Exact Soliton and Kink Solutions for New (3+1)-Dimensional Nonlinear Modified Equations of Wave Propagation

1 Introduction

In the context of nonlinear (3+1)-dimensional equations, studies are flourishing because these equations describe the real features in a variety of science, technology, electrodynamics, fluid, wave propagation, and engineering areas [1-8]. Nonlinear evolution equations have been used to describe physical phenomena in fluid mechanics, plasma waves, chemical physics and solid state physics. Typical (1+1)-dimensional equations, such as the KdV equation, the mKdV equation, the BBM equation, the Boussinesq equation, the Gardner equation, and the KP equation, possess soliton solutions. The exact solutions of these equations, such as the travelling wave solutions and the soliton solutions, play a significant role in understanding various qualitative and quantitative features of nonlinear scientific phenomena. In the open literature, a set of systematic methods have been developed to obtain explicit solutions for nonlinear (1+1) and (2+1)-dimensional equations. The resulting solutions involve generic phase shifts and wave frequencies containing many existing choices [9-14].

In the recent studies in this direction, physicists, engineers, and mathematicians mostly focus their studies on the (1+1)-dimensional, such as the Korteweg-de Vires (KdV), the modified Korteweg-de Vires (mKdV) equations, and the (2+1)-dimensional integrable models such as the Kadomtsev-Petviashivili (KP) and the Nizhnik-Novikov-Veselov (NNV) equations. In other words, the solitary waves in these dimensions have been studied extensively in theoretical and experimental features. For higher dimensions, the situation is less clear. However, the physical space-time models are (3+1)-dimensional and considered to be realistic. This promotes physicists and mathematicians to invest more work to find higher-dimensional models in some ways [1,12]. Other works have been conducted employing powerful methods, such as the symmetries method, to develop (3+1)-dimensional integrable models. The use of the recursion operator is the most interesting algorithm used to establish more higher-dimensional integrable models.

A variety of powerful methods have been used to study the nonlinear evolution equations, for the analytic and numerical solutions. Examples of these methods are the Hirota bilinear method, the Hietarinta approach, the Bäcklund transformation method, multiple exp-function method, Darboux transformation, Pfaffian technique, the inverse scattering method, the Painlevé analysis, the generalized symmetry method and other methods. The inverse scattering method of integrable problems is more general than the Hirota’s bilinear method which yields special solutions. Moreover, the inverse scattering method is more complex and requires cumbersome works, whereas the Hirota’s bilinear method is mainly algebraic. The computer symbolic systems such as Maple and Mathematica allow us to perform complicated and tedious calculations.

The KdV equation is given by

that describes shallow water waves of long wavelength and small amplitude. It has been also used to model acoustic solitons in plasmas, internal gravity waves in the oceans and even blood pressure pulses. The term u_t describes the time evolution of the wave propagating in one direction, and this describes the KdV equation an evolution equation. However, The nonlinear term uu_x describes steepening of the wave, and the linear dispersive term u_xxx accounts for the spreading of the wave.

Many attempts have been invested to improve the KdV model. Benjamin-Bona-Mahony [1] introduced the regularized long-wave equation, or the BBM equation [1] that reads

replaces the third-order derivative in the Korteweg-de Vries (KdV) equation (1) by a mixed derivative −u_xxt, which results in a bounded dispersion relation. The Benjamin-Bona-Mahony (BBM) equation (2) can be used to describe the behavior of an undular bore, in water, which comprises a smooth wavefront followed by a train of solitary waves.

The aforementioned equations give a rich information in the solitary waves theory, fluid mechanics and electrodynamics. Researchers also introduced modified forms of these equations given by

and

known as the modified Korteweg-de Vries (mKdV) equation, and the modified Benjamin-Bona-Mahony (mBBM) equation respectively. However, other equations in (2+1)-dimensions and (3+1)-dimensions have been introduced and examined in the literature.

Motivated by the fact that the (3+1)-dimensional equations possess a rich development of scientific phenomena, it is then normal to propose two new sets of (3+1)-dimensional nonlinear modified equations. The first set stems mainly from the development made by Hereman [2, 3] where he proposed a new (3+1)-dimensional modified KdV given in the form

Using the same sense used for (5), we introduce two more (3+1)-dimensional modified KdV equations given by

and

Following the same way of formulation, we introduce a second set of (3+1)-dimensional equations, that possess the modified BBM equation and include the following forms

and

which will be called (3+1)-dimensional modified BBM equations.

Our aim is to study the systems proposed above. We plan to derive soliton solutions and other solutions if exist. Moreover, we will determine travelling wave solutions by using the hyperbolic and trigonometric ansatze. The computer symbolic system Maple will be used to perform the computational work.

2 The (3+1)-dimensional modified KdV equations

We study the first model of the (3+1)-dimensional modified KdV equations [2,3] that reads

We introduce the wave variable

where k, r and s are constants, and ω is the dispersion relation. A variety of ansatze will be applied to derive a set of solutions characterized by distinct physical structures.

(i) Using the sech / csch method:

The sech method admits the use of the solution in the form

where R and M are are parameters that will be determined. Using first the balance method to determine M, we find that solutions exist only if M = 1. Substituting (13) with M = 1, into (11), collecting the coefficients of coshⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find

This will give

The bell shaped soliton solution is thus given by

We also assume the solution takes the form

Proceeding as presented earlier we obtain

This gives the singular solution

(ii) Using the tanh / coth method:

The tanh method admits the use of the solution in the form

where R is a parameter that will be determined. Substituting (20) into (11), collecting the coefficients of tanhⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find

This will give

The kink solution is thus given by

We may also assume the solution takes the form

Proceeding as presented earlier we obtain

This gives the singular solution

(iii) Using the sec / csc method:

The sec ansatz sets the solution in the form

where R is a parameter that will be determined. Substituting (27) into (11), collecting the coefficients of cosⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find

This will give

The exact solution is thus given by

We also assume the solution takes the form

Proceeding as presented earlier we obtain

This gives the singular solution

(iv) Using the tan / cot method:

The tan ansatz assumes the solution in the form

where R is a parameter that will be determined. Proceeding as before, collecting the coefficients of tanⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find

The periodic solution is thus given by

In a like manner, we can derive the exact solution

Concerning the other two forms of the (3+1)-dimensional modified KdV equations

and

we can follow the same approach as presented for the first model. Based on this, we just list two solitary waves for each model. For Eq. (38) one obtains the soliton solution

and the kink solution

However, for Eq. (39) one obtains the soliton solution

and the kink solution

Some main conclusions can be made here. Hereman [2,3] introduced one (3+1)-dimensional modified KdV equation with one solitary wave solution. However, in this work we introduced two more equations, and we obtained a variety of solitary wave solutions and periodic solutions for each model. Moreover, we showed that each model gives rise to soliton and kink solutions as well.

3 The (3+1)-dimensional modified BBM equations

We study the first model of the (3+1)-dimensional modified BBM equations that reads

We next use the wave variable

where k, r and s are constants, and ω is the dispersion relation. To achieve our goal of determining a variety of solutions with distinct physical structures, we will follow the analysis used before for the set of modified (3+1)-dimensional modified KdV equations.

(i) Using the sech / csch method:

As presented earlier, the sech method introduces the solution in the form

where R and M are parameters that will be determined. Using the balance method as applied earlier, we find that solutions exist only for M = 1. Substituting (46) with M = 1 into (44), collecting the coefficients of coshⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find

This will give

This in turn gives the bell shaped soliton solution by

We also assume the solution takes the form

Proceeding as presented earlier we obtain

This gives the singular solution

(ii) Using the tanh / coth method:

The tanh method sets the solution in the form

Substituting (53) into (44), collecting the coefficients of coshⁱ(ξ), i = 0, 2, and equating each coefficient to zero, and solving the resulting equations we find

The kink solution is thus given by

We may also assume the solution takes the form

Proceeding as presented earlier we obtain the singular solution

(iii) Using the sec / csc method:

The sec ansatz sets the solution in the form

Substituting (58) into (44), collecting the coefficients of cosⁱ(ξ), i = 0, 2, and proceeding as before we find

We also assume the solution takes the form

Proceeding as presented earlier we obtain the singular solution

(iv) Using the tan / cot method:

The tan ansatz assumes the solution in the form

where R is a parameter that will be determined. Proceeding as before, collecting the coefficients of cosⁱ(ξ), i = 0, 2, and equating each coefficient to zero we find the periodic solution

In a like manner, we can derive the exact solution

Concerning the other two forms of the (3+1)-dimensional modified BBM equations

and

we can follow the same approach as presented for the first model. Based on this, we just list two solitary waves for each model. For Eq. (65) one obtains the soliton solution

and the kink solution

However, for Eq. (66) one obtains the soliton solution

and the kink solution

4 Discussion

In this work, we presented two sets of (3+1)-dimensional modified KdV equation and the modified Benjamin-Bona-Mahoney equation. The dispersion relations for these equations were found to be distinct. The constraints that guarantee the existence of soliton and kink solutions were examined and found to be related to the coefficients k, r and s of the space variables x, y, and z respectively.