Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model

Mohammed Banikhalid; Amirah Azmi; Marwan Alquran; Mohammed Ali

doi:10.1515/nleng-2024-0005

Enjoy 40% off

academic books on De Gruyter Brill *

Article Open Access

Conditions for the existence of soliton solutions: An analysis of coefficients in the generalized Wu–Zhang system and generalized Sawada–Kotera model

Mohammed Banikhalid , Amirah Azmi , Marwan Alquran and Mohammed Ali

Published/Copyright: May 29, 2024

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information

From the journal Nonlinear Engineering Volume 13 Issue 1

Abstract

Exploring nonlinear equations and systems with predetermined coefficient values constrains the depth of understanding of the dynamics inherent in various applications and phenomena represented by such equations. On the contrary, exploration of nonlinear models with free coefficients offers avenues for improved development and ongoing refinement. In light of this, this study aimed to reassess the Wu–Zhang (WZ) system and Sawada–Kotera (SK) model by introducing arbitrary coefficients. Our goal is to identify the constraints necessary to ensure the existence of soliton solutions. Through the application of two distinct approaches, namely, the sine–cosine function method and tanh–coth expansion method, we systematically examine the conditions that facilitate the emergence of solitons within the WZ system and SK model. The insights gained from this analysis are supported by the presentation of 2D and 3D plots, providing a visual depiction of the propagation characteristics exhibited by the obtained solutions. The findings of the current work on conditions for the existence of soliton solutions for both generalized Wu–Zhang and generalized Sawada–Kotera models are novel and presented here for the first time.

Keywords: generalized Wu–Zhang system; generalized Sawada–Kotera model; sine–cosine function method; tanh–coth expansion method

MSC 2010: 26A33; 35F25; 35C10

1 Introduction

The exploration of nonlinear equations and systems with predetermined coefficient values has traditionally been a common approach in scientific inquiry. However, this conventional method imposes limitations on the depth of our understanding of the dynamic processes inherent in diverse applications and phenomena represented by such equations. In contrast, exploring nonlinear models with free coefficients presents an opportunity for enhanced development and continual refinement.

This study delves into the reevaluation of the generalized Wu–Zhang (gWZ) system and generalized Sawada–Kotera (gSK) equation by introducing arbitrary coefficients. The primary objective is to identify the constraints necessary to guarantee the existence of soliton solutions within the gWZ system and gSK model, which are governed by the following equations.

The gWZ reads as

(1.1) u t − α u u x − β w x = 0 ,

(1.2) w t + γ ( u w ) x + δ u x x x = 0 ,

where u = u ( x , t ) and w = w ( x , t ) , with t and x representing the time and space coordinates, respectively. The parameters α and γ pertain to the nonlinearity, β denotes the perturbation factor, δ signifies the dispersion parameter, and the subscripts indicate the partial derivatives concerning the independent coordinate variables x and t .

The gSK reads as

(1.3) ψ t + a ( ( ψ ψ x x ) x + ( ψ 3 ) x ) + b ψ x x x x x = 0 ,

where ψ = ψ ( x , t ) , and a and b refer to the nonlinearity and dispersion parameters, respectively.

The aforementioned WZ Systems (1.1) and (1.2) have been investigated in many occasions for only the case where the assigned parameters α , β , γ , and δ are − 1 , − 1 , 1 , and 1 3 , respectively. For instance, the extended tanh and Hirota methods were used to construct singular solitary wave, periodic, and multi-soliton solutions [1]. In another similar study but with graphical analysis, four methods known as the csch-function method, tan–cot method, extended tanh–coth expansion, and the modified simple equation method, were used to retrieve optical solutions [2]. Rational solutions in terms of sinh and cosh functions were obtained by means of exponential rational function method [3]. Moreover, when the time derivative is replaced by conformable type [4], the first integral method was implemented to extract fractional soliton and soliton-like solutions. Finally, in the study by Mirzazadeh et al. [5], the Lie group formalism was applied to derive symmetries of the Wu–Zhang (WZ) system, and the mapping approach was adapted to report shock-wave solutions.

The Sawada–Kotera (SK) model has been extensively examined in previous studies, primarily focusing on the specific scenario where the assigned parameters are a = 15 and b = 1 . For example, using a simplified Hirota bilinear approach, researchers have derived solitary and periodic wave solutions for the fifth-order SK, encompassing one-soliton solutions, periodic two-soliton solutions, and singular periodic soliton solutions [6,7]. Additionally, employing three different wave-type methods has led to the identification of lump, breather, and two-wave solutions [8]. Moreover, the exp-function method has also been applied, resulting in the discovery of singular-like soliton solutions for the SK model [9].

In the literature, two main approaches are employed to derive solutions for nonlinear partial differential equations yielding either unidirectional wave solutions or multi-soliton solutions. Techniques such as the Backlund transform [10–12], the nonlinear transform method [13,14], and the Hirota direct method [15–17] are primarily used for extracting singular and nonsingular multi-soliton solutions. On the other side, approaches known as test methods, including the sine–cosine function method [18–20], tanh–coth expansion method [21,22], Kudryashov-expansion [23–25], ( G ′ ∕ G ) -expansion [26,27], exp-expansion method [28,29], rational sine–cosine method [30–32], and others, are designed to generate single-wave motion and traveling periodic waves.

In this study, we employ the sine-function method and the tanh–coth expansion scheme to ascertain the constraint conditions on the coefficient parameters α , β , γ , and δ necessary for ensuring the existence of exact solutions to the generalized WZ system. Additionally, we investigate the relationship between a and b to determine the conditions for solutions to exist in the gSK model.

2 Explicit traveling wave solutions to gWZ

We may reduce the gWZ system into a single nonlinear equation by introducing a new function θ = θ ( x , t ) such that

(2.1) u = θ x .

From (1.1), we deduce that

(2.2) w = 1 β θ t − α 2 θ x 2 .

Substitution of (2.1) and (2.2) into (2.1) leads to the following new equation:

(2.3) θ t t − ( α − γ ) θ x θ x t + γ θ t θ x x − 3 α γ 2 θ x 2 θ x x + β δ θ x x x x = 0 .

Now, via the linear transformation z = x − c t , we convert the nonlinear partial differential equation (2.3) into an ordinary differential equation defined as

(2.4) c 2 v ′ + c 2 ( α − 2 γ ) v ′ 2 − α γ 2 v ′ 3 + β δ v ‴ = 0 ,

where v = v ( z ) = θ ( x , t ) . Moreover, if we let V ( z ) = v ′ ( z ) , (2.4) takes the following form:

(2.5) c 2 V + c 2 ( α − 2 γ ) V 2 − α γ 2 V 3 + β δ V ″ = 0 ,

where V = V ( z ) . Next, we solve (2.5) by the applicable trial sine-function method.

2.1 Sine-function method

In this section, we explore the conditions necessary for the coefficients α , β , γ , and δ to ensure the presence of solutions in the gWZ system, represented in terms of powers of the sine function. To achieve this, we let

(2.6) V ( z ) = A sin n ( μ z ) .

Inserting (2.6) into (2.5) and using the identity cos 2 ( μ z ) = 1 − sin 2 ( μ z ) , we obtain

(2.7) 0 = A β δ μ 2 ( n − 1 ) n sin n − 2 ( μ z ) + A ( c 2 − β δ μ 2 n 2 ) sin n ( μ z ) + 1 2 A 2 c ( α − 2 γ ) sin 2 n ( μ z ) − 1 2 α γ A 3 sin 3 n ( μ z ) .

Setting the exponents 3 n and n − 2 equal, which is the only case that provides nontrivial solution, and equating the coefficients of the corresponding powers of sin ( μ z ) to zero, we arrive at the following algebraic system:

(2.8) 0 = n + 1 , 0 = A β δ μ 2 ( n − 1 ) n − α γ A 3 2 , 0 = A ( c 2 − β δ μ 2 n 2 ) , 0 = 1 2 A 2 c ( α − 2 γ ) .

From the aforementioned system, we observe that n = − 1 , and the fourth equation holds true only when α = 2 γ . Consequently, the following outcomes emerge:

(2.9) A = ∓ 2 μ β δ α γ , c = ∓ β δ μ .

Without loss of generality, we examine a specific instance of (2.9), where A = 2 μ β δ α γ and c = − β δ μ . Returning to Formulas (2.1) and (2.2), the solution to the gWZ system, employing the sine-function method, is determined as

(2.10) u 1 ( x , t ) = 2 β δ μ γ csc ( μ ( x + β δ μ t ) ) , w 1 ( x , t ) = δ μ 2 γ csc 2 ( μ ( x + β δ μ t ) ) × ( 2 sin ( μ ( x + β δ μ t ) ) − 2 ) .

To verify the validity of the derived solution presented in (2.10), we substitute it into the original System (1.1). The system is found to be valid whenever α = 2 γ .

Corollary 1

The solutions obtained in (2.10) satisfy the gWZ system under the following two conditions:

α = 2 γ ,
β δ > 0 .

Corollary 2

When employing the proposed solution V ( z ) = A cos n ( μ z ) , the result is the identical algebraic system (2.8). Therefore, using the predetermined values, A = 2 μ β δ α γ and c = − β δ μ , an additional solution is derived for gWZ, defined as:

(2.11) u 2 ( x , t ) = 2 β δ μ γ sec ( μ ( x + β δ μ t ) ) , w 2 ( x , t ) = δ μ 2 γ sec 2 ( μ ( x + β δ μ t ) ) × ( 2 cos ( μ ( x + β δ μ t ) ) − 2 ) .

Figure 1 shows the propagation characteristics of both u 1 ( x , t ) and w 1 ( x , t ) for specific parameter values of β , δ , γ , and μ . Notably, u 1 exhibits a periodic pattern characterized by convex–concave shapes, while the motion of w 1 is characterized by a concave–periodic pattern.

$Figure 1 Physical structures of (a) u 1 ( x , t ) {u}_{1}\left(x,t) and (b) w 1 ( x , t ) {w}_{1}\left(x,t) , where β = δ = 2 , γ = μ = 1 \beta =\delta =2,\gamma =\mu =1 .$

Figure 1

Physical structures of (a) u 1 ( x , t ) and (b) w 1 ( x , t ) , where β = δ = 2 , γ = μ = 1 .

2.2 Tanh–coth expansion method

The aim here is to investigate whether the constraints on the model’s parameters, as outlined in Corollary 1, hold on when employing an alternative method. In particular, we adapt the use of tanh–coth expansion approach. The extended tanh–coth expansion method proposes the solution in the form of finite sum of negative-positive powers in the new dependent variable Y = Y ( z ) , i.e.,

(2.12) V ( z ) = ∑ j = − n n a j Y j ,

where Y = Y ( z ) satisfies the auxiliary differential equation Y ′ = μ ( 1 − Y 2 ) . It is known, that this auxiliary equation has the solution Y = tanh ( μ z ) or Y = coth ( μ z ) . The value of the index n can be established by a balancing procedure, where we compare the order of higher linear derivatives to that of other nonlinear terms. Given that V ( z ) in (2.12) is a polynomial of degree n and Y ′ is a polynomial of degree 2, it follows that the degree of V k ( z ) is k n and the degree of V ( m ) ( z ) is n + m . By checking (2.5), we find that n + 2 equals 3 n , which leads to n = 1 . Thus, (2.12) is restricted to

(2.13) V ( z ) = A + B Y + F Y .

Differentiation of (2.13) with the assistance of Y ′ = μ ( 1 − Y 2 ) results in

(2.14) V ″ ( z ) = 2 μ 2 ( Y 2 − 1 ) ( B Y 4 − F ) Y 3 .

To find the values of A , B , F , c , and μ , we substitute both Eqs. (2.13) and (2.14) into Eq. (2.5) and set the coefficients of Y j to zero for j = − 3 , − 2 , … , 3 . This leads to the task of solving the following algebraic system:

(2.15) 0 = 2 β δ F μ 2 − α γ F 3 2 , 0 = 1 2 F 2 ( c ( α − 2 γ ) − 3 α γ A ) , 0 = − 1 2 F ( 3 α γ A 2 − 2 A c ( α − 2 γ ) + 4 β δ μ 2 + 3 α γ B F − 2 c 2 ) , 0 = − α γ A 3 2 + 1 2 A 2 c ( α − 2 γ ) + A ( c 2 − 3 α γ B F ) + B c F ( α − 2 γ ) , 0 = − 1 2 B ( 3 α γ A 2 − 2 A c ( α − 2 γ ) + 4 β δ μ 2 + 3 α γ B F − 2 c 2 ) , 0 = 1 2 B 2 ( c ( α − 2 γ ) − 3 α γ A ) , 0 = 2 β B δ μ 2 − α γ B 3 2 .

Solving (2.15) results in the following findings:

Case 1.1: A = 0 , B = ∓ 2 2 β δ α μ , F = 0 , c = ∓ 2 β δ μ , and γ = α 2 . If we consider B = 2 2 β δ α μ and c = − 2 β δ μ , the solution of gWZ is

(2.16) u 3 ( x , t ) = 2 2 β δ α μ tanh ( μ ( x + 2 β δ μ t ) ) , w 3 ( x , t ) = 4 δ μ 2 α tanh ( μ ( x + 2 β δ μ t ) ) × ( 1 − tanh ( μ ( x + 2 β δ μ t ) ) ) .

Case 1.2: A = B = 0 , F = ∓ 2 2 β δ α μ , c = ∓ 2 β δ μ , and γ = α 2 . If we consider F = 2 2 β δ α μ and c = − 2 β δ μ , the solution of gWZ is

(2.17) u 4 ( x , t ) = 2 2 β δ α μ coth ( μ ( x + 2 β δ μ t ) ) , w 4 ( x , t ) = 4 δ μ 2 α coth ( μ ( x + 2 β δ μ t ) ) × ( 1 − coth ( μ ( x + 2 β δ μ t ) ) ) .

Case 1.3: A = 0 , B = F = ∓ 2 2 β δ α μ , c = ∓ 2 2 β δ μ , and γ = α 2 . If we consider B = F = 2 2 β δ α μ and c = − 2 2 β δ μ , the solution of gWZ is

(2.18) u 5 ( x , t ) = 2 2 β δ μ α ( coth ( μ ( x + 2 2 β δ μ t ) ) + tanh ( μ ( x + 2 2 β δ μ t ) ) ) , w 5 ( x , t ) = 16 μ 2 δ α coth ( 2 μ ( x + 2 2 β δ μ t ) ) × ( 1 − coth ( 2 μ ( x + 2 2 β δ μ t ) ) ) .

Remark 1

The solutions presented in (2.16)–(2.18) are in accordance with the gWZ system when α = 2 γ . This agrees with the findings reported in Corollary 1.

Figures 2 and 3 depict the propagation type of both u 3 ( x , t ) , w 3 ( x , t ) and u 4 ( x , t ) , w 4 ( x , t ) for particular values of β , δ , α , and μ . It can be observed that u 3 and w 3 admit two types of regular-kink shapes, while u 4 and w 4 are characterized by two types of singular-kinks.

$Figure 2 Physical structures of (a) u 3 ( x , t ) {u}_{3}\left(x,t) and (b) w 3 ( x , t ) {w}_{3}\left(x,t) , where β = 2 \beta =2 , δ = 2 \delta =2 , α = 1 \alpha =1 , and μ = 1 5 \mu =\frac{1}{5} .$

Figure 2

Physical structures of (a) u 3 ( x , t ) and (b) w 3 ( x , t ) , where β = 2 , δ = 2 , α = 1 , and μ = 1 5 .

$Figure 3 Physical structures of (a) u 4 ( x , t ) {u}_{4}\left(x,t) and (b) w 4 ( x , t ) {w}_{4}\left(x,t) , where β = 2 \beta =2 , δ = 2 \delta =2 , α = 1 \alpha =1 , and μ = 1 5 \mu =\frac{1}{5} .$

Figure 3

Physical structures of (a) u 4 ( x , t ) and (b) w 4 ( x , t ) , where β = 2 , δ = 2 , α = 1 , and μ = 1 5 .

3 Explicit solutions to gSK

In this section, we apply the proposed earlier techniques to explore the constraint conditions on the two embedded coefficients a and b , which admit explicit solutions to gSK. First, we convert the gSK into an ordinary differential equation via the linear transformation s = x − r t , where r stands for the wave speed. As a result, we reach at the following corresponding equation:

(3.1) r Ψ − a ( Ψ Ψ ″ + Ψ 3 ) − b Ψ ( 4 ) = 0 ,

where Ψ ( s ) = ψ ( x , t ) . Next, we solve (3.1) by the sine-function method.

3.1 Sine-function method

The exponent-sine solution of (3.1) is of the form

(3.2) Ψ ( s ) = A sin n ( μ s ) .

Then, we substitute (3.2) into (3.1) and simplify as needed to obtain

(3.3) − 8 b μ 4 n ( n 3 − 6 n 2 + 11 n − 6 ) + 16 b μ 4 n ( n 3 − 3 n 2 + 4 n − 2 ) sin 2 ( μ s ) + 8 ( r − b μ 4 n 4 ) sin 4 ( μ s ) − 8 a A μ 2 ( n − 1 ) n sin n + 2 ( μ s ) + 8 a A μ 2 n 2 sin n + 4 ( μ s ) − 8 a A 2 sin 2 n + 4 ( μ s ) = 0 .

To identify potential values for the parameter n , we equate two distinct exponents related to the base function sin ( μ s ) . Specifically, we set 2 n + 4 equal to 0, leading to the determination that n = − 2 . Consequently, the following sub-equations arise from (3.3):

(3.4) 0 = 8 ( a A 2 + 6 a A μ 2 + 120 b μ 4 ) , 0 = 32 μ 2 ( a A + 30 b μ 2 ) , 0 = 8 ( r − 16 b μ 4 ) .

By solving the aforementioned system, we obtain

(3.5) A = − 2 μ 2 , r = 16 b μ 4 , a = 15 b .

Accordingly, the general solution of gSK, labeled as ψ 1 ( x , t ) , is

(3.6) ψ 1 ( x , t ) = − 2 μ 2 csc 2 ( μ ( x − 16 b μ 4 t ) ) .

Corollary 3

The solution obtained in (3.6) satisfies the gSK equation under the condition a = 15 b .

Corollary 4

By considering the suggested solution Ψ ( s ) = A cos n ( μ s ) , the result is the identical algebraic System (3.4). Therefore, an additional solution is derived for gSK, defined as:

(3.7) ψ 2 ( x , t ) = − 2 μ 2 sec 2 ( μ ( x − 16 b μ 4 t ) ) .

Figure 4, shows 3D and 2D propagation of ψ 1 ( x , t ) for specific parameter values of μ and b . In particular, ψ 1 exhibits a periodic motion characterized by concave shapes.

$Figure 4 Physical structures of ψ 1 ( x , t ) {\psi }_{1}\left(x,t) : (i) 3D propagation and (ii) 2D profiles, where b = 1 b=1 and μ = 0.7 \mu =0.7 .$

Figure 4

Physical structures of ψ 1 ( x , t ) : (i) 3D propagation and (ii) 2D profiles, where b = 1 and μ = 0.7 .

3.2 Tanh–coth expansion method

By balancing the nonlinear term Ψ 3 with the highest derivative Ψ ( 4 ) , the tanh–coth expansion solution of (3.1) is

(3.8) Ψ ( s ) = A + B Y + F Y 2 + G Y + R Y 2 ,

where Y = tanh ( μ s ) or Y = coth ( μ s ) . To determine A , B , F , G , R , μ , r , and the constraint relation on a and b , we insert (3.8) into (3.1) and set each coefficient of Y i : i = − 6 , − 5 , … , 4 , 5 , 6 to zero, we obtain the following system:

(3.9) 0 = − R ( a R ( 6 μ 2 + R ) + 120 b μ 4 ) , 0 = − G ( a R ( 8 μ 2 + 3 R ) + 24 b μ 4 ) , 0 = 240 b μ 4 R − a ( R ( 6 A μ 2 + 3 A R − 8 μ 2 R ) + G 2 ( 2 μ 2 + 3 R ) ) , 0 = 40 b G μ 4 − a ( 2 G ( A μ 2 + 3 A R − 5 μ 2 R ) + 3 B R ( 2 μ 2 + R ) + G 3 ) , 0 = − a ( 3 A 2 R + 3 A G 2 − 8 A μ 2 R + 2 B G μ 2 + 6 B G R + 3 F R 2 + 8 F μ 2 R − 2 G 2 μ 2 + 2 μ 2 R 2 ) + R ( r − 136 b μ 4 ) , 0 = − a ( 3 A 2 G + 6 A B R − 2 A G μ 2 + 3 B G 2 − 10 B μ 2 R + 4 F G μ 2 + 6 F G R + 2 G μ 2 R ) + G ( r − 16 b μ 4 ) , 0 = − a ( A 3 + 2 A ( 3 B G + F μ 2 + 3 F R + μ 2 R ) + 3 B 2 R − 4 B G μ 2 + 3 F G 2 − 16 F μ 2 R ) + A r + 16 b μ 4 ( F + R ) , 0 = − a ( 3 A 2 B − 2 A B μ 2 + 6 A F G + 3 B 2 G + 2 B F μ 2 + 6 B F R + 4 B μ 2 R − 10 F G μ 2 ) + B ( r − 16 b μ 4 ) , 0 = − a ( 3 A 2 F + A ( 3 B 2 − 8 F μ 2 ) − 2 B 2 μ 2 + 2 B G ( 3 F + μ 2 ) + F ( 2 F μ 2 + 3 F R + 8 μ 2 R ) ) + F ( r − 136 b μ 4 ) , 0 = 40 b B μ 4 − a ( 2 B ( 3 A F + A μ 2 − 5 F μ 2 ) + B 3 + 3 F G ( F + 2 μ 2 ) ) , 0 = 240 b F μ 4 − a ( F ( 3 A F + 6 A μ 2 − 8 F μ 2 ) + B 2 ( 3 F + 2 μ 2 ) ) , 0 = − B ( a F ( 3 F + 8 μ 2 ) + 24 b μ 4 ) , 0 = − F ( a F ( F + 6 μ 2 ) + 120 b μ 4 ) .

Solving the aforementioned system yields the constraint condition

(3.10) a = 15 b ,

along with the following seven cases:

Case 2.1: { A = 2 μ 2 , B = G = R = 0 , F = − 2 μ 2 , r = 16 a μ 4 15 } .
Case 2.2: { A = 1 15 ( 15 μ 2 − 105 μ 2 ) , B = G = R = 0 , F → − 2 μ 2 , { r → 2 15 ( 11 a μ 4 + 105 a μ 4 ) } .
Case 2.3: { A = 1 15 ( 105 μ 2 + 15 μ 2 ) , B = G = R = 0 , F = − 2 μ 2 , r = 1 15 ( − 2 ) ( 105 a μ 4 − 11 a μ 4 ) } .
Case 2.4: { A = 4 μ 2 , B = G = 0 , F = R = − 2 μ 2 , { r = 256 a μ 4 15 .
Case 2.5: { A = − 4 7 15 μ 2 , B = G = 0 , F = R = − 2 μ 2 , r = 32 15 ( 11 a μ 4 + 105 a μ 4 ) } .
Case 2.6: { A = 4 7 15 μ 2 , B = G = 0 , F = R = → − 2 μ 2 , r = 1 15 ( − 32 ) ( 105 a μ 4 − 11 a μ 4 ) } .
Case 2.7: { A = 2 μ 2 , B = F = G = 0 , R = − 2 μ 2 , r = 16 a μ 4 15 } .

Accordingly, we report more seven solutions to the gSK, labeled as ψ 3 ( x , t ) – ψ 9 ( x , t ) , and given by

(3.11) ψ 3 ( x , t ) = 2 μ 2 sech 2 μ x − 16 15 a μ 4 t , ψ 4 ( x , t ) = − 1 15 μ 2 30 tanh 2 μ x − 2 15 ( 105 + 11 ) a μ 5 t + 105 − 15 , ψ 5 ( x , t ) = 1 15 μ 2 − 30 tanh 2 2 15 ( 105 − 11 ) a μ 5 t + μ x + 105 + 15 , ψ 6 ( x , t ) = − 8 μ 2 csch 2 2 μ x − 256 15 a μ 4 t , ψ 7 ( x , t ) = − 2 μ 2 tanh 2 μ x − 32 15 ( 105 + 11 ) a μ 5 t − 2 μ 2 coth 2 μ x − 32 15 ( 105 + 11 ) a μ 5 t − 4 7 15 μ 2 , ψ 8 ( x , t ) = − 2 μ 2 tanh 2 32 15 ( 105 − 11 ) a μ 5 t + μ x − 2 μ 2 coth 2 32 15 ( 105 − 11 ) a μ 5 t + μ x + 4 7 15 μ 2 , ψ 9 ( x , t ) = − 2 μ 2 csch 2 μ x − 16 15 a μ 4 t .

Remark 2

The functions ψ 3 , ψ 4 , and ψ 5 exhibit the propagation of unidirectional bell-shaped waves, whereas ψ 6 – ψ 9 propagate as a singular cusp wave, as illustrated in Figures 5 and 6.

$Figure 5 Physical structures of ψ 3 ( x , t ) {\psi }_{3}\left(x,t) : (i) 3D propagation and (ii) 2D profiles, where a = 0.1 a=0.1 and μ = 1 \mu =1 .$

Figure 5

Physical structures of ψ 3 ( x , t ) : (i) 3D propagation and (ii) 2D profiles, where a = 0.1 and μ = 1 .

$Figure 6 Physical structures of ψ 7 ( x , t ) {\psi }_{7}\left(x,t) : (i) 3D propagation and (ii) 2D profiles, where a = 0.1 a=0.1 and μ = 0.5 \mu =0.5 .$

Figure 6

Physical structures of ψ 7 ( x , t ) : (i) 3D propagation and (ii) 2D profiles, where a = 0.1 and μ = 0.5 .

4 Conclusion

In conclusion, our research has introduced two nonlinear models, namely the gWZ system and the gSK equation, featuring arbitrary coefficients. We have uncovered the essential conditions governing these coefficients to ensure the existence of novel solutions for both applications. In particular, we achieved the following two main findings:

Solitary wave solutions of singular-periodic types and kink types are exhibited in the gWZ system when the coefficients of the model maintain the two conditions α = 2 γ and β δ > 0 .
The gSK model accommodates the propagation of periodic, bell-shaped, and cusp waves under the condition that the model’s parameters a and b maintain the relationship a = 15 b .

The current study incorporated graphical analyses to present the physical structures underlying the propagation of the obtained solutions. The implications of our findings are significant, offering potential advancements in understanding the behavior of the two models and their applications across diverse disciplines such as optics, plasma, fluid dynamics, and engineering applications.

Funding information: This work is not funded.
Author contributions: The authors contributed equally to this work.
Conflict of interest: The authors declare that they have no competing interests.
Ethical approval: Not applicable.
Data availability statement: Not applicable.

References

[1] Inc M, Kilic B, Karatas E, Al Qurashi MM, Baleanu D, Tchier F. On soliton solutions of the Wu-Zhang system. Open Phys. 2016;14:76–80. 10.1515/phys-2016-0004Search in Google Scholar

[2] Jawad AJM, Al-Azzawi FJI. Soliton solutions of Wu-Zhang system of evolution equations. Num Comp Meth Sci Eng. 2019;1(1):1–11. Search in Google Scholar

[3] Kaplan M, Mayeli P, Hosseini K. Exact traveling wave solutions of the Wu-Zhang system describing (1+1)-dimensional dispersive long wave. Opt Quant Electron. 2017;49:404. 10.1007/s11082-017-1231-0Search in Google Scholar

[4] Eslami M, Rezazadeh H. The first integral method for Wu-Zhang system with conformable time-fractional derivative. Calcolo. 2016;53:475–85. 10.1007/s10092-015-0158-8Search in Google Scholar

[5] Mirzazadeh M, Ekici M, Eslami M, Krishnan EV, Kumar S, Biswas A. Solitons and other solutions to Wu-Zhang system. Nonlinear Anal Model Control. 2017;22(4):441–58. 10.15388/NA.2017.4.2Search in Google Scholar

[6] Liu C, Dai Z. Exact soliton solutions for the fifth-order Sawada–Kotera equation. Appl Math Comput. 2008;206(1):272–75. 10.1016/j.amc.2008.08.028Search in Google Scholar

[7] Hossain AKMKS, Akbar MA. Multi-soliton solutions of the Sawada-Kotera equation using the Hirota direct method: Novel insights into nonlinear evolution equations. Partial Differ Equ Appl Math. 2023;8:100572. 10.1016/j.padiff.2023.100572Search in Google Scholar

[8] Guo Y, Li D, Wang J. The new exact solutions of the Fifth-Order Sawada–Kotera equation using three wave method. Appl Math Lett. 2019;94:232–37. 10.1016/j.aml.2019.03.001Search in Google Scholar

[9] Matinfar M, Aminzadeh M, Nemati M. Exp-function method for the exact solutions of Sawada-Kotera equation. Indian J Pure Appl Math. 2014;45:111–20. 10.1007/s13226-014-0054-ySearch in Google Scholar

[10] Yusuf A, Sulaiman TA, Abdeljabbar A, Alquran M. Breather waves, analytical solutions and conservation laws using Lie-Bäcklund symmetries to the (2+1)-dimensional Chaffee-Infante equation. J Ocean Eng Sci. 2023;8(2):145–51. 10.1016/j.joes.2021.12.008Search in Google Scholar

[11] Zhang Y, Chen DY. A modified Bäcklund transformation and multi-soliton solution for the Boussinesq equation. Chaos Solitons Fractals. 2005;23(1):175–81. 10.1016/j.chaos.2004.04.006Search in Google Scholar

[12] Mao H, Miao Y. Backlund transformation and nonlinear superposition formula for the two-component short pulse equation. J Phys A Math Theor. 2022;55:475207. 10.1088/1751-8121/aca4acSearch in Google Scholar

[13] Alquran M, Alhami R. Analysis of lumps, single-stripe, breather-wave, and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method. Nonlinear Dyn. 2022;109:1985–92. 10.1007/s11071-022-07509-0Search in Google Scholar

[14] Alhami R, Alquran M. Extracted different types of optical lumps and breathers to the new generalized stochastic potential-KdV equation via using the Cole-Hopf transformation and Hirota bilinear method. Opt Quant Electron. 2022;54:553. 10.1007/s11082-022-03984-2Search in Google Scholar

[15] Ma WX. N-Soliton solutions and the Hirota conditions in (1+1)-dimensions. Int J Nonlinear Sci Numer Simul. 2022;23(1):123–33. 10.1515/ijnsns-2020-0214Search in Google Scholar

[16] Ma WX. N-Soliton solutions and the Hirota conditions in (2+1)-dimensions. Opt Quantum Electron. 2020;52:511. 10.1007/s11082-020-02628-7Search in Google Scholar

[17] Ma WX. Soliton solutions by means of Hirota bilinear forms. Partial Differ Equ Appl Math. 2022;5:100220. 10.1016/j.padiff.2021.100220Search in Google Scholar

[18] Wazwaz AM. Exact solutions of compact and noncompact structures for the KP-BBM equation. Appl Math Comput. 2005;169(1):700–12. 10.1016/j.amc.2004.09.061Search in Google Scholar

[19] Wazwaz AM. Compact and noncompact physical structures for the ZK-BBM equation. Appl Math Comput. 2005;169(1):713–25. 10.1016/j.amc.2004.09.062Search in Google Scholar

[20] Alquran M. Solitons and periodic solutions to nonlinear partial differential equations by the sine-cosine method. Appl Math Inf Sci. 2012;6(1):85–8. Search in Google Scholar

[21] Alquran M. Physical properties for bidirectional wave solutions to a generalized fifth-order equation with third-order time-dispersion term. Results Phys. 2021;28:104577. 10.1016/j.rinp.2021.104577Search in Google Scholar

[22] Alquran M, Jaradat I, Yusuf A, Sulaiman TA. Heart-cusp and bell-shaped-cusp optical solitons for an extended two-mode version of the complex Hirota model: application in optics. Opt Quant Electron. 2021;53:26. 10.1007/s11082-020-02674-1Search in Google Scholar

[23] Kudryashov NA. One method for finding exact solutions of nonlinear differential equations. Commun Nonlinear Sci Numer Simul. 2012;17(6):2248–53. 10.1016/j.cnsns.2011.10.016Search in Google Scholar

[24] Alquran M. Classification of single-wave and bi-wave motion through fourth-order equations generated from the Ito model. Phys Scr. 2023;98:085207. 10.1088/1402-4896/ace1afSearch in Google Scholar

[25] Alquran M. Optical bidirectional wave-solutions to new two-mode extension of the coupled KdV-Schrodinger equations. Opt Quant Electron. 2021;53:588. 10.1007/s11082-021-03245-8Search in Google Scholar

[26] Bin Z. (G′∕G)-Expansion method for solving fractional partial differential equations in the theory of mathematical physics. Commun Theor Phys. 2012;58:623. 10.1088/0253-6102/58/5/02Search in Google Scholar

[27] Miah MM, Ali HMS, Akbar MA, Wazwaz AM. Some applications of the (G’/G, 1/G)-expansion method to find new exact solutions of NLEEs. Eur Phys J Plus. 2017;132:252. 10.1140/epjp/i2017-11571-0Search in Google Scholar

[28] He JH, Wu XH. Exp-function method for nonlinear wave equations. Chaos Solitons Fractals. 2006;30:700–8. 10.1016/j.chaos.2006.03.020Search in Google Scholar

[29] Jaradat I, Alquran M. A variety of physical structures to the generalized equal-width equation derived from Wazwaz-Benjamin-Bona-Mahony model. J Ocean Eng Sci. 2022;7(3):244–7. 10.1016/j.joes.2021.08.005Search in Google Scholar

[30] Alquran M, Ali M, Jadallah H. New topological and non-topological unidirectional-wave solutions for the modified-mixed KdV equation and bidirectional-waves solutions for the Benjamin Ono equation using recent techniques. J Ocean Eng Sci. 2022;7(2):163–9. 10.1016/j.joes.2021.07.008Search in Google Scholar

[31] Alquran M. New interesting optical solutions to the quadratic-cubic Schrodinger equation by using the Kudryashov-expansion method and the updated rational sine-cosine functions. Opt Quant Electron. 2022;54:666. 10.1007/s11082-022-04070-3Search in Google Scholar

[32] Mahak N, Akram G. Exact solitary wave solutions by extended rational sine-cosine and extended rational sinh-cosh techniques. Phys Scr. 2019;94:115212. 10.1088/1402-4896/ab20f3Search in Google Scholar

Received: 2024-01-25

Revised: 2024-03-03

Accepted: 2024-03-25

Published Online: 2024-05-29

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

Articles in the same Issue

https://doi.org/10.1515/nleng-2024-0005

Keywords for this article

generalized Wu–Zhang system; generalized Sawada–Kotera model; sine–cosine function method; tanh–coth expansion method

Creative Commons

BY 4.0