived based on the bilinear frame, and the rational analytical solution can be introduced as follows:

(3.12) Σ 3 = τ 1 ( y ) m 2 e m 2 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + m 2 τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) τ 1 ( y ) e m 2 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + m 2 τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) ,

(3.13) Γ 3 = d d y τ 1 ( y ) m 2 e m 2 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + m 2 τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) τ 1 ( y ) e m 2 x + n 1 ( y ) + ( ( d ( y ) ) 2 + m 2 2 ) t + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + m 2 τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) .

Figures 4, 5, 6 show the analysis of the treatment of extended homoclinic (heteroclinic) breather wave solution with two-wave solution, where graphs of Γ 3 are given with the following selected parameters:

(3.14) m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , n 1 ( y ) = y , d ( y ) = 1 , e ( y ) = y ,

(3.15) m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , n 1 ( y ) = y , d ( y ) = 2 1 + y 2 , e ( y ) = y + y 3 ,

(3.16) m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , n 1 ( y ) = y , d ( y ) = cos ( y ) , e ( y ) = y + y 3 ,

given in Eq. (3.13). By the aforementioned parameters, the structural property homoclinic breather wave solutions are presented in Figures 46 with density plot with different times. It presents a type of interaction solutions between exponential and trigonometric waves.

Figure 4 
                     Plot of the extended homoclinic breather wave solution (3.14) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       3
                                    
                                 
                              
                              {\Gamma }_{3}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 4

Plot of the extended homoclinic breather wave solution (3.14) ( Γ 3 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 5 
                     Plot of the extended homoclinic breather wave solution (3.15) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       3
                                    
                                 
                              
                              {\Gamma }_{3}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 5

Plot of the extended homoclinic breather wave solution (3.15) ( Γ 3 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 6 
                     Plot of the extended homoclinic breather wave solution (3.7) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       3
                                    
                                 
                              
                              {\Gamma }_{3}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 6

Plot of the extended homoclinic breather wave solution (3.7) ( Γ 3 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

3.1.4 Set IV solutions

(3.17) m l = m l , l = 1 , 2 , p 1 ( y ) = m 1 2 + d ( y ) 2 , p 2 ( y ) = m 2 2 + d ( y ) 2 , τ i ( y ) = τ i ( y ) , i = 1 , 2 , 3 , 4 ,

where m l , l = 1 , 2 , and n l ( y ) , l = 1 , 2 are free functions of y and unknown parameters. They are derived based on the bilinear frame, and the rational analytical solution can be introduced as follows:

(3.18) Σ 4 = τ 1 ( y ) m 1 e a 1 τ 2 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 3 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) d ( y ) + τ 4 ( y ) m 2 e a 2 τ 1 ( y ) e a 1 + τ 2 ( y ) cos ( d ( y ) ( x + e ( y ) ) ) + τ 3 ( y ) sin ( d ( y ) ( x + e ( y ) ) ) + τ 4 ( y ) e a 2 ,

(3.19) Γ 4 = M 4 + M 1 + M 2 + M 3 + M 5 M 6 N 1 ( N 5 + N 6 + N 4 + N 3 + N 2 ) M 6 2 ,

S 1 = sin ( d ( y ) ( x + e ( y ) ) ) , S 2 = cos ( d ( y ) ( x + e ( y ) ) ) ,

M 1 = S 1 d d y τ 2 ( y ) d ( y ) + τ 2 ( y ) d d y d ( y ) + S 2 τ 3 ( y ) d d y d ( y ) + d ( y ) d d y τ 3 ( y ) ,

M 2 = τ 2 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) ,

M 3 = τ 3 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) d ( y ) ,

M 4 = e a 1 m 1 τ 1 ( y ) 2 d ( y ) d d y d ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) ,

M 5 = e a 2 m 2 2 d ( y ) d d y d ( y ) τ 4 ( y ) t + τ 4 ( y ) d d y n 2 ( y ) + d d y τ 4 ( y ) ,

M 6 = τ 1 ( y ) e a 1 + τ 2 ( y ) S 2 + τ 3 ( y ) S 1 + τ 4 ( y ) e a 2 ,

N 1 = τ 1 ( y ) m 1 e a 1 τ 2 ( y ) S 1 d ( y ) + τ 3 ( y ) S 2 d ( y ) + τ 4 ( y ) m 2 e a 2 ,

N 2 = e a 2 2 d ( y ) d d y d ( y ) τ 4 ( y ) t + τ 4 ( y ) d d y n 2 ( y ) + d d y τ 4 ( y ) ,

N 3 = τ 3 ( y ) S 2 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) ,

N 4 = τ 2 ( y ) S 1 d d y d ( y ) ( x + e ( y ) ) + d ( y ) d d y e ( y ) ,

N 5 = e a 1 τ 1 ( y ) 2 d ( y ) d d y d ( y ) t + τ 1 ( y ) d d y n 1 ( y ) + d d y τ 1 ( y ) ,

N 6 = d d y τ 2 ( y ) S 2 + d d y τ 3 ( y ) S 1 .

a 1 = m l x + n l ( y ) + ( ( d ( y ) ) 2 + m l 2 ) t , l = 1 , 2 .

Figures 7, 8, 9 show the analysis of the treatment of extended homoclinic (heteroclinic) breather wave solution with two-wave solution, where graphs of Γ 4 are given with the following selected parameters:

(3.20) m 1 = 1 , m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , τ 4 ( y ) = 2 , n 1 ( y ) = y , n 2 ( y ) = 3 y , d ( y ) = 1 , e ( y ) = ln ( 1 + y 2 ) ,

(3.21) m 1 = 1 , m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , τ 4 ( y ) = 2 , n 1 ( y ) = y , n 2 ( y ) = 3 y , d ( y ) = 2 1 + y 2 , e ( y ) = y + y 3 ,

(3.22) m 1 = 1 , m 2 = 2 , τ 1 ( y ) = 1 , τ 2 ( y ) = 2 , τ 3 ( y ) = 3 , τ 4 ( y ) = 2 , n 1 ( y ) = y , n 2 ( y ) = 3 y , d ( y ) = cos ( y ) , e ( y ) = y + y 3 ,

given in Eq. (3.4). By the aforementioned parameters, the structural property homoclinic breather wave solutions are presented in Figures 79 with the density plot with different times. It shows a type of interaction solutions between exponential and trigonometric waves.

Figure 7 
                     Plot of the extended homoclinic breather wave solution (3.20) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       4
                                    
                                 
                              
                              {\Gamma }_{4}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 7

Plot of the extended homoclinic breather wave solution (3.20) ( Γ 4 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 8 
                     Plot of the extended homoclinic breather wave solution (3.21) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       4
                                    
                                 
                              
                              {\Gamma }_{4}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 8

Plot of the extended homoclinic breather wave solution (3.21) ( Γ 4 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

Figure 9 
                     Plot of the extended homoclinic breather wave solution (3.22) (
                           
                              
                              
                                 
                                    
                                       Γ
                                    
                                    
                                       4
                                    
                                 
                              
                              {\Gamma }_{4}
                           
                        ) such as density plot for four cases 
                           
                              
                              
                                 t
                                 =
                                 0
                                 ,
                                 t
                                 =
                                 0.1
                                 ,
                                 0.5
                                 ,
                                 1
                              
                              t=0,t=0.1,0.5,1
                           
                        , respectively.
Figure 9

Plot of the extended homoclinic breather wave solution (3.22) ( Γ 4 ) such as density plot for four cases t = 0 , t = 0.1 , 0.5 , 1 , respectively.

3.2 Interaction periodic and dark soliton

Here, we utilize to formulate the new exact solutions to the ( 2 + 1 )-dimensional MDWW system. Consider the following function for studying the interaction periodic and dark soliton solution:

(3.23) h = τ 1 ( y ) e a 1 + τ 2 ( y ) tan ( a 2 ) + τ 3 ( y ) tanh ( a 3 ) + τ 4 ( y ) e a 4 ,

a l = m l x + n l ( y ) + p l ( y ) t , l = 1 , 2 , 3 , 4 .

Afterward, the values m l , n l ( y ) , p l ( y ) , and ( l = 1 , 2 , 3 , 4 ) will be found. By substituting Eq. (3.23) into (2.15) and taking the coefficients of the power of each exp ( a l ) , l = 1 , 2 and tan ( a 2 ) , tanh ( a 3 ) to be zero, we yield a system of equations (algebraic) (these are not collected here for minimalist) for m l , n l ( y ) , p l ( y ) , ( l = 1 , 2 , 3 , 4 ) . These algebraic equations with the help of the emblematic computation software like, Maple, give the solutions by using Σ = ( ln h ) x and Γ = ( ln h ) x y :

Σ = τ 1 ( y ) m 1 e a 1 + τ 2 (