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Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice

  • Syed T. R. Rizvi , Aly R. Seadawy EMAIL logo and Samia Ahmed
Published/Copyright: December 7, 2023
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Nonlinear Engineering
From the journal Nonlinear Engineering Volume 12 Issue 1

Abstract

In this article, the equation showing the cold bosonic atoms in a zig-zag optical lattice model for some breathers, M-shaped solution and lump soliton solution, homoclinic breather pulses, breather lump pulses, periodic-cross kink wave, kink cross-rational propagation, and interaction between lump periodic and kink wave was examined. Some M-shaped solution, M-shaped interaction with periodic and kink, M-shaped interaction with rogue and kink, M-shaped rational solution, M-shaped rational solution with one kink, M-shaped rational solution with two kink, solutions for lump soliton waves, lump one kink waves, lump two kink waves, periodic-cross lump wave propagation, periodic wave propagation, rogue wave propagation, and multiwave propagation were also acquired. Likewise, our solution was also graphically presented, and also their stability was checked.

Keywords: breathers; homoclinic breather; rogue wave; lump soliton; M-shaped solution; multiwave

1 Introduction

Researcher’s attention has been drawn to breathers that are solitary waves. There are two types of breathers: standing breathers and travelling breathers. A discrete solution that changes in amplitude over time is the standing breather [1–4]. If the main frequency of the breather and all of its multipliers are outside of the phonon spectrum of the lattice, then breathers must exist in discrete lattices. Different manifestations of nonlinear waves such as breathers and periodic and kink solitons have been predicated theoretically using experimental findings in nonlinear optics, fluids, plasmas, and other areas [5–9]. Breathers, lump soliton, M-shaped solutions, and their interactions with others grap interest of many researches [10–16]. Rizvi et al. were able to find solutions for breathers, periodic-cross kink, multi-wave, and M-shaped interactions problems for cold bosonic atoms in a zig zag optical lattice (CBAZZ) [17]. Additionally, they obtained certain lump soliton, lump periodic, and lump-multisoliton solutions [18]. Seadawy et al. found some solutions for multiwave, rogue wave, periodic wave, kink wave, homoclinic breather, and some nonlinearities [19]. Ahmed et al. found some lump soliton, breather, and rogue wave solutions for the nonlinear chain of atoms [20].

In this article, the Bose–Hubbard model has been used to analyse the energy spectrum and inherent localized modes associated with modulation instability of boson chains. In Heisenberg ferromagnetic spin chains, the author used an oblique magnetic field to control the quantum breathers. The following equation governs the CBAZZ phenomenon [21]:

(1) i χ t − ( s − 2 A 1 − 2 A 2 ) χ − d 2 ( A 1 + 4 A 2 ) χ x x + v ( m − 1 ) ∣ χ 2 ∣ χ ,

where u represents the boson–boson interaction. When u > 0 , this interaction is repulsive and u < 0 interaction is attractive. A 1 and A 2 denote the first nearest and second nearest hopping, respectively. To solve Eq. (1), we use the following transformation:

(2) χ ( x , t ) = F ( μ ) e i ( c x + w t ) ,

where μ = g x + h t .

Substitute Eq. (2) into Eq. (1). We have real and imaginary parts, respectively,

(3) [ c 2 d 2 ( A 1 + 4 A 2 ) + w + s − 2 ( A 1 + A 2 ) ] F + v ( m − 1 ) F 3 + g 2 d 2 ( A 1 + 4 A 2 ) F ′ ′ = 0 ,

(4) [ − h + 2 c g d 2 ( A 1 + 4 A 2 ) ] F ′ = 0 .

Now, we execute the following transformation for different solutions [22]:

(5) F = 2 ( log p ) μ ,

(6) 2 A 1 c 2 d 2 p 2 p ′ + 2 A 1 d 2 g 2 p 2 p ′ ′ ′ + 4 A 1 d 2 g 2 p ′ 3 − 6 A 1 d 2 g 2 p p ′ p ″ − 4 A 1 p 2 p ′ + 8 A 2 c 2 d 2 p 2 p ′ + 8 A 2 d 2 g 2 p 2 p ′ ′ ′ + 16 A 2 d 2 g 2 p ′ 3 − 24 A 2 d 2 g 2 p p ′ p ″ − 4 A 2 p 2 p ′ + 8 m v p ′ 3 + 2 s p 2 p ′ − 8 v p ′ 3 + 2 w p 2 p ′ .

Using Eq. (6), study the following wave solutions.

2 Homoclinic breather pulses

For homoclinic breather pulses, we use the following ansatz [23]:

(7) p = e − r ( n 1 μ + n 2 ) + q 1 e n ( r 3 μ + r 4 ) + q 2 cos ( r 1 ( n 5 μ + n 6 ) ) .

Substituting Eq. (7) into Eq. (6), then we obtain some values for solution:

(8) n 1 = − c 2 d 2 A 1 + 4 c 2 d 2 A 2 − 2 A 1 − 2 A 2 + w + s A 1 + 4 A 2 d g r , n 3 = − − 2 c 2 d 2 A 1 − 8 c 2 d 2 A 2 + 4 A 1 + 4 A 2 − 2 w − 2 s A 1 + 4 A 2 d g r , r 1 = 0 .

For finding the solution for χ in Eq. (2), substituting Eq. (8) into Eq. (7), and then inserting into Eq. (5), Eq. (2) becomes

(9) χ 1 ( x , t ) = 2 e i ( c x + t w ) q 1 2 c 2 d 2 A 1 + 8 c 2 d 2 A 2 − 4 A 1 − 4 A 2 + 2 w + 2 s A 1 + 4 A 2 e Γ 1 d g − − c 2 d 2 A 1 − 4 c 2 d 2 A 2 + 2 A 1 + 2 A 2 − w − s A 1 + 4 A 2 e Γ 2 d g e Γ 2 + q 1 e Γ 1 + q 2 ,

where Γ 1 = r ( t h + g x ) 2 c 2 d 2 A 1 + 8 c 2 d 2 A 2 − 4 A 1 − 4 A 2 + 2 w + 2 s A 1 + 4 A 2 d g r + n 4 and Γ 2 = − r ( t h + g x ) − c 2 d 2 A 1 − 4 c 2 d 2 A 2 + 2 A 1 + 2 A 2 − w − s A 1 + 4 A 2 d g r + n 2 .

3 Periodic-cross rational waves

For periodic-cross rational waves, we use the given ansatz [24]:

(10) p = a 1 2 + a 2 2 + q 1 cos ( K 1 ) + q 2 cosh ( K 2 ) + r 5 , a 1 = r 1 μ + r 2 , a 2 = r 3 μ + r 4 , K 1 = n 1 μ + n 2 , and K 2 = n 3 μ + n 4 ,

Substituting Eq. (10) into Eq. (6), then we obtain some values for solution:

(11) r 2 = − r 3 r 4 r 1 , n 3 = − A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w A 1 + 4 A 2 d g .

Substituting Eq. (11) into Eq. (10) and then inserting into Eq. (5), Eq. (2) becomes

(12) χ 2 ( x , t ) = 2 e i ( c x + t w ) Π − n 1 q 1 sin ( Θ 2 ) + 2 r 1 r 1 ( g x + h t ) − r 3 r 4 r 1 + 2 r 3 ( r 3 ( g x + h t ) + r 4 ) q 2 cosh ( Θ 1 ) + q 1 cos ( Θ 2 ) + r 1 ( g x + h t ) − r 3 r 4 r 1 2 + ( r 3 ( g x + h t ) + r 4 ) 2 + r 5 ,

where Π = q 2 − A 1 c 2 d 2 + 2 A 1 − 4 A 2 c 2 d 2 + 2 A 2 − s − w A 1 + 4 A 2 sinh ( Θ 1 ) d g , Θ 1 = ( g x + h t ) − A 1 c 2 d 2 + 2 A 1 − 4 A 2 c 2 d 2 + 2 A 2 − s − w A 1 + 4 A 2 d g + n 4 and Θ 2 = n 1 ( g x + h t ) + n 2 .

4 Kink cross-rational propagation

For the kink cross-rational propagation, we apply the given p [24]:

(13) p = e − K 1 + q 1 e K 1 + a 1 2 + a 2 2 + r 5 , a 1 = r 1 μ + r 2 , μ 2 = r 3 μ + r 4 , and K 1 = n 1 μ + n 2 ,

Substituting Eq. (13) into Eq. (6), then we obtain some values for solution:

(14) r 5 = − A 1 c 2 d 2 r 4 2 − 3 A 1 d 2 g 2 r 1 2 − 2 A 1 r 4 2 + 4 A 2 c 2 d 2 r 4 2 − 12 A 2 d 2 g 2 r 1 2 − 2 A 2 r 4 2 + r 4 2 s + r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w , n 1 = 0 , r 2 = 0 , and r 3 = 0 .

Substituting Eq. (14) into Eq. (13) and then inserting into Eq. (5), then Eq. (2) becomes

(15) χ 3 ( x , t ) = 4 r 1 2 e i ( c x + t w ) ( g x + h t ) ϒ + r 1 2 ( g x + h t ) 2 + e n 2 q 1 + e − n 2 + r 4 2 ,

where ϒ = − A 1 c 2 d 2 r 4 2 + 3 A 1 d 2 g 2 r 1 2 + 2 A 1 r 4 2 − 4 A 2 c 2 d 2 r 4 2 + 12 A 2 d 2 g 2 r 1 2 + 2 A 2 r 4 2 − r 4 2 s − r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w .

5 M-shaped interaction with periodic and kink (M-SPK)

For M-SPK, we study the given transformation [22]:

(16) p = a 1 2 + a 2 2 + + q 1 cos ( K 1 ) + q 2 e K 2 + r 5 , a 1 = r 1 μ + r 2 , a 2 = r 3 μ + r 4 , K 1 = n 1 μ + n 2 , and K 2 = n 3 μ + n 4 .

Substituting Eq. (16) into Eq. (6), then we obtain some values for solution:

(17) n 1 = − − A 1 c 2 d 2 + 2 A 1 − 4 A 2 c 2 d 2 + 2 A 2 − s − w A 1 + 4 A 2 d g , r 2 = − r 3 r 4 r 1 .

Substituting Eq. (17) into Eq. (16) and then inserting into Eq. (5), then Eq. (2) becomes

(18) χ 4 ( x , t ) = 2 e i ( c x + t w ) Δ + n 3 q 2 e n 3 ( g x + h t ) + n 4 + 2 r 1 r 1 ( g x + h t ) − r 3 r 4 r 1 + 2 r 3 ( r 3 ( g x + h t ) + r 4 ) q 1 cos ( Ω ) + q 2 e n 3 ( g x + h t ) + n 4 + r 1 ( g x + h t ) − r 3 r 4 r 1 2 + ( r 3 ( g x + h t ) + r 4 ) 2 + r 5 ,

where Δ = − q 1 A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w A 1 + 4 A 2 sin ( Ω ) d g , Ω = ( g x + h t ) A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w A 1 + 4 A 2 d g + n 2 .

6 M-shaped interaction with rogue and kink (M-SRK)

For M-SRK, we consider the following p [23]:

(19) p = a 1 2 + a 2 2 + + q 1 cosh ( K 1 ) + q 2 e K 2 + r 5 , a 1 = r 1 μ + r 2 , a 2 = r 3 μ + r 4 , K 1 = n 1 μ + n 2 , and K 2 = n 3 μ + n 4 .

Substituting Eq. (19) into Eq. (6), then we obtain some values for solution:

(20) n 1 = − A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w A 1 + 4 A 2 d g , r 2 = − r 3 r 4 r 1 .

Substituting Eq. (20) into Eq. (19) and then inserting into Eq. (5), then Eq. (2) becomes

(21) χ 5 ( x , t ) = 2 e i ( c x + t w ) ξ + n 3 q 2 e n 3 ( g x + h t ) + n 4 + 2 r 1 r 1 ( g x + h t ) − r 3 r 4 r 1 + 2 r 3 ( r 3 ( g x + h t ) + r 4 ) q 1 cosh ( θ ) + q 2 e n 3 ( g x + h t ) + n 4 + r 1 ( g x + h t ) − r 3 r 4 r 1 2 + ( r 3 ( g x + h t ) + r 4 ) 2 + r 5 ,

where ξ = q 1 − A 1 c 2 d 2 + 2 A 1 − 4 A 2 c 2 d 2 + 2 A 2 − s − w A 1 + 4 A 2 sinh ( θ ) d g , θ = ( g x + h t ) − A 1 c 2 d 2 + 2 A 1 − 4 A 2 c 2 d 2 + 2 A 2 − s − w A 1 + 4 A 2 d g + n 2 .

7 M-shaped rational solution (M-SRS)

For M-SRS, we use the following transformation [24]:

(22) p = a 1 2 + a 2 2 + r 5 , a 1 = r 1 μ + r 2 , and a 2 = r 3 μ + r 4 .

Substituting Eq. (22) into Eq. (6), then we obtain some values for solution:

(23) r 2 = − r 3 r 4 r 1 , and r 5 = − r 4 2 ( r 1 2 + r 3 2 ) r 1 2 .

Substituting Eq. (23) into Eq. (22) and then inserting into Eq. (5), then Eq. (2) becomes

(24) χ 6 ( x , t ) = 2 e i ( c x + t w ) 2 r 1 r 1 ( g x + h t ) − r 3 r 4 r 1 + 2 r 3 ( r 3 ( g x + h t ) + r 4 ) r 1 ( g x + h t ) − r 3 r 4 r 1 2 + ( r 3 ( g x + h t ) + r 4 ) 2 − r 4 2 ( r 1 2 + r 3 2 ) r 1 2 .

8 M-shaped rational solution with one kink (M-SR1K)

For M-SR1K, we assume the given ansatz [24]:

(25) p = a 1 2 + a 2 2 + q 1 e K 1 + r 5 , a 1 = r 1 μ + r 2 , a 2 = r 3 μ + r 4 , and K 1 = n 1 μ + n 2 .

Substituting Eq. (25) into Eq. (6), then we obtain some values for solution:

(26) r 5 = − A 1 c 2 d 2 r4 2 − 3 A 1 d 2 g 2 r 1 2 − 2 A 1 r 4 2 + 4 A 2 c 2 d 2 r 4 2 − 12 A 2 d 2 g 2 r 1 2 − 2 A 2 r 4 2 + r 4 2 s + r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w , n 1 = r 2 = r 3 = 0 .

Substituting Eq. (26) into Eq. (25) and then insering into Eq. (5), then Eq. (2) becomes

(27) χ 7 ( x , t ) = 4 r 1 2 e i ( c x + t w ) ( g x + h t ) − A 1 c 2 d 2 r 4 2 + 3 A 1 d 2 g 2 r 1 2 + 2 A 1 r 4 2 − 4 A 2 c 2 d 2 r 4 2 + 12 A 2 d 2 g 2 r 1 2 + 2 A 2 r 4 2 − r 4 2 s − r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w + r 1 2 ( g x + h t ) 2 + e n 2 q 1 + r 4 2 .

9 M-shaped rational solution with two kinks (M-SR2K)

For M-SR2K, we suppose the following p [24]:

(28) p = a 1 2 + a 2 2 + q 1 e K 1 + q 2 e K 2 + r 5 , a 1 = r 1 μ + r 2 , a 2 = r 3 μ + r 4 , K 1 = n 1 μ + n 2 , and K 2 = n 3 μ + r 4 .

Substituting Eq. (28) into Eq. (6) then we obtain some values for solution:

(29) r 2 = r 3 = n 3 = 0 , r 5 = − A 1 c 2 d 2 r 4 2 − 3 A 1 d 2 g 2 r 1 2 − 2 A 1 r 4 2 + 4 A 2 c 2 d 2 r 4 2 − 12 A 2 d 2 g 2 r 1 2 − 2 A 2 r 4 2 + r 4 2 s + r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w .

Substituting Eq. (29) into Eq. (28) and then inserting into Eq. (5), then Eq. (2) becomes

(30) χ 8 ( x , t ) = 2 e i ( c x + t w ) ( n 1 q 1 e n 1 ( g x + h t ) + n 2 + 2 r 1 2 ( g x + h t ) ) Ψ + q 1 e n 1 ( g x + h t ) + n 2 + r 1 2 ( g x + h t ) 2 + e n 4 q 2 + r 4 2 ,

where Ψ = − A 1 c 2 d 2 r 4 2 + 3 A 1 d 2 g 2 r 1 2 + 2 A 1 r 4 2 − 4 A 2 c 2 d 2 r 4 2 + 12 A 2 d 2 g 2 r 1 2 + 2 A 2 r 4 2 − r 4 2 s − r 4 2 w A 1 c 2 d 2 − 2 A 1 + 4 A 2 c 2 d 2 − 2 A 2 + s + w .

10 Stability

Now, we find out the stability using the Hamiltonian method [25]:

(31) ϒ = 1 2 ∫ − c c χ 2 ( x ) d x .

Now, we verify the stability as:

(32) ∂ Γ ∂ C > 0 ,

where ϒ and C represent the momentum and velocity, respectively (Table 1).

Table 1

Stability χ i ( x , t ) , where ( i = 1 , 2 , 3 , … , 8 . )

Solution Stability Values of variables
χ 1 ( x , t ) Stable c = 2.5 , d = ‒ 4.2 , g = 0.3 , A 1 = 1.5 , A 2 = 1.2 , n 2 = 3.5 , n 4 = ‒ 0.6 , q 1 = ‒ 1.5 , q 2 = ‒ 2 , r = ‒ 0.7 , w = ‒ 0.2 , h = 3.3 , s = 1.05
χ 2 ( x , t ) Stable A 1 = 13 , A 2 = 7.5 , c = ‒ 0.3 , d = 1.5 , g = 0.01 , h = 3 , n 1 = 10 , n 2 = ‒ 0.2 , n 4 = 1.4 , q 1 = 0.1 , q 2 = 2 , r 1 = 2.5 , r 3 = 3 , r 4 = 8.5 , r 5 = 0.05 , s = 0.9 , w = 0.02
χ 3 ( x , t ) Stable A 1 = 0.05 , A 2 = 4.5 , c = 0.02 , d = 0.2 , g = 2.7 , h = 1.2 , n 2 = 1.5 , q 1 = 1.5 , r 1 = ‒ 1.5 , r 4 = 0.04 , s = ‒ 3.3 , w = 1.2
χ 4 ( x , t ) Stable A 1 = 2.7 , A 2 = 6 , c = 1 , d = 0.70 , g = 2.5 , h = 5.2 , n 2 = 0.5 , n 3 = 5 , n 4 = 3 , q 1 = 4.5 , q 2 = 1.7 , r 1 = 0.5 , r 3 = 0.3 , r 4 = ‒ 1.04 , r 5 = 1.5 , s = 1.09 , w = 0.2
χ 5 ( x , t ) Stable A 1 = 2.05 , A 2 = 2 , c = 1.4 , d = 0.7 , g = 1.5 , h = 2 , n 2 = 0.05 , n 3 = ‒ 0.5 , n 4 = 0.4 , q 1 = 1.5 , q 2 = 1 , r 1 = 5 , r 3 = 3.3 , r 4 = 1.4 , r 5 = 0.5 , s = 1.9 , w = ‒ 0.5
χ 6 ( x , t ) Unstable c = 7 , g = 0.3 , h = 0.1 , r 1 = 5 , r 3 = 0.4 , r 4 = 6 , w = 0.5
χ 7 ( x , t ) Stable A 1 = 3 , A 2 = 2.5 , c = 2 , d = 1.1 , g = 0.3 , h = 0.5 , n 2 = 2.2 , q 1 = 0.1 , r 1 = 1.5 , r 4 = 3 , s = 0.2 , w = ‒ 2
χ 8 ( x , t ) Stable A 1 = 1.3 , A 2 = 2 , c = 3.5 , d = 1.5 , g = 0.1 , h = 5.5 , n 1 = 1.6 , n 2 = 0.2 , n 4 = 4.2 , q 1 = 0.5 , q 2 = 8 , r 1 = 7.5 , r 4 = 0.4 , s = 0.7 , w = ‒ 1.2

11 Transformation for traveling wave solution

To solve Eq. (1), the following transformation is given as [26]:

(33) χ ( x , t ) = Q ( x , t ) e i ϕ ( x , t ) ,

where ϕ ( x , t ) = − b x + k t + γ .

Substituting Eq. (33) into Eq. (1), we obtain real and imaginary part, respectively, as:

(34) ( k − b 2 + s − 2 A 1 − 2 A 2 ) Q + d 2 ( A 1 + 4 A 2 ) Q x x − v ( m − 1 ) Q 3 = 0 ,

(35) Q t + 2 b d 2 ( A 1 + 4 A 2 ) Q x = 0 .

Now, we find different solutions using the following transformation:

(36) Q ( x , t ) = 2 u ( ln p ) x ,

By substituting Eq. (36) into Eq. (34) and Eq. (35), we obtain the following bilinear forms, respectively:

(37) 4 A 1 d 2 u p x 3 − 6 A 1 d 2 u p p x x p x + 2 A 1 d 2 u p 2 p x x x − 4 A 1 u p 2 p x + 16 A 2 d 2 u p x 3 − 24 A 2 d 2 u p p x x p x + 8 A 2 d 2 u p 2 p x x x − 4 A 2 u p 2 p x + 2 b 2 u p 2 p x + 2 k u p 2 p x − 8 m u 3 v p x 3 + 2 s u p 2 p x + 8 u 3 v p x 3 ,

(38) − 4 A 1 b d 2 u p x 2 + 4 A 1 b d 2 u p p x x − 16 A 2 b d 2 u p x 2 + 16 A 2 b d 2 u p p x x − 2 u p t ( x , t ) p x + 2 u p p x t .

12 Breather lump pulses

For breather lump pulses, we use the given ansatz [26]:

(39) P = e − l V 1 + w 1 e l V 1 + w 2 cos ( l 1 Λ 2 ) + r 6 ,

where

V 1 = r 1 x + r 2 t + r 3 and V 2 = r 4 x + r 5 t .

Insert Eq. (39) into Eq. (37) and Eq. (38). After inserting Eq. (39) in both equations, we obtain some values of the parameters, which are given as:

(40) r 1 = − 2 A 1 + 2 A 2 − b 2 − k − s 4 m v − 4 v l u , r 2 = − 2 b d 2 ( A 1 + 4 A 2 ) − 2 A 1 + 2 A 2 − b 2 − k − s 4 m v − 4 v l u , r 5 = − 2 A 1 b d 2 r 4 − 8 A 2 b d 2 r 4 , r 6 = 0 , w 1 = 0 .

Substitute Eq. (40) into Eq. (39) and then in Eq. (36) to have solution of Eq. (33):

(41) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) − − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v e Ψ u − l 1 r 4 w 2 sin ( l 1 ( t ( − 2 A 1 b d 2 r 4 − 8 A 2 b d 2 r 4 ) + r 4 x ) ) e Ψ + w 2 cos ( l 1 ( t ( − 2 A 1 b d 2 r 4 − 8 A 2 b d 2 r 4 ) + r 4 x ) ) ,

where

Ψ = − l − 2 b d 2 t ( A 1 + 4 A 2 ) − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v l u + x − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v l u + r 3 .

13 Periodic wave propagation

For periodic wave propagation, we consider the following ansatz [26]:

(42) p = V 1 2 + V 2 2 + n 1 cos ( λ ) + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , and λ = z 1 x + z 2 t .

Insert Eq. (42) into Eq. (37) and Eq. (38). After inserting Eq. (42) in both equations, we obtain some values of the parameters, which are given as:

(43) r 1 = − 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 d , r 2 = − d r 4 r 5 − 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 .

Substitute Eq. (43) into Eq. (42) and then in Eq. (36) to have solution of Eq. (33):

(44) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) 2 − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 ( ς ) d + 2 r 4 ( r 4 x + r 5 t + r 6 ) − w 1 z 1 sin ( t z 2 + x z 1 ) ( ς ) 2 + ( r 4 x + r 5 t + r 6 ) 2 + r 7 + w 1 cos ( t z 2 + x z 1 ) ,

where

ς = − d r 4 r 5 t − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 + x − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 d + r 3 .

14 Periodic-cross kink waves

For periodic-cross kink wave, we use the given ansatz [27]:

(45) p = e − V 1 + z 1 e V 1 + z 2 cos ( V 2 ) + z 3 cosh ( V 3 ) + r 8 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t , V 3 = r 6 x + r 7 t .

Insert Eq. (45) into Eq. (37) and Eq. (38). After inserting Eq. (45) in both equations, we obtain some values of the parameters, which are given as:

(46) r 1 = − 2 A 1 + 2 A 2 − b 2 − k − s 4 m v − 4 v u , r 2 = b d 2 ( 2 A 1 2 + 10 A 1 A 2 − A 1 b 2 − A 1 k − A 1 s + 8 A 2 2 − 4 A 2 b 2 − 4 A 2 k − 4 A 2 s ) 2 ( m − 1 ) u v − 2 A 1 + 2 A 2 − b 2 − k − s 4 m v − 4 v , r 4 = − 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 d , r 5 = − 2 A 1 b d − 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 − 8 A 2 b d − 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 , r 6 = 0 , r 8 = 0 , and z 1 = 0 .

Substitute Eq. (46) into Eq. (45) and then in Eq. (36) to have solution of Eq. (33):

(47) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) − − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v e Λ u − z 2 − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 sin ( Ω ) d e Λ + z 2 cos ( Ω ) + z 3 cosh ( r 7 t ) ,

where

Λ = − b d 2 t ( 2 A 1 2 + 10 A 1 A 2 − A 1 b 2 − A 1 k − A 1 s + 8 A 2 2 − 4 A 2 b 2 − 4 A 2 k − 4 A 2 s ) 2 ( m − 1 ) u v − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v − x − 2 A 1 − 2 A 2 + b 2 + k + s 4 m v − 4 v u − r 3 ,

Ω = t − 2 A 1 b d − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 − 8 A 2 b d − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 + x − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 d .

15 Periodic-cross lump wave propagation

For periodic-cross lump wave propagation, we consider the following transformation [27]:

(48) g = V 1 2 + V 2 2 + w 1 cos ( C 1 ) + w 2 cosh ( C 2 ) + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , C 1 = z 1 x + z 2 t , and C 2 = z 3 x + z 4 t ,

Insert Eq. (48) into Eq. (37) and Eq. (38). After inserting Eq. (48) in both equations, we obtain some values of the parameters, which are given as:

(49) r 2 = − r 5 r 1 and z 3 = − − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 d .

Substitute Eq. (49) into Eq. (48) and then in Eq. (36) to have solution of Eq. (33):

(50) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) w 2 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 sinh ( Φ 1 ) d + 2 r 1 ( τ ) + 2 r 4 ( r 4 x + r 5 t + r 6 ) − w 1 z 1 sin ( Φ 2 ) w 2 cosh ( Φ 1 ) + ( τ ) 2 + ( r 4 x + r 5 t + r 6 ) 2 + r 7 + w 1 cos ( Φ 2 ) ,

where Φ 1 = x 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 d + t z 4 , Φ 2 = t z 2 + x z 1 and τ = − r 4 r 5 t r 1 + r 1 x + r 3 .

16 Interaction between lump periodic and kink pulses

For interaction between lump periodic and kink pulses, we assume the given ansatz [27]:

(51) P = V 1 2 + V 2 2 + w 1 e C 1 + w 2 cos ( λ ) + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , C 1 = z 1 x + z 2 t , and λ = j 1 x + j 2 t .

Insert Eq. (51) into Eqs (37) and (38). After inserting Eq. (51) in both equations we obtain some values of the parameters, which are given as:

(52) r 7 = 3 ( 3 A 1 d 2 r 1 2 + 3 A 1 d 2 r 4 2 + 12 A 2 d 2 r 1 2 + 12 A 2 d 2 r 4 2 − 8 m r 1 2 u 2 v − 8 m r 4 2 u 2 v + 8 r 1 2 u 2 v + 8 r 4 2 u 2 v ) A 1 d 2 j 1 2 + 2 A 1 + 4 A 2 d 2 j 1 2 + 2 A 2 − b 2 − k − s , r 3 = 0 , r 6 = 0 .

Substitute Eq. (52) into Eq. (51) and then in Eq. (36) to have solution of Eq. (33):

(53) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) ( − j 1 w 2 sin ( j 1 x + j 2 t ) + 2 r 1 ( r 1 x + r 2 t ) + 2 r 4 ( r 4 x + r 5 t ) + w 1 z 1 e t z 2 + x z 1 ) ρ + w 2 cos ( j 1 x + j 2 t ) + ( r 1 x + r 2 t ) 2 + ( r 4 x + r 5 t ) 2 + w 1 e t z 2 + x z 1 ,

where

ρ = 3 ( 3 A 1 d 2 r 1 2 + 3 A 1 d 2 r 4 2 + 12 A 2 d 2 r 1 2 + 12 A 2 d 2 r 4 2 − 8 m r 1 2 u 2 v − 8 m r 4 2 u 2 v + 8 r 1 2 u 2 v + 8 r 4 2 u 2 v ) A 1 d 2 j 1 2 + 2 A 1 + 4 A 2 d 2 j 1 2 + 2 A 2 − b 2 − k − s .

17 Rogue wave propagation

For rogue wave propagation, we suppose that [28]:

(54) p = V 1 2 + V 2 2 + w 1 cosh ( λ ) + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , and λ = z 1 x + z 2 t .

Insert Eq. (54) into Eqs (37) and (38). After inserting Eq. (54) into both equations, we obtain some values of the parameters, which are given as:

(55) r 1 = − r 4 r 5 r 2 , z 1 = − − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 d , z 2 = − 2 A 1 b d − − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 − 8 A 2 b d − − 2 A 1 − 2 A 2 + b 2 + k + s A 1 + 4 A 2 .

Substitute Eq. (55) into Eq. (54) and then in Eq. (36) to have solution of Eq. (33):

(56) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) w 1 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 sinh ( Π ) d − 2 r 4 r 5 − r 4 r 5 x r 2 + r 2 t + r 3 r 2 + 2 r 4 ( r 4 x + r 5 t + r6 ) w 1 cosh ( Π ) + − r 4 r 5 x r 2 + r 2 t + r 3 2 + ( r 4 x + r 5 t + r 6 ) 2 + r 7 ,

where

Π = t − 2 A 1 b d 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 − 8 A 2 b d 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 + x 2 A 1 + 2 A 2 − b 2 − k − s A 1 + 4 A 2 d .

18 Multiwave propagation

For multiwave propagation, we use the following p [29]:

(57) p = ν 0 cosh ( λ 1 ) + ν 1 cos ( λ 2 ) + ν 2 cosh ( λ 3 ) + r 10 ,

where

λ 1 = r 1 x + r 2 t + r 3 , λ 2 = r 4 x + r 5 t + r 6 , and λ 3 = r 7 x + r 8 t + r 9 .

Insert Eq. (57) into Eq. (37) and Eq. (38). After inserting Eq. (57) in both equations, we obtain some values of the parameters, which are given as:

(58) r 1 = − 2 A 1 + 2 A 2 − b 2 − k − s 2 A 1 + 8 A 2 d , r 7 = − 2 A 1 + 2 A 2 − b 2 − k − s 2 A 1 + 8 A 2 d .

Substitute Eq. (58) into Eq. (57) and then in Eq. (36) to have solution of Eq. (33):

(59) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) ν 0 − 2 A 1 − 2 A 2 + b 2 + k + s 2 A 1 + 8 A 2 sinh x − 2 A 1 − 2 A 2 + b 2 + k + s 2 A 1 + 8 A 2 d + r 2 t + r 3 d − ν 1 r 4 sin ( r 4 x + r 5 t + r 6 ) ν 0 cosh x − 2 A 1 − 2 A 2 + b 2 + k + s 2 A 1 + 8 A 2 d + r 2 t + r 3 + r 10 + ν 1 cos ( r 4 x + r 5 t + r 6 ) .

19 Lump soliton waves

For lump soliton waves, we use the given ansatz [26]:

(60) p = V 1 2 + V 2 2 + r 7 ,

V 1 = r 1 x + r 2 t + r 3 , and V 2 = r 4 x + r 5 t + r 6 ,

Insert Eq. (60) into Eq. (37) and Eq. (38). After inserting Eq. (60) in both equations, we obtain some values of the parameters, which are given as:

(61) r 2 = 4 b d 2 r 1 r 4 2 ( A 1 + 4 A 2 ) 9 r 1 2 + r 4 2 , r 5 = − 2 b d 2 r 4 ( A 1 + 4 A 2 ) ( 3 r 1 2 + r 4 2 ) 9 r 1 2 + r 4 2 , r 3 = 0 , and r 6 = 0 .

Substitute Eq. (61) into Eq. (60) and then in Eq. (36) to have solution of Eq. (33):

(62) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) 2 r 1 4 b d 2 r 1 r 4 2 t ( A 1 + 4 A 2 ) 9 r 1 2 + r 4 2 + r 1 x + 2 r 4 r 4 x − 2 b d 2 r 4 t ( A 1 + 4 A 2 ) ( 3 r 1 2 + r 4 2 ) 9 r 1 2 + r 4 2 4 b d 2 r 1 r 4 2 t ( A 1 + 4 A 2 ) 9 r 1 2 + r 4 2 + r 1 x 2 + r 4 x − 2 b d 2 r 4 t ( A 1 + 4 A 2 ) ( 3 r 1 2 + r 4 2 ) 9 r 1 2 + r 4 2 2 + r 7 .

20 Lump soliton with one kink wave

For lump soliton with one kink wave, we study the following ansatz [29]:

(63) p = V 1 2 + V 2 2 + n 1 e C 1 + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , and C 1 = z 1 x + z 2 t .

Insert Eq. (63) into Eqs (37) and (38). After inserting Eq. (63) in both equations, we obtain some values of the parameters, which are given as:

(64) r 2 = − r 4 r 5 r 1 , r 3 = r 1 ( 3 A 1 d 2 z 1 2 + 4 A 1 + 12 A 2 d 2 z 1 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 ( z 1 ( A 1 d 2 z 1 2 − 2 A 1 + 4 A 2 d 2 z 1 2 − 2 A 2 + b 2 + k + s ) ) , r 6 = r 4 ( 3 A 1 d 2 z 1 2 + 4 A 1 + 12 A 2 d 2 z 1 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 ( z 1 ( A 1 d 2 z 1 2 − 2 A 1 + 4 A 2 d 2 z 1 2 − 2 A 2 + b 2 + k + s ) ) .

Substitute Eq. (64) into Eq. (63) and then in Eq. (36) to have solution of Eq. (33):

(65) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) ( 2 r 1 ( Δ 1 ) + 2 r 4 ( Δ 2 ) + w 1 z 1 e t z 2 + x z 1 ) ( Δ 1 ) 2 + ( Δ 2 ) 2 + r 7 + w 1 e t z 2 + x z 1 ,

where

Δ 1 = r 1 ( 3 A 1 d 2 z 1 2 + 4 A 1 + 12 A 2 d 2 z 1 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 z 1 ( A 1 d 2 z 1 2 − 2 A 1 + 4 A 2 d 2 z 1 2 − 2 A 2 + b 2 + k + s ) − r 4 r 5 t r 1 + r 1 x

and

Δ 2 = r 4 ( 3 A 1 d 2 z 1 2 + 4 A 1 + 12 A 2 d 2 z 1 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 z 1 ( A 1 d 2 z 1 2 − 2 A 1 + 4 A 2 d 2 z 1 2 − 2 A 2 + b 2 + k + s ) + r 4 x + r 5 t .

21 Lump soliton with two kink waves

For lump soliton with two kink waves, we use the given transformation [30,35]:

(66) p = V 1 2 + V 2 2 + n 1 e C 1 + n 2 e C 2 + r 7 ,

where

V 1 = r 1 x + r 2 t + r 3 , V 2 = r 4 x + r 5 t + r 6 , C 1 = z 1 x + z 2 t , and C 2 = z 3 x + z 4 t ,

Insert Eq. (66) in Eqs (37) and (38). After inserting Eq. (66) in both equations, we obtain some values of the parameters, which are given as:

(67) r 2 = − r 4 r 5 r 1 , r 3 = r 1 ( 3 A 1 d 2 z 3 2 + 4 A 1 + 12 A 2 d 2 z 3 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 ( z 3 ( A 1 d 2 z 3 2 − 2 A 1 + 4 A 2 d 2 z 3 2 − 2 A 2 + b 2 + k + s ) ) , r 6 = r 4 ( 3 A 1 d 2 z 3 2 + 4 A 1 + 12 A 2 d 2 z 3 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 ( z 3 ( A 1 d 2 z 3 2 − 2 A 1 + 4 A 2 d 2 z 3 2 − 2 A 2 + b 2 + k + s ) ) .

Substitute Eq. (66) into Eq. (65) and then in Eq. (36) to have solution of Eq. (33):

(68) χ ( x , t ) = 2 u e i ( − b x + γ + k t ) ( 2 r 1 ( Ψ 1 ) + 2 r 4 ( Ψ 2 ) + w 1 z 1 e t z 2 + x z 1 + w 2 z 3 e t z 4 + x z 3 ) ( Ψ 1 ) 2 + ( Ψ 2 ) 2 + r 7 + w 1 e t z 2 + x z 1 + w 2 e t z 4 + x z 3 ,

where

Ψ 1 = r 1 ( 3 A 1 d 2 z 3 2 + 4 A 1 + 12 A 2 d 2 z 3 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 z 3 ( A 1 d 2 z 3 2 − 2 A 1 + 4 A 2 d 2 z 3 2 − 2 A 2 + b 2 + k + s ) − r 4 r 5 t r 1 + r 1 x

and

Ψ 2 = r 4 ( 3 A 1 d 2 z 3 2 + 4 A 1 + 12 A 2 d 2 z 3 2 + 4 A 2 − 2 b 2 − 2 k − 2 s ) 2 z 3 ( A 1 d 2 z 3 2 − 2 A 1 + 4 A 2 d 2 z 3 2 − 2 A 2 + b 2 + k + s ) + r 4 x + r 5 t .

22 Results and discussion

We were able to successfully construct the desired type of solutions that express wave discrepancy by choosing the appropriate values for the parameters. From Figures 1, 2, 3, 4, 5, 6, 7, 8, we presented 3D, 2D, contour, density plot, and stream plot, and in the remaining Figures 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, we have shown 3D, 2D, contour, (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot respectively. In Figure 1, we obtain bright-dark face using values c = − 0.5 , d = 2 , g = 0.3 , A 1 = 0.5 , A 2 = 0.2 , n 2 = 5 , n 4 = 6 , q 1 = 1.5 , q 2 = − 0.2 , r = 0.7 , w = 2 , h = 7 , and s = 0.1 . According to Eq. (12), periodic-cross rational solution in Figure 2 with one stripe soliton is presented. Figures 3 and 4 are bright and dark face and varies amplitude is prosecute by Eqs (15) and (18), respectively. We have M-shaped rational solution that has M-shaped bright-dark face shown in Figures 5–8. As illustrated in Figure 9 with the bright face, Eq. (41) provides the breather lump pulses. In Eq. (44) examines the behaviour of periodic wave propagation with bell-shaped form in Figure 10. Periodic-cross kink solution in Figure 11 with some bright-dark faces is indicated by Eq. (49). In Figure 12, periodic-cross lump wave propagation with bright lump is shown in Eq. (50). Some periodic and kink waves are shown in Figure 13 by using Eq. (53). Figures 15 and 16 are bright and dark faces and varies amplitude by Eqs (56) and (59). Figure 16 represent bell shaped form and various oscillations by Eq. (62). Eqs (65) and (68) examine the behaviour of a lump with one kink and two kink solitons with bright-dark faces in Figures 17 and 18.

Figure 1 Graphical representation of solution χ 1 ( x , t ) {\chi }_{1}\left(x,t) in Eq. (9) is presented via c = 2.5 c=2.5 , d = ‒ 4.2 d=&#x2012;4.2 , g = 0.3 g=0.3 , A 1 = 1.5 {A}_{1}=1.5 , A 2 = 1.2 {A}_{2}=1.2 , n 2 = 3.5 {n}_{2}=3.5 , n 4 = ‒ 0.6 {n}_{4}=&#x2012;0.6 , q 1 = ‒ 1.5 {q}_{1}=&#x2012;1.5 , q 2 = ‒ 2 {q}_{2}=&#x2012;2 , r = ‒ 0.7 r=&#x2012;0.7 , w = ‒ 0.2 w=&#x2012;0.2 , h = 3.3 h=3.3 , and s = 1.05 . s=1.05. (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.
Figure 1

Graphical representation of solution χ 1 ( x , t ) in Eq. (9) is presented via c = 2.5 , d = ‒ 4.2 , g = 0.3 , A 1 = 1.5 , A 2 = 1.2 , n 2 = 3.5 , n 4 = ‒ 0.6 , q 1 = ‒ 1.5 , q 2 = ‒ 2 , r = ‒ 0.7 , w = ‒ 0.2 , h = 3.3 , and s = 1.05 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.

Figure 2 Wave profile of solution χ 2 ( x , t ) {\chi }_{2}\left(x,t) in Eq. (12) is presented via A 1 = 13 {A}_{1}=13 , A 2 = 7.5 {A}_{2}=7.5 , c = ‒ 0.3 c=&#x2012;0.3 , d = 1.5 d=1.5 , g = 0.01 g=0.01 , h = 3 h=3 , n 1 = 10 {n}_{1}=10 , n 2 = ‒ 0.2 {n}_{2}=&#x2012;0.2 , n 4 = 1.4 {n}_{4}=1.4 , q 1 = 0.1 {q}_{1}=0.1 , q 2 = 2 {q}_{2}=2 , r 1 = 2.5 {r}_{1}=2.5 , r 3 = 3 {r}_{3}=3 , r 4 = 8.5 {r}_{4}=8.5 , r 5 = 0.05 {r}_{5}=0.05 , s = 0.9 s=0.9 , and w = 0.02 w=0.02 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.
Figure 2

Wave profile of solution χ 2 ( x , t ) in Eq. (12) is presented via A 1 = 13 , A 2 = 7.5 , c = ‒ 0.3 , d = 1.5 , g = 0.01 , h = 3 , n 1 = 10 , n 2 = ‒ 0.2 , n 4 = 1.4 , q 1 = 0.1 , q 2 = 2 , r 1 = 2.5 , r 3 = 3 , r 4 = 8.5 , r 5 = 0.05 , s = 0.9 , and w = 0.02 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.

Figure 3 Graphical shape profile of solution χ 3 ( x , t ) {\chi }_{3}\left(x,t) in Eq. (15) is presented via A 1 = 0.05 {A}_{1}=0.05 , A 2 = 4.5 {A}_{2}=4.5 , c = 0.02 c=0.02 , d = 0.2 d=0.2 , g = 2.7 g=2.7 , h = 1.2 h=1.2 , n 2 = 1.5 {n}_{2}=1.5 , q 1 = 1.5 {q}_{1}=1.5 , r 1 = ‒ 1.5 {r}_{1}=&#x2012;1.5 , r 4 = 0.04 {r}_{4}=0.04 , s = ‒ 3.3 s=&#x2012;3.3 , and w = 1.2 w=1.2 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.
Figure 3

Graphical shape profile of solution χ 3 ( x , t ) in Eq. (15) is presented via A 1 = 0.05 , A 2 = 4.5 , c = 0.02 , d = 0.2 , g = 2.7 , h = 1.2 , n 2 = 1.5 , q 1 = 1.5 , r 1 = ‒ 1.5 , r 4 = 0.04 , s = ‒ 3.3 , and w = 1.2 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.

Figure 4 Graphical shapes of solution χ 4 ( x , t ) {\chi }_{4}\left(x,t) in Eq. (18) is shown as A 1 = 2.7 {A}_{1}=2.7 , A 2 = 6 {A}_{2}=6 , c = 1 c=1 , d = 0.70 d=0.70 , g = 2.5 g=2.5 , h = 5.2 h=5.2 , n 2 = 0.5 {n}_{2}=0.5 , n 3 = 5 {n}_{3}=5 , n 4 = 3 {n}_{4}=3 , q 1 = 4.5 {q}_{1}=4.5 , q 2 = 1.7 {q}_{2}=1.7 , r 1 = 0.5 {r}_{1}=0.5 , r 3 = 0.3 {r}_{3}=0.3 , r 4 = ‒ 1.04 {r}_{4}=&#x2012;1.04 , r 5 = 1.5 {r}_{5}=1.5 , s = 1.09 s=1.09 , and w = 0.2 w=0.2 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.
Figure 4

Graphical shapes of solution χ 4 ( x , t ) in Eq. (18) is shown as A 1 = 2.7 , A 2 = 6 , c = 1 , d = 0.70 , g = 2.5 , h = 5.2 , n 2 = 0.5 , n 3 = 5 , n 4 = 3 , q 1 = 4.5 , q 2 = 1.7 , r 1 = 0.5 , r 3 = 0.3 , r 4 = ‒ 1.04 , r 5 = 1.5 , s = 1.09 , and w = 0.2 . (a) 3D plot, (b) 2D plot, (c) contour plot, (d) density plot, and (e) stream plot, respectively.

Figure 5 Dynamical wave profile of solution χ 5 ( x , t ) {\chi }_{5}\left(x,t) in Eq. (21) is presented as A 1 = 2.05 {A}_{1}=2.05 , A 2 = 2 {A}_{2}=2 , c = 1.4 c=1.4 , d = 0.7 d=0.7 , g = 1.5 g=1.5 , h = 2 h=2 , n 2 = 0.05 {n}_{2}=0.05 , n 3 = ‒ 0.5 {n}_{3}=&#x2012;0.5 , n 4 = 0.4 {n}_{4}=0.4 , q 1 = 1.5 {q}_{1}=1.5 , q 2 = 1 {q}_{2}=1 , r 1 = 5 {r}_{1}=5 , r 3 = 3.3 {r}_{3}=3.3 , r 4 = 1.4 {r}_{4}=1.4 , r 5 = 0.5 {r}_{5}=0.5 , s = 1.9 s=1.9 , and w = ‒ 0.5 w=&#x2012;0.5 .
Figure 5

Dynamical wave profile of solution χ 5 ( x , t ) in Eq. (21) is presented as A 1 = 2.05 , A 2 = 2 , c = 1.4 , d = 0.7 , g = 1.5 , h = 2 , n 2 = 0.05 , n 3 = ‒ 0.5 , n 4 = 0.4 , q 1 = 1.5 , q 2 = 1 , r 1 = 5 , r 3 = 3.3 , r 4 = 1.4 , r 5 = 0.5 , s = 1.9 , and w = ‒ 0.5 .

Figure 6 Graphical shape of solution χ 6 ( x , t ) {\chi }_{6}\left(x,t) in Eq. (24) is shown as c = 7 c=7 , g = 0.3 g=0.3 , h = 0.1 h=0.1 , r 1 = 5 {r}_{1}=5 , r 3 = 0.4 {r}_{3}=0.4 , r 4 = 6 {r}_{4}=6 , and w = 0.5 w=0.5 .
Figure 6

Graphical shape of solution χ 6 ( x , t ) in Eq. (24) is shown as c = 7 , g = 0.3 , h = 0.1 , r 1 = 5 , r 3 = 0.4 , r 4 = 6 , and w = 0.5 .

Figure 7 Graphical wave profile of solution χ 7 ( x , t ) {\chi }_{7}\left(x,t) in Eq. (27) is presented via A 1 = 3 {A}_{1}=3 , A 2 = 2.5 {A}_{2}=2.5 , c = 2 c=2 , d = 1.5 d=1.5 , g = ‒ 0.3 g=&#x2012;0.3 , h = 0.5 h=0.5 , n 2 = 5 {n}_{2}=5 , q 1 = 0.1 {q}_{1}=0.1 , r 1 = 1.5 {r}_{1}=1.5 , r 4 = ‒ 3 {r}_{4}=&#x2012;3 , s = 0.2 s=0.2 , and w = ‒ 2 w=&#x2012;2 .
Figure 7

Graphical wave profile of solution χ 7 ( x , t ) in Eq. (27) is presented via A 1 = 3 , A 2 = 2.5 , c = 2 , d = 1.5 , g = ‒ 0.3 , h = 0.5 , n 2 = 5 , q 1 = 0.1 , r 1 = 1.5 , r 4 = ‒ 3 , s = 0.2 , and w = ‒ 2 .

Figure 8 Dynamical wave profile of solution χ 8 ( x , t ) {\chi }_{8}\left(x,t) in Eq. (30) is shown as A 1 = 1.3 {A}_{1}=1.3 , A 2 = 2 {A}_{2}=2 , c = 3.5 c=3.5 , d = 1.5 d=1.5 , g = 0.1 g=0.1 , h = 5.5 h=5.5 , n 1 = 1.6 {n}_{1}=1.6 , n 2 = 0.2 {n}_{2}=0.2 , n 4 = 4.2 {n}_{4}=4.2 , q 1 = 0.5 {q}_{1}=0.5 , q 2 = 8 {q}_{2}=8 , r 1 = 7.5 {r}_{1}=7.5 , r 4 = 0.4 {r}_{4}=0.4 , s = 0.7 s=0.7 , and w = ‒ 1.2 w=&#x2012;1.2 .
Figure 8

Dynamical wave profile of solution χ 8 ( x , t ) in Eq. (30) is shown as A 1 = 1.3 , A 2 = 2 , c = 3.5 , d = 1.5 , g = 0.1 , h = 5.5 , n 1 = 1.6 , n 2 = 0.2 , n 4 = 4.2 , q 1 = 0.5 , q 2 = 8 , r 1 = 7.5 , r 4 = 0.4 , s = 0.7 , and w = ‒ 1.2 .

Figure 9 Graphical presentation of solution χ ( x , t ) \chi \left(x,t) of Eq. (41) is shown as A 1 = ‒ 1.6 {A}_{1}=&#x2012;1.6 , A 2 = 2.5 {A}_{2}=2.5 , b = ‒ 0.6 b=&#x2012;0.6 , γ = 0.05 \gamma =0.05 , d = 2 d=2 , k = 0.5 k=0.5 , l = ‒ 2 l=&#x2012;2 , l 1 = 1.5 {l}_{1}=1.5 , m = 0.7 m=0.7 , r 2 = 0.3 {r}_{2}=0.3 , r 3 = 0.2 {r}_{3}=0.2 , r 4 = 4.2 {r}_{4}=4.2 , r 6 = 0.4 {r}_{6}=0.4 , s = ‒ 0.02 s=&#x2012;0.02 , u = 0.5 u=0.5 , v = 1.7 v=1.7 , and w 2 = 2.2 {w}_{2}=2.2 .
Figure 9

Graphical presentation of solution χ ( x , t ) of Eq. (41) is shown as A 1 = ‒ 1.6 , A 2 = 2.5 , b = ‒ 0.6 , γ = 0.05 , d = 2 , k = 0.5 , l = ‒ 2 , l 1 = 1.5 , m = 0.7 , r 2 = 0.3 , r 3 = 0.2 , r 4 = 4.2 , r 6 = 0.4 , s = ‒ 0.02 , u = 0.5 , v = 1.7 , and w 2 = 2.2 .

Figure 10 Graphical appearance of solution χ ( x , t ) \chi \left(x,t) of Eq. (44) is shown as A 1 = 1.4 {A}_{1}=1.4 , A 2 = 0.6 {A}_{2}=0.6 , b = 2 b=2 , γ = 0.3 \gamma =0.3 , d = 1.4 d=1.4 , k = 0.4 k=0.4 , m = 0.2 m=0.2 , r 3 = 3.3 {r}_{3}=3.3 , r 4 = 2.2 {r}_{4}=2.2 , r 5 = 1.2 {r}_{5}=1.2 , r 6 = 8.5 {r}_{6}=8.5 , r 7 = 0.5 {r}_{7}=0.5 , s = 0.2 s=0.2 , u = 1.3 u=1.3 , v = 5 v=5 , w 1 = 0.5 {w}_{1}=0.5 , z 1 = 5 {z}_{1}=5 , and z 2 = 0.02 {z}_{2}=0.02 .
Figure 10

Graphical appearance of solution χ ( x , t ) of Eq. (44) is shown as A 1 = 1.4 , A 2 = 0.6 , b = 2 , γ = 0.3 , d = 1.4 , k = 0.4 , m = 0.2 , r 3 = 3.3 , r 4 = 2.2 , r 5 = 1.2 , r 6 = 8.5 , r 7 = 0.5 , s = 0.2 , u = 1.3 , v = 5 , w 1 = 0.5 , z 1 = 5 , and z 2 = 0.02 .

Figure 11 Dynamical wave profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (49) is shown as A 1 = 4 {A}_{1}=4 , A 2 = 0.6 {A}_{2}=0.6 , b = 1 b=1 , γ = 0.8 \gamma =0.8 , d = 1.4 d=1.4 , k = 1.7 k=1.7 , m = ‒ 0.5 m=&#x2012;0.5 , r 3 = 0.03 {r}_{3}=0.03 , r 7 = ‒ 0.5 {r}_{7}=&#x2012;0.5 , s = 0.5 s=0.5 , u = 1.3 u=1.3 , v = 0.9 v=0.9 , z 2 = 2.2 {z}_{2}=2.2 , and z 3 = 3.3 . {z}_{3}=3.3.
Figure 11

Dynamical wave profile of solution χ ( x , t ) of Eq. (49) is shown as A 1 = 4 , A 2 = 0.6 , b = 1 , γ = 0.8 , d = 1.4 , k = 1.7 , m = ‒ 0.5 , r 3 = 0.03 , r 7 = ‒ 0.5 , s = 0.5 , u = 1.3 , v = 0.9 , z 2 = 2.2 , and z 3 = 3.3 .

Figure 12 Graphical profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (50) is shown as A 1 = 1.4 {A}_{1}=1.4 , A 2 = 0.6 {A}_{2}=0.6 , b = ‒ 2 b=&#x2012;2 , γ = ‒ 0.9 \gamma =&#x2012;0.9 , d = 1.4 d=1.4 , k = ‒ 0.4 k=&#x2012;0.4 , r 1 = 2.2 {r}_{1}=2.2 , r 3 = 3 {r}_{3}=3 , r 4 = ‒ 1.2 {r}_{4}=&#x2012;1.2 , r 5 = 5 {r}_{5}=5 , r 6 = ‒ 8.5 {r}_{6}=&#x2012;8.5 , r 7 = 1.7 {r}_{7}=1.7 , s = 0.2 s=0.2 , u = 1.3 u=1.3 , v = ‒ 5 v=&#x2012;5 , w 1 = 0.5 {w}_{1}=0.5 , w 2 = ‒ 2.2 {w}_{2}=&#x2012;2.2 , z 1 = 0.25 {z}_{1}=0.25 , z 2 = ‒ 1.5 {z}_{2}=&#x2012;1.5 , and z 4 = 4.5 {z}_{4}=4.5 .
Figure 12

Graphical profile of solution χ ( x , t ) of Eq. (50) is shown as A 1 = 1.4 , A 2 = 0.6 , b = ‒ 2 , γ = ‒ 0.9 , d = 1.4 , k = ‒ 0.4 , r 1 = 2.2 , r 3 = 3 , r 4 = ‒ 1.2 , r 5 = 5 , r 6 = ‒ 8.5 , r 7 = 1.7 , s = 0.2 , u = 1.3 , v = ‒ 5 , w 1 = 0.5 , w 2 = ‒ 2.2 , z 1 = 0.25 , z 2 = ‒ 1.5 , and z 4 = 4.5 .

Figure 13 Dynamical profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (53) is shown as A 1 = 1.15 {A}_{1}=1.15 , A 2 = 9 {A}_{2}=9 , b = 7 b=7 , γ = 0.3 \gamma =0.3 , d = 2 d=2 , j 1 = 2.3 {j}_{1}=2.3 , j 2 = 6 {j}_{2}=6 , k = 2 k=2 , m = ‒ 5.5 m=&#x2012;5.5 , r 1 = ‒ 4.5 {r}_{1}=&#x2012;4.5 , r 2 = 0.2 {r}_{2}=0.2 , r 4 = ‒ 0.4 {r}_{4}=&#x2012;0.4 , r 5 = 5 {r}_{5}=5 , s = ‒ 0.1 s=&#x2012;0.1 , u = 2 u=2 , v = 0.5 v=0.5 , w 1 = 1.1 {w}_{1}=1.1 , w 2 = ‒ 2.2 {w}_{2}=&#x2012;2.2 , z 1 = 0.1 {z}_{1}=0.1 , and z 2 = ‒ 0.1 . {z}_{2}=&#x2012;0.1.
Figure 13

Dynamical profile of solution χ ( x , t ) of Eq. (53) is shown as A 1 = 1.15 , A 2 = 9 , b = 7 , γ = 0.3 , d = 2 , j 1 = 2.3 , j 2 = 6 , k = 2 , m = ‒ 5.5 , r 1 = ‒ 4.5 , r 2 = 0.2 , r 4 = ‒ 0.4 , r 5 = 5 , s = ‒ 0.1 , u = 2 , v = 0.5 , w 1 = 1.1 , w 2 = ‒ 2.2 , z 1 = 0.1 , and z 2 = ‒ 0.1 .

Figure 14 Graphical wave profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (56) is shown as A 1 = 1.5 {A}_{1}=1.5 , A 2 = 2.2 {A}_{2}=2.2 , b = 2 b=2 , γ = 0.5 \gamma =0.5 , d = ‒ 0.2 d=&#x2012;0.2 , k = ‒ 5 k=&#x2012;5 , r 2 = ‒ 2.5 {r}_{2}=&#x2012;2.5 , r 3 = 1.3 {r}_{3}=1.3 , r 4 = ‒ 4.4 {r}_{4}=&#x2012;4.4 , r 5 = 5.2 {r}_{5}=5.2 , r 6 = ‒ 1.6 {r}_{6}=&#x2012;1.6 , r 7 = 7 {r}_{7}=7 , s = ‒ 1.6 s=&#x2012;1.6 , u = 1.2 u=1.2 , w 1 = 4.2 {w}_{1}=4.2 , and z 2 = 1.2 . {z}_{2}=1.2.
Figure 14

Graphical wave profile of solution χ ( x , t ) of Eq. (56) is shown as A 1 = 1.5 , A 2 = 2.2 , b = 2 , γ = 0.5 , d = ‒ 0.2 , k = ‒ 5 , r 2 = ‒ 2.5 , r 3 = 1.3 , r 4 = ‒ 4.4 , r 5 = 5.2 , r 6 = ‒ 1.6 , r 7 = 7 , s = ‒ 1.6 , u = 1.2 , w 1 = 4.2 , and z 2 = 1.2 .

Figure 15 Dynamical wave profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (59) is shown as A 1 = 4 {A}_{1}=4 , A 2 = 6 {A}_{2}=6 , b = 2 b=2 , γ = 2 \gamma =2 , d = 4 d=4 , k = 0.9 k=0.9 , ν 0 = 0.04 {\nu }_{0}=0.04 , ν 1 = 1.2 {\nu }_{1}=1.2 , r 10 = 1.7 {r}_{10}=1.7 , r 2 = 2.2 {r}_{2}=2.2 , r 3 = 3.3 {r}_{3}=3.3 , r 4 = 1.2 {r}_{4}=1.2 , r 5 = 1.5 {r}_{5}=1.5 , r 6 = 6.3 {r}_{6}=6.3 , s = 0.2 s=0.2 , and u = 1.3 . u=1.3.
Figure 15

Dynamical wave profile of solution χ ( x , t ) of Eq. (59) is shown as A 1 = 4 , A 2 = 6 , b = 2 , γ = 2 , d = 4 , k = 0.9 , ν 0 = 0.04 , ν 1 = 1.2 , r 10 = 1.7 , r 2 = 2.2 , r 3 = 3.3 , r 4 = 1.2 , r 5 = 1.5 , r 6 = 6.3 , s = 0.2 , and u = 1.3 .

Figure 16 Graphical profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (62) is shown as A 1 = 1.5 {A}_{1}=1.5 , A 2 = 3.6 {A}_{2}=3.6 , b = ‒ 1.9 b=&#x2012;1.9 , γ = 0.7 \gamma =0.7 , d = 2.5 d=2.5 , k = 4.4 k=4.4 , m = 5 m=5 , r 1 = 3 {r}_{1}=3 , r 4 = ‒ 0.6 {r}_{4}=&#x2012;0.6 , r 6 = 2 {r}_{6}=2 , r 7 = 7.2 {r}_{7}=7.2 , s = 2.7 s=2.7 , u = 2.2 u=2.2 , and v = 1.1 . v=1.1.
Figure 16

Graphical profile of solution χ ( x , t ) of Eq. (62) is shown as A 1 = 1.5 , A 2 = 3.6 , b = ‒ 1.9 , γ = 0.7 , d = 2.5 , k = 4.4 , m = 5 , r 1 = 3 , r 4 = ‒ 0.6 , r 6 = 2 , r 7 = 7.2 , s = 2.7 , u = 2.2 , and v = 1.1 .

Figure 17 Dynamical presentation of solution χ ( x , t ) \chi \left(x,t) of Eq. (65) is shown as A 1 = 1.1 {A}_{1}=1.1 , A 2 = ‒ 6 {A}_{2}=&#x2012;6 , b = ‒ 1.7 b=&#x2012;1.7 , γ = 0.03 \gamma =0.03 , d = 2 d=2 , k = ‒ 6.1 k=&#x2012;6.1 , m = 3.3 m=3.3 , r 1 = 2.1 {r}_{1}=2.1 , r 4 = ‒ 7.2 {r}_{4}=&#x2012;7.2 , r 5 = 0.6 {r}_{5}=0.6 , r 7 = ‒ 2.4 {r}_{7}=&#x2012;2.4 , s = 2 s=2 , u = 2 u=2 , v = ‒ 0.1 v=&#x2012;0.1 , w 1 = 0.1 {w}_{1}=0.1 , z 1 = 0.1 {z}_{1}=0.1 , and z 2 = 5 . {z}_{2}=5.
Figure 17

Dynamical presentation of solution χ ( x , t ) of Eq. (65) is shown as A 1 = 1.1 , A 2 = ‒ 6 , b = ‒ 1.7 , γ = 0.03 , d = 2 , k = ‒ 6.1 , m = 3.3 , r 1 = 2.1 , r 4 = ‒ 7.2 , r 5 = 0.6 , r 7 = ‒ 2.4 , s = 2 , u = 2 , v = ‒ 0.1 , w 1 = 0.1 , z 1 = 0.1 , and z 2 = 5 .

Figure 18 Graphical shape profile of solution χ ( x , t ) \chi \left(x,t) of Eq. (68) is shown as A 1 = ‒ 1.4 A1=&#x2012;1.4 , A 2 = 0.6 {A}_{2}=0.6 , b = 1 b=1 , γ = 2.5 \gamma =2.5 , d = 0.02 d=0.02 , k = ‒ 1.5 k=&#x2012;1.5 , m = 3.3 m=3.3 , r 1 = ‒ 6.5 {r}_{1}=&#x2012;6.5 , r 4 = 0.6 {r}_{4}=0.6 , r 5 = ‒ 5 {r}_{5}=&#x2012;5 , r 7 = 2 {r}_{7}=2 , s = 0.02 s=0.02 , u = 0.2 u=0.2 , v = 0.1 v=0.1 , w 1 = 0.1 {w}_{1}=0.1 , w 2 = 2.5 {w}_{2}=2.5 , z 1 = 1.25 {z}_{1}=1.25 , z 2 = 5 {z}_{2}=5 , z 3 = 3.3 {z}_{3}=3.3 , and z 4 = ‒ 4 . {z}_{4}=&#x2012;4.
Figure 18

Graphical shape profile of solution χ ( x , t ) of Eq. (68) is shown as A 1 = ‒ 1.4 , A 2 = 0.6 , b = 1 , γ = 2.5 , d = 0.02 , k = ‒ 1.5 , m = 3.3 , r 1 = ‒ 6.5 , r 4 = 0.6 , r 5 = ‒ 5 , r 7 = 2 , s = 0.02 , u = 0.2 , v = 0.1 , w 1 = 0.1 , w 2 = 2.5 , z 1 = 1.25 , z 2 = 5 , z 3 = 3.3 , and z 4 = ‒ 4 .

23 Conclusion

In this article, we investigate some solutions for cold bosonic atoms in a zig-zag optical lattice phenomenon such as homoclinic breather, periodic-cross and kink cross-rational waves, M-SPK, M-SRK, M-SRS, M-SR1K, and M-SR2K. we also categorized lump soliton waves solution, multi- and rogue wave propagation, and periodic wave propagation. We also tested the stability of our solution and represented it in a table.

Acknowledgements

The authors extend their appreciation to the Deputyship for Research and Innovation, Ministry of Education in Saudi Arabia, for funding this research work through project number: 445-9-410. Furthermore, the authors would like to extend their appreciation to Taibah University for its supervision support.

  1. Funding information: The authors state no funding involved.

  2. Author contributions: Aly Seadawy: methodology and supervision. Syed Rizvi: validation, conceptualization and software. Samia Ahmed: investigation and writing – reviewing and editing.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-08-12
Revised: 2023-09-21
Accepted: 2023-09-21
Published Online: 2023-12-07

© 2023 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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https://doi.org/10.1515/nleng-2022-0337
Keywords for this article
breathers; homoclinic breather; rogue wave; lump soliton; M-shaped solution; multiwave
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  19. Investigation of magnetized convection for second-grade nanofluids via Prabhakar differentiation
  20. Influence of the blade size on the dynamic characteristic damage identification of wind turbine blades
  21. Cilia and electroosmosis induced double diffusive transport of hybrid nanofluids through microchannel and entropy analysis
  22. Semi-analytical approximation of time-fractional telegraph equation via natural transform in Caputo derivative
  23. Analytical solutions of fractional couple stress fluid flow for an engineering problem
  24. Simulations of fractional time-derivative against proportional time-delay for solving and investigating the generalized perturbed-KdV equation
  25. Pricing weather derivatives in an uncertain environment
  26. Variational principles for a double Rayleigh beam system undergoing vibrations and connected by a nonlinear Winkler–Pasternak elastic layer
  27. Novel soliton structures of truncated M-fractional (4+1)-dim Fokas wave model
  28. Safety decision analysis of collapse accident based on “accident tree–analytic hierarchy process”
  29. Derivation of septic B-spline function in n-dimensional to solve n-dimensional partial differential equations
  30. Development of a gray box system identification model to estimate the parameters affecting traffic accidents
  31. Homotopy analysis method for discrete quasi-reversibility mollification method of nonhomogeneous backward heat conduction problem
  32. New kink-periodic and convex–concave-periodic solutions to the modified regularized long wave equation by means of modified rational trigonometric–hyperbolic functions
  33. Explicit Chebyshev Petrov–Galerkin scheme for time-fractional fourth-order uniform Euler–Bernoulli pinned–pinned beam equation
  34. NASA DART mission: A preliminary mathematical dynamical model and its nonlinear circuit emulation
  35. Nonlinear dynamic responses of ballasted railway tracks using concrete sleepers incorporated with reinforced fibres and pre-treated crumb rubber
  36. Two-component excitation governance of giant wave clusters with the partially nonlocal nonlinearity
  37. Bifurcation analysis and control of the valve-controlled hydraulic cylinder system
  38. Engineering fault intelligent monitoring system based on Internet of Things and GIS
  39. Traveling wave solutions of the generalized scale-invariant analog of the KdV equation by tanh–coth method
  40. Electric vehicle wireless charging system for the foreign object detection with the inducted coil with magnetic field variation
  41. Dynamical structures of wave front to the fractional generalized equal width-Burgers model via two analytic schemes: Effects of parameters and fractionality
  42. Theoretical and numerical analysis of nonlinear Boussinesq equation under fractal fractional derivative
  43. Research on the artificial control method of the gas nuclei spectrum in the small-scale experimental pool under atmospheric pressure
  44. Mathematical analysis of the transmission dynamics of viral infection with effective control policies via fractional derivative
  45. On duality principles and related convex dual formulations suitable for local and global non-convex variational optimization
  46. Study on the breaking characteristics of glass-like brittle materials
  47. The construction and development of economic education model in universities based on the spatial Durbin model
  48. Homoclinic breather, periodic wave, lump solution, and M-shaped rational solutions for cold bosonic atoms in a zig-zag optical lattice
  49. Fractional insights into Zika virus transmission: Exploring preventive measures from a dynamical perspective
  50. Rapid Communication
  51. Influence of joint flexibility on buckling analysis of free–free beams
  52. Special Issue: Recent trends and emergence of technology in nonlinear engineering and its applications - Part II
  53. Research on optimization of crane fault predictive control system based on data mining
  54. Nonlinear computer image scene and target information extraction based on big data technology
  55. Nonlinear analysis and processing of software development data under Internet of things monitoring system
  56. Nonlinear remote monitoring system of manipulator based on network communication technology
  57. Nonlinear bridge deflection monitoring and prediction system based on network communication
  58. Cross-modal multi-label image classification modeling and recognition based on nonlinear
  59. Application of nonlinear clustering optimization algorithm in web data mining of cloud computing
  60. Optimization of information acquisition security of broadband carrier communication based on linear equation
  61. A review of tiger conservation studies using nonlinear trajectory: A telemetry data approach
  62. Multiwireless sensors for electrical measurement based on nonlinear improved data fusion algorithm
  63. Realization of optimization design of electromechanical integration PLC program system based on 3D model
  64. Research on nonlinear tracking and evaluation of sports 3D vision action
  65. Analysis of bridge vibration response for identification of bridge damage using BP neural network
  66. Numerical analysis of vibration response of elastic tube bundle of heat exchanger based on fluid structure coupling analysis
  67. Establishment of nonlinear network security situational awareness model based on random forest under the background of big data
  68. Research and implementation of non-linear management and monitoring system for classified information network
  69. Study of time-fractional delayed differential equations via new integral transform-based variation iteration technique
  70. Exhaustive study on post effect processing of 3D image based on nonlinear digital watermarking algorithm
  71. A versatile dynamic noise control framework based on computer simulation and modeling
  72. A novel hybrid ensemble convolutional neural network for face recognition by optimizing hyperparameters
  73. Numerical analysis of uneven settlement of highway subgrade based on nonlinear algorithm
  74. Experimental design and data analysis and optimization of mechanical condition diagnosis for transformer sets
  75. Special Issue: Reliable and Robust Fuzzy Logic Control System for Industry 4.0
  76. Framework for identifying network attacks through packet inspection using machine learning
  77. Convolutional neural network for UAV image processing and navigation in tree plantations based on deep learning
  78. Analysis of multimedia technology and mobile learning in English teaching in colleges and universities
  79. A deep learning-based mathematical modeling strategy for classifying musical genres in musical industry
  80. An effective framework to improve the managerial activities in global software development
  81. Simulation of three-dimensional temperature field in high-frequency welding based on nonlinear finite element method
  82. Multi-objective optimization model of transmission error of nonlinear dynamic load of double helical gears
  83. Fault diagnosis of electrical equipment based on virtual simulation technology
  84. Application of fractional-order nonlinear equations in coordinated control of multi-agent systems
  85. Research on railroad locomotive driving safety assistance technology based on electromechanical coupling analysis
  86. Risk assessment of computer network information using a proposed approach: Fuzzy hierarchical reasoning model based on scientific inversion parallel programming
  87. Special Issue: Dynamic Engineering and Control Methods for the Nonlinear Systems - Part I
  88. The application of iterative hard threshold algorithm based on nonlinear optimal compression sensing and electronic information technology in the field of automatic control
  89. Equilibrium stability of dynamic duopoly Cournot game under heterogeneous strategies, asymmetric information, and one-way R&D spillovers
  90. Mathematical prediction model construction of network packet loss rate and nonlinear mapping user experience under the Internet of Things
  91. Target recognition and detection system based on sensor and nonlinear machine vision fusion
  92. Risk analysis of bridge ship collision based on AIS data model and nonlinear finite element
  93. Video face target detection and tracking algorithm based on nonlinear sequence Monte Carlo filtering technique
  94. Adaptive fuzzy extended state observer for a class of nonlinear systems with output constraint
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