New solitary wave solutions and stability analysis of the Benney-Luke and the Phi-4 equations in mathematical physics

1.   Introduction

Due to their very broad spectrum of applicability in nonlinear science, nonlinear evolution equations (NLEEs) were very significant elements. Nonlinear physical phenomena are among the most important areas of research in science and engineering, such as plasma physics, fluid mechanics, gas dynamics, elasticity, relativity, chemical responses, ecology, optical fiber, solid state physics, biomechanics to mention few. All these equations are fundamentally controlled by NLEEs ^{[1,2,3,4,5,6,7,8,9,10]}. NLEEs are frequently used to demonstrate separate wave motion.

It has been gaining more concentration ever since the arrival of the solitary wave in science aspects. Extracting precisely travelling wave alternatives to NLEEs is therefore essential. That's because getting accurate alternatives to NLEEs offers us the freedom to present data about the characteristics of complicated physical phenomenon. Thus, in the assessment of nonlinear physical phenomenon, the development of precise traveling wave solutions to NLEEs has become a concern. Several analytical methods were used to develop wave travel alternatives for NLEEs ^{[11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31]}. Full soliton stability is not yet mathematically or physically well understood. Although these solitons, owing to resonance with the continuous spectrum, have a natural tendency to leak energy, they can still withstand this inclination and stay strong. In this work, we will provide the exact travelling wave solutions and some dispersion relations for the governing equations ^[33].

As follows, the paper is organized. Section Ⅱ presents the method descriptions. Section Ⅲ discusses the method's applications to the governing equations. Analysis of stability is being studied in Section Ⅳ. The paper is concluded by Section Ⅴ.

2.   Description of the method

In this section, we will state the main steps of GERFM as follows ^[34]:

1. Let us take into account the NPDE in the form:

Using the transformations $\psi = {\psi}(\xi)$ and $\xi = \sigma x-l t$, Eq.(2.2) is reduced to following ODE as:

where the values of $\sigma$ and $l$ will be found later.

2. Suppose that solution of Eq. (2.2) is expressed by a finite series as:

where

The values of constants $p_i, q_i(1 \leq i \leq 4)$, $A_0, A_k$ and $B_k (1 \leq k \leq M)$ are determined, in such a way that solution (2.3) always persuade Eq. (2.2). By considering the homogenous balance principle the value of $M$ is determined.

3. Putting Eq. (2.3) into Eq. (2.2) and rearranging the terms in Eq. (2.2) lead to an algebraic equations $P(Z_1, Z_2, Z_3, Z_4) = 0$ in terms of $Z_i = e^{q_i \xi}$ with $i = 1, \ldots, 4$. Equating the coefficients of $P$ to zero, a system of nonlinear equations in terms of $p_i, q_i(1 \leq i\leq 4)$, and $\sigma, l, A_0, A_k$ and $B_k (1 \leq k \leq M)$ is reached.

4. By solving the above system of equations using any symbolic computation software, the values of $p_i, q_i(1 \leq i \leq 4)$, $A_0, A_k$, and $B_k(1 \leq k \leq M)$ are determined, replacing these values in Eq. (2.3) provides us the soliton solutions of Eq. (2.1).

3.   Application of the method

3.1.   The Benny-Luke equation

Consider the BLE of the form ^[33]

In order to find the solutions of Eq. (3.1), we utilize

where $K$ and $L$ are arbitrary constants to be determined.

If we use transformation (3.2) in Eq. (3.1), after an integration along with neglecting constant of integration, the following nonlinear ODE is obtained

Balancing the terms of $\textbf{u}'''$ and $\left(\textbf{u}'\right)^2$ in Eq. (3.3) gives $M+3 = 2(M+1)$, and $M = 1$. Using $M = 1$ along with Eqs. (2.3) and (2.4), one gets:

Using a methodology similar to the one adopted in Subsection 2, we get some solutions of (3.1), as bellows:

Family 1: We attain $p = [ -1,-1 , 1, -1]$ and $q = [1,-1,1,-1]$, so we will obtain

Case 1:

So, the solitary wave solutions of Eq. (3.1) takes the form of

Thus the solution of (3.1) is obtained as

where $\xi = K\left(x-{\frac { \sqrt {4 {K}^{2}\gamma-1}}{\sqrt {4 \delta {K}^{2}-1}}}t\right)$.

Case 2:

So, the solitary wave solutions of Eq. (3.1) takes the form of

Therefore the solution of (3.1) is attained as

where $\xi = K\left(x+ {\frac { \sqrt {16 {K}^{2}\gamma-1}}{\sqrt {16 \delta {K}^{2}-1}}}t\right)$.

Case 3:

So, the solitary wave solutions of Eq. (3.1) takes the form of

Thus we attain the solution of (3.1) as follows

where $\xi = K\left(x+ {\frac { \sqrt {4 {K}^{2}\gamma-1}}{\sqrt {4 \delta {K}^{2}-1}}}t\right)$.

Case 4:

So, the solitary wave solutions of Eq. (3.1) takes the form of

And hence we attain the solution of (3.1) in the form

where $\xi = K\left(x-{\frac { \sqrt {16 {K}^{2}\gamma-1}}{\sqrt {16 \delta {K}^{2}-1}}}t\right)$.

Family 2: We attain $p = [-i, -i, 1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Inserting these values in Eqs.(3.4) and (3.10), one gets

Therefore we attain the solution of (3.1) as follows

where $\xi = K\left(x-{\frac { \sqrt {4 {K}^{2}\gamma+1}}{\sqrt {4 \delta {K}^{2}+1}}}t\right)$.

Case 2:

Plugging these values in Eqs.(3.4) and (3.10), one gets

Thus the solution of (3.1) is given by

where $\xi = K\left(x+ {\frac { \sqrt {16 {K}^{2}\gamma+1}}{\sqrt {16 \delta {K}^{2}+1}}}t\right)$.

Family 3: We attain $p = [-2-i, –2+i, 1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Imposing these values in Eqs. (3.4) and (3.13), one gets

where $E_1 = \left(2\cos \left(\xi \right) - \sin\left(\xi\right) \right)$, we then reach the solution of (3.1 stated as

where $\xi = K\left(x-{\frac { \sqrt {4 {K}^{2}\gamma+1}}{\sqrt {4\delta {K}^{2}+1}}}t\right)$.

Case 2:

Inserting these values in Eqs. (3.4) and (3.13), one reaches

where $E_{2} = A_{{0}} \left(4 {K}^{2}\gamma+ 1 \right) \left(4\delta {K}^{2}+1 \right).$ Therefore we attain the solution of (3.1) as

where $\xi = K\left(x+{\frac { \sqrt {4 {K}^{2}\gamma+1}}{\sqrt {4 \delta {K}^{2}+1}}}t\right)$.

Family 4: We attain $p = [1-i, 1+i, -1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Putting these values in Eqs.(3.4) and (3.16), one gets

where $E_3 = A_{{0}} \left(4 {K}^{2}\gamma+1 \right) \left(4 \delta {K}^ {2}+1 \right).$ Therefore we attain the solution of (3.1) in the following form

where $\xi = K\left(x-{\frac { \sqrt {4 {K}^{2}\gamma+1}}{\sqrt {4 \delta {K}^{2}+1}}}t\right)$.

Case 2:

So, the solitary wave solutions of Eq. (3.1) takes the form of

$E_4 = A_{{0}} \left(4{K}^{2}\gamma+1 \right) \left(4\delta {K}^{2}+1 \right).$ Therefore one can reach the solution of (3.1) as comes next

where $\xi = K\left(x- {\frac { \sqrt {4 {K}^{2}\gamma+1}}{\sqrt {4 \delta {K}^{2}+1}}}t\right)$.

Family 5: We attain $p = [ -1,3,1,- 1]$ and $q = [1,-1,1,-1]$, so we will obtain

Case 1:

Imposing these values in Eqs. (3.4) and (3.19), one gets

where $E_5 = \sinh \left(\xi \right) \left(4{K}^{2}\gamma-1 \right) \left(4\delta {K}^{2}-1 \right)$. Therefore we attain the solution of (3.1) as

where $\xi = K\left(x- {\frac { \sqrt {4 {K}^{2}\gamma-1}}{\sqrt {4 \delta {K}^{2}-1}}}t\right)$.

Family 6: We attain $p = [ -3,-1 , 1, 1]$ and $q = [1,-1,1,-1]$, so we will obtain

Case 1:

Putting these values in Eqs.(3.4) and (3.21), one gets

$E_6 = \left(4 \delta {K}^{2}-1 \right) A_{{0}} \left(4 {K}^{2}\gamma-1 \right).$ Therefore we reach the solution of (3.1) as follows

where $\xi = K\left(x+ {\frac { \sqrt {4 {K}^{2}\gamma-1}}{\sqrt {4 \delta {K}^{2}-1}}} t\right)$.

Family 7: We attain $p = [ -3,-2, 1, 1]$ and $q = [0,1,0,1]$, so we will obtain

Case 1:

Plugging these values in Eqs.(3.4) and (3.23), one attains

where $E_7 = \left({K}^{2}\gamma-1 \right) \left(3/2+{{\rm e}^{\xi}} \right).$ Therefore we attain the solution of (3.1) as

where $\xi = K\left(x- {\frac { \sqrt {{K}^{2}\gamma-1}}{\sqrt {\delta {K}^{ 2}-1}}} t\right)$.

Family 8: We attain $p = [ 1,0,1, 1]$ and $q = [1,0,1,0]$, so we will obtain

Case 1:

Putting these values in Eqs.(3.4) and (3.25), one reaches

Therefore we attain the solution of (3.1) as comes next

where $\xi = K\left(x+ {\frac { \sqrt {{K}^{2}\gamma-1}}{\sqrt {\delta {K}^{2 }-1}}}t\right)$.

Family 9: We attain $p = [ 2,1, 1, 1]$ and $q = [1,0,1,0]$, so we will obtain

Case 1:

Putting these values in Eqs.(3.4) and (3.27), we have

Therefore we reach the solution of (3.1) as comes next

where $\xi = K\left(x- {\frac { \sqrt {{K}^{2}\gamma-1}}{\sqrt {\delta {K}^{2}-1}}}t\right)$.

Family 10: We attain $p = [ -1,0, 1, 1]$ and $q = [0,1,0,1]$, so we will obtain

Case 1:

Employing these values to Eqs. (3.4) and (3.29), one reaches

Therefore we attain the solution of (3.1) is as comes next

where $\xi = K\left(x+ {\frac { \sqrt {{K}^{2}\gamma-1}}{\sqrt {\delta {K}^{2 }-1}}} t\right)$.

3.2.   The Phi-4 equation

Now, let us Consider the PE given by ^[33]

To solve Eq. (3.31), we again apply the travelling wave transformation (3.2). Then Eq. (3.31) turns to the following nonlinear ODE as

Balancing the terms of $\textbf{u}''$ and $\textbf{u}^3$ in Eq. (3.32) gives $3N = N+2$, so $N = 1$. So, the solution of Eq. (3.31) will be as:

Family 1: We attain $p = [1+i, 1-i, 1, -1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Putting these values into Eqs. (3.33) and (3.34), one gets

Therefore we attain the solution of (3.31) given by

Case 2:

Inserting these values into Eqs. (3.33) and (3.34), one gets

Therefore we attain the solution of (3.31) stated as

Family 2: We attain $p = [1-i, 1-i, 1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Putting these values into Eqs. (3.33) and (3.37), one gets

Therefore we attain the solution of (3.31) as comes next

Family 3: We attain $p = [-2-i, -2+i, 1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Imposing these values into Eqs. (3.33) and (3.39), one gets

Hence we reach the solution of (3.31) as follows

Family 4: We attain $p = [1-i, -1-i, -1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Putting these values into Eqs. (3.33) and (3.41), one gets

Therefore we attain the solution of (3.31) as comes next

Family 5: We attain $p = [2-i, -2-i, -1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Inserting these values into Eqs. (3.33) and (3.43), one gets

Therefore we attain the solution of (3.31) given as

Family 6: We attain $p = [i, -i, 1, 1]$ and $q = [i, -i, i, -i]$, so we will obtain

Case 1:

Employing these values into Eqs. (3.33) and (3.45), one gets

Therefore we attain the solution of (3.31) is as comes next

Family 7: We attain $p = [ -1,-1 , 1, -1]$ and $q = [1,-1,1,-1]$, so we will obtain

Case 1:

Putting these values into Eqs. (3.33) and (3.47), one gets

Therefore we attain the solution of (3.31) as follows

4.   Stability analysis

In this section, the stability analysis ^[35,36,37] for the governing equations that is (3.1) and (3.31) will be analyzed.

4.1.   Stability for Eq. (3.1)

Consider the perturbed solution of the form ^[37]

it can be easily seen that, any constant $P_{0}$ is a steady state solution for (3.1). Plugging (4.1) into (3.1), one arrives at

linearizing the above equation in $\tau$, we reach

Assume that (4.3) has a solution of the form

where $k$ is the normalized wave number, plugging (4.4) into (4.3)

solving for $\omega$, we obtain

From (4.6), one can see that the real part is negative for all $k$ values, then any superposition of the solutions will appear to decay. Thus, the dispersion is stable.

4.2.   Stability for Eq. (3.31)

In a similar manner, consider the perturbed solution of the form

it is plainly to see that in (4.7) any constant $P_{1}$ is a steady state solution for (3.31). Plugging (4.7) into (3.31), one gets

linearizing the above equation in $\tau$, one gets

Suppose again that (4.9) has a formal solution of the form (4.4), inserting (4.4) into (4.9), one gets

solving for $\omega$ from the immediate equation, we acquire

From (4.11), it can be seen that the real part is negative for all $k, m, P^{2}_{1}$ values. Thus, any superposition of the solutions will appear to decay. Hence, the dispersion is stable.

5.   Conclusion

In this work, we present new solitary wave solutions for the BLE and PE. We applied the new GERFM to reach such solutions. Moreover, the stability for the governing equations is investigated via the aspect of linear stability analysis. It has been proved that, both the governing equations are stable. These new families of solutions are shown the power, effectiveness and fruitfulness of this method. These fresh solutions have many applications in physics and other physical sciences branches. Other nonlinear PDEs involving mathematical physics and other different branches of physical sciences can also be solved through this method.

Acknowledgments

We would like to express our thanks to the anonymous referees who help us improved this paper.

Conflict of interest

All authors declared that there is no conflict of interest in this paper.