## Abstract

The variant Boussinesq equation has significant application in propagating long waves on the surface of the liquid layer under gravity action. In this article, the improved Bernoulli subequation function (IBSEF) method and the new auxiliary equation (NAE) technique are introduced to establish general solutions, some fundamental soliton solutions accessible in the literature, and some archetypal solitary wave solutions that are extracted from the broad-ranging solution to the variant Boussinesq wave equation. The established soliton solutions are knowledgeable and obtained as a combination of hyperbolic, exponential, rational, and trigonometric functions, and the physical significance of the attained solutions is speculated for the definite values of the included parameters by depicting the 3D profiles and interpreting the physical incidents. The wave profile represents different types of waves associated with the free parameters that are related to the wave number and velocity of the solutions. The obtained solutions and graphical representations visualize the dynamics of the phenomena and build up the mathematical foundation of the wave process in dissipative and dispersive media. It turns out that the IBSEF method and the NAE are powerful and might be used in further works to find novel solutions for other types of nonlinear evolution equations ascending in physical sciences and engineering.

## 1 Introduction

Since many processes in science, technology, and engineering are modeled through nonlinear evolution equations (NLEEs), the investigation of closed-form analytical solutions of NLEEs is very important. Closed-form solutions provide further physical information and help us to understand the processes of the associated physical systems. Consequently, their studies are of fundamental importance due to the effective application of the analytical solutions in various fields, such as plasma physics, solid-state physics, neural physics, chaos, diffusion process, reaction process, optical fibers, nonlinear optics, quantum mechanics, mathematical biology, propagation of shallow water waves, and electromagnetic theory [1,2,3,4,5,6,7,8,9,10,11]. On account of this, researchers established several techniques and tried to examine various types of NLEEs with the help of computer algebra like Maple, MATLAB, and Mathematica. However, not all models are solvable by a single method. Consequently, several methods [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31] have been established by mathematicians, physicists, and engineers.

Among the approaches, the improved Bernoulli subequation function (IBSEF) method [32,33] and the new auxiliary equation (NAE) [34] method are effective, direct, and compatible algebraic methods for establishing exact soliton solutions to NLEEs. In 1871, the Boussinesq equation was derived to describe certain physical processes. Since then, several generalizations and some variants of this model have been established in the literature. To the best of our understanding, through different schemes, several researchers have investigated the variant Boussinesq equation, namely: Gao and Tian [35] examined it through the generalized tanh method; Yan and Zhang [36] used an improved sine–cosine method and Wu elimination method; Jabbari *et al*. [37] investigated it by the homotopy analysis method; Ul-Hassan [38] applied exp-function method; Manafian *et al.* [39] implemented the improved tanh(
*et al.* [40] searched based on the generalized unified method; and Alharbi and Almatrafi [41] exploited the improved exp(
*et al.* [43] studied the long nonlinear internal waves between analytical and numerical wave models in shallow water. The (*G*′/*G*)-expansion procedure was used to achieve the exact and explicit solitary wave solutions to the model in the presence of dispersive coefficient.

Nevertheless, the stated model has not been investigated through the IBSEF method and the NAE approach. Therefore, the purpose of this article is to accomplish broad-ranging and adequate standard soliton solutions to the variant Boussinesq equations by putting into use the proposed methods. Besides, we analyze various types of waves for different values of the free parameters of the obtained solutions illustrated in the 3D plot *via* MATLAB and highlight the significance of wave number, velocity, and speed of the solutions in changing the nature of the wave profile.

## 2 Description of two proposed methods

This section introduces and analyzes the IBSEF method and the NAE method in detail.

### 2.1 The IBSEF method

To illustrate the IBSEF method, we take into consideration the NLEE associated with two independent variables,

where

Under the traveling wave variable

where

where

where

The values of

Equalizing the coefficients of

With Mathematica software, we can unravel the system of algebraic equations to determine the values of

Thus, by embedding the values of the parameters

### 2.2 The NAE method

Suppose the NLEE [38]

where

Step 1: To format into an ordinary differential equation of partial differential equation (2.2.1), we need to choose a wave variable as

where

where prime means the derivative with respect to

Step 2: In harmony with the NAE method, the exact soliton solution of Eq. (2.2.3) is assigned to be

where

Step 3: The balancing principle needs to be applied to find the value of the positive integer

Step 4: Eq. (2.2.3) provides a polynomial of

The values of

## 3 Solution analysis

In this section, we have established some standard and broad-ranging explicit wave solutions through the aforementioned methods of the variant Boussinesq equation, which has the following form [39]:

To convert the nonlinear model (3.1), we used the wave transformation as follows:

Using Eq. (3.2) into Eq. (3.1) and integrating, we obtain

Eq. (3.3) gives

By replacing the value of

### 3.1 Analysis of solutions through the IBSEF method

To apply the procedure of the IBSEF method, we gain the correspondence for *q*, *p*, and
*u*″ and

Picking

where

The substitution of Eq. (3.1.1) together with Eq. (2.1.5) in Eq. (3.6) yields a polynomial in

For

**Case 1(a):** By making use of the values stated in (3.1.2) along with (2.1.8) in Eq. (3.1.1), the subsequent exponential function solution to the variant Boussinesq equation is established as follows:

where

Rewrite the solution function (3.1.4) into hyperbolic form

where

Let us choose

and we obtain a part of the solution of (3.1) by putting

For

and we undertake from (3.5)

where

On the other hand, by setting

and based on Eq. (3.5), we obtain

where

**Case 2(a):** Considering the value of the parameter arranged in (3.1.3) with the help of Eq. (2.1.8) in Eq. (3.1.1), we accomplish the following solution:

which can be converted to hyperbolic form as follows:

where

Setting

and on behalf of Eq. (3.5),

Choose

and Eq. (3.5) yields

When

and from Eq. (3.5),

where

For

**Case 1(b):** By substituting the values of the parameters arranged in (3.1.2) along with Eqs. (2.1.2) and (2.1.9) in Eq. (3.1.1), we obtain the exponential function solution to the variant Boussinesq equation of the exponential form:

where

Eq. (3.1.20) can be written as

Choosing

and another solution of Eq. (3.1) obtained by captivating (3.5) is

If we pick

and Eq. (3.5) gives

When

and using Eq. (3.5) gives

where

**Case 2(b):** By using Eq. (3.1.3) with the help of Eqs. (2.1.2) and (2.1.9) from Eq. (3.1.1), we accomplish the following solution:

where

Calculating the result

When

and

Applying the value

and

where

It is important to note that the wave solutions of the variant Boussinesq equation found here are functional and useful and were not proven in the earlier research. The solutions derived earlier might be fruitful in investigating unidirectional wave propagation in nonlinear media and dispersive relativistic one-particle theory, *etc*.

### 3.2 Analysis of solutions through the NAE method

To find the solutions to the stated equation through the NAE method, we gain the value of

Using the NAE approach, substitute the value of

where

Based on the solution (3.2.1) with help of (2.2.5) from Eq. (3.6), we assert

Equalizing the like power of

Set 1:

Set 2:

Based on the above values of the parameters in Eq. (3.2.4), as follows:

By substituting the values of the constants arranged in (3.2.3) and (2.2.5) into Eq. (3.2.1), we obtain the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

where

or

and

where

By substituting the values of the constant arranged in (3.2.4) and (2.2.5) into Eq. (3.2.1), we obtain the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

When

By combining (3.2.3) and (2.2.5) in Eq. (3.2.1), we accomplish the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

Using (3.2.4) and (2.2.5) into Eq. (3.2.1), we achieve the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

When

Employing (3.2.3) together with (2.2.5) into Eq. (3.2.1), we achieve the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

Considering (3.2.4) together with (2.2.5) into Eq. (3.2.1), we attain the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) in the subsequent form

and

or

and

where

When

Inserting (3.2.3) together with (2.2.5) into Eq. (3.2.1), we secure the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

By involving (3.2.4) along with (2.2.5) into Eq. (3.2.1), we derive the solution of stated Eqs. (3.1)–(3.2.1) and (3.5) as follows:

and

or

and

where

When

Appling (3.2.3) along with (2.2.5) into Eqs. (3.2.1) and (3.5) gives

and

or

and

Substituting (3.2.4) along with (2.2.5) into Eqs. (3.2.1) and (3.5) gives

and

or

and

where

When

Setting (3.2.3) together with (2.2.5) into Eq. (3.2.1), we derive the solutions of (3.1) from Eqs. (3.2.1) and (3.5), which provides

and

or

and

By means of (3.2.4) together with (2.2.5) from Eq. (3.2.1), we derive the solutions of (3.1) in the subsequent form

and

or

and

where

By means of (3.2.3) together with (2.2.5) into Eq. (3.2.1), we gain the solutions of (3.1) from Eqs. (3.2.1) and (3.5), which gives

and

Using (3.2.4) along with (2.2.5) into Eq. (3.2.1), we derive the solutions of (3.1) from Eqs. (3.2.1) and (3.5), which provides

and

where

When

and

Providing (3.2.4) along with (2.2.5) into Eq. (3.2.1), we attain the solution functions (3.2.1) and (3.5) as shown below

and

where

When

Applying the parametric values stated in (3.2.3) into Eqs. (3.2.1) and (3.5) represent the solutions

and

When

Substituting the value assigned in (3.2.3) and Eq. (2.2.5) into Eq. (3.2.1) yields the solution

Using the value

Applying (3.2.4) and Eq. (2.2.5) to Eqs. (3.2.1) and (3.5) gives

and

respectively, where

When

Selecting (3.2.3) and (2.2.5) for Eq. (3.2.1) and after simplification, we obtain

Choosing

and from Eq. (3.5),

where

If

and (3.5) gives

Based on (3.2.4) with the help of (2.2.5), after computation, the solution function (3.2.1) becomes

For

and from Eq. (3.5), we obtain

where

Selecting

and obtain from Eq. (3.5) as

where

When

We attain the required solution of (3.1) by considering (3.2.3) and (2.2.5) into (3.2.1) and (3.5) as follows:

and

where

Again, by selecting the values of (3.2.4) and (2.2.5) and putting into Eqs. (3.2.1) and (3.5), we obtain

and

where

When

Proceeding in this way, for (3.2.3) we preserve

and

where

Also for (3.2.4), we gain the required solution of (3.1) as follows:

and

where

When

and

From the above investigation, we have found various types of solutions of the variant Boussinesq equation, such as hyperbolic function, trigonometric solution, exponential function solution, and rational solution.

## 4 Result discussions and wave profile description

In this section, the obtained traveling wave solutions of the stated equations are represented in the figures and reviewed the natures of these waves for dissimilar values of the free parameters with the aid of software MATLAB.

### 4.1 Wave profile interpretations of attained solutions using the IBSEF method

In this article, we have found 16 solutions (whereas Manafian *et al*. [39] established 15 solutions) using the IBSEF approach, which are illustrative and have rich structured wave profiles that might be helpful in describing the unidirectional wave propagation in nonlinear media with dispersion relativistic one-particle theory. It is important to note that the wave solutions of the variant Boussinesq equation found here are functional and useful and were not proven in the earlier research.

For

The profile of the solution

For very small values of