1. Introduction

Among various mathematical models, nonlinear evolution equations (NLEEs) are very useful. NLEEs can model different kinds of scientific phenomena in different fields including physics, chemistry, marine, coastal and ocean engineering, fluid dynamics, and plasma physics. NLEEs can be seen in “the system of equations for the ion sound wave under the action of the ponderomotive force due to high-frequencies field”. Also, the “Langmuir wave” is considered as an important type of NLEEs. As a result, the study of NLEEs has attracted lots of researchers’ interests. Finding exact solutions of NLEEs is very important in nonlinear phenomena. There are different categories for solutions of NLEEs such as periodic-, traveling-wave-, cross-kink-wave, and soliton- solutions. Among these solutions, solitons have received much more attention in applied sciences than others, and this is because of their special properties. As a matter of fact, solitons have been appeared in different systems in nature such as shallow water waves, plasma, optical waves, matter-waves in Bose-Einstein condensates, marine, ocean and coastal waves, and ultra-short pulses in nonlinear optics. Solitons are known for keeping their shape and width unaltered even after collision with other similar solitons. There are different kinds of solitons, but we can categorize them into three original types, namely bright, dark, and singular solitons. A bright-soliton is a pulse on a zero intensity background while a dark-soliton appears as an intensity dip in an infinitely extended constant background. Further any explode-decay mode soliton is a singular soliton.

One of the attractive nonlinear evolution equations which is described the movement of water by small-amplitude and long-wave is the Boussinesq equation. Besides, Boussinesq type equations predict wave transformations in coastal fields for time dependent waves as well. For the first time in 1967, a set of equations for variable water depth is presented by Peregrine [¹]. These are effective equations for shallow water. As a matter of fact, they are the standard Boussinesq equations that are used in ocean and coastal sciences. These are effective equations to investigate shallow waters. As a matter of fact, they are the standard Boussinesq equations which are used in ocean and coastal engineering. Mathematical modelling of tsunami and tidal oscillations can be done by these equations. Further, these equations are very applicable in study of another subjects in different areas such as: the dynamics of thin slimy layers which have free surfaces, non-linear strings, shape memory in making metal compositions, paired electrical circuits, continuum limit of lattice dynamics, and wave-propagation in elastic rods (see [²-⁴]).

Consider the following standard nonlinear Boussinesq equation [⁵, ⁶]:

It’s well recognized that the investigation of propagation of waves on the surface of water is a dynamic research field in the nonlinear science. The nonlinear Boussinesq equation describes the physical phenomena in study of the dynamics for thin layers that have viscosity and free surface [⁵, ⁶]. This equation and its related type equations are also important to describe some another physical phenomena like the nonlinear lattice-waves, acoustic-waves, ion sound-waves in a plasma, the shape memory in metal compositions, the propagation of waves in elastic rods, the paired electrical circuits, and vibrations in a nonlinear string [⁷-⁹]. Moreover, it is useful for the realistic applications in the percolation of water in porous media of a horizontal layer of material, large scale atmospheric and jet stream [?, ¹⁰-¹³]. One may encounter with shallow water waves’ dynamics in different fields, for example in sea and seashore, in rivers and lakes or the other similar regions [¹¹].

The Boussinesq equation which makes the surface gravity waves is a useful model for simulating wave propagation in the long waves in ocean and seashore regions. The numerical computation is also accomplished by the Boussinesq models in the areas of wave propagation in long waves in shallow water. In ocean and coastal engineering, oceanographers and seashore engineers used the models for simulating of surface water waves in shallow seas and seaports, dune modelling, ocean basin-scale tsunami propagation, wave overtopping and inundation, and near shore wave processes [¹⁵-¹⁹].

In the last two decades, different kinds of Boussinesq equations were expanded and studied in scientific researches. In the present manuscript, we employ the ansätz method for obtaining closed form soliton solutions of the following four variants of Boussinesq equation:

Indeed, we derive some new families of analytical solutions for the Boussinesq-type models Eqs. (1)-(4). It must be noted that these equations are presented in Ref. [²⁰] for the first time. In addition, Eqs. (1)-(4) are non-integrable ones. Further Eqs. (1) and (4) contain spatial dispersion. In comparison, Eqs. (2) and (3) have spatial and temporal dispersion whereas the second and the third ones contain spatial-temporal dispersion. Besides, in Eq. (1) there are the second order dissipative term u _xx and the fourth order spatial term u _xxxx : We can have similar descriptions for terms of other equations, for example, the fourth order derivative term u _xxxx in Eq. (1) has changed to the fourth order derivative term u _xxtt in Eq. (2).

These equations were studied to find their singular- and soliton-solutions, the Hirota’s direct method obtained some solutions of these kinds as well, and other methods found more solutions for Eqs. (1)-(4) with different physical bases (see [²⁰-²²]).

The structure of this study is organized as follows: The analytical solutions with graphical representations of all solutions are presented in Sec. 2. In Sec. 3, we addressed the physical explanation for the behavior of all reported soliton solutions. The conclusion is given in the end.

2. Analytical solutions

In this present work, different kinds of solitary wave solutions are obtained for Eqs. (1)-(4). To do this, we consider three types of solutions with the general hyperbolic trigonometric forms given by

where ν = kx - ct. By these, we obtain three different families of one-soliton solutions for Eqs. (1)-(4). These families are bright, dark, and singular soliton solutions.

3. Physical discussion

It is worthy to have a physical explanation for the presented solutions of all types of Boussinesq equations. The Boussinesq equation is valid for water waves for weakly nonlinear and relatively long waves in fluid dynamics. In computer models the Boussinesq-type equations are applied for simulating of water waves in shallow seashores and seas in coastal sciences. But the Boussinesq equation is applicable to fairy long waves, which means that when the wave length in comparison with its depth of water is not short. In this work, we have obtained some different families for the Boussinesq-type equations. All obtained solutions can be categorized into three types, namely, bright-, dark-, and singular-soliton ones. We encounter a bright-soliton when the energy of the soliton reaches to its maximum value at the center of the wave and then this energy at infinity goes to 0. In this case, soliton group velocity u exceeds some thresholds. Solutions (12), (25), (40), and (52) are bright-soliton ones which are plotted in Figs. 1, 4, 7 and 10.

Figure 1 a) The bright-soliton solution u₁₁ in (12) for k = 1, b) the corresponding 2D plot for t = 0, c) graphical representation with -7 ≤ x ≤ 10, and t = 1, 2, 3. 

Figure 2 a) The dark-soliton solution u₁₂ in (15) for k = 0.5, b) the corresponding 2D plot for t = 0, c) graphical representation with -7 ≤ x ≤ 8; and t = 1, 2, 3. 

Figure 3 a) The singular-soliton solution u₁₃ in (18) for k = 0.1, b) the corresponding 2D plot for t = 0, c) Graphical representation with -7 ≤ x ≤ 10; and t = 1, 2, 3. 

Figure 4 a) The bright-soliton solution u₂₁ in (25) for k = 0.5, b) The corresponding 2D plot for t = 0, c) graphical representation with -10 ≤ x ≤ 12, and t = 1, 2. 3. 

Figure 5 a) The dark-soliton solution u₂₂ in (25) for k = 0.5, b) the corresponding 2D plot for t = 0, c) graphical representation with -7 ≤ x ≤ 10, and t = 1, 2, 3. 

Figure 6 a) The singular-soliton solution u₂₃ in (32) for k = 0.5, b) the corresponding 2D plot for t = 0, c) graphical representation with -15 ≤ x ≤ 10, and t = 1, 2, 3. 

Figure 7 a) The bright-soliton solution u₃₁ in (32) for k = 0.5, b) the corresponding 2D plot for t = 0, c) graphical representation with -10 ≤ x ≤ 12, and t = 1, 2, 3. 

Figure 8 a) The dark-soliton solution u₃₂ in (43) for k = 1, b) the corresponding 2D plot for t = 0, c) graphical representation with -5 ≤ x ≤ 10, and t = 1, 2, 3. 

Figure 9 a) The singular-soliton solution u₃₃ in (46) for k = 1, b) the corresponding 2D plot for t = 0, c) graphical representation with -15 ≤ x ≤ 10, and t = 1, 2, 3. 

Figure 10 a) The bright-soliton solution u₄₁ in (52) for k = 1, b) the corresponding 2D plot for t = 0, c) graphical representation with -7 ≤ x ≤ 10, and t = 1, 2, 3. 

A dark soliton is a solitary wave that is generated by cutting a portion of a continuous wave. As a matter of fact, a dark soliton is an amplitude dip in the continuous wave. It is generally known that dark soliton is a localized surface which is an amplitude dip, that causes a temporary decrease in wave amplitude, and can usually be seen in normal dispersion areas. Furthermore, because of its unique properties in homogeneous optical fibers, a dark-soliton is widely studied. In a homogeneous background, the intensity profile of the dark soliton shows a dip or hole-soliton. Dark-solitons have been found to be both stable and robust to losses. Soliton energy is greatest at infinity in dark-solitons, and there is a gap in the center. All solutions (15), (28), (43), and (55) are dark-soliton ones which are plotted in Figs. 2, 5, 8 and 11.

Figure 11 a) The dark-soliton solution u₄₂ in (55) for k = 1, b) the corresponding 2D plot for t = 0, c) graphical representation with -10 ≤ x ≤ 5; and t = 1, 2, 3. 

Nakamura [²³] has shown that a certain nonlinear evolution equation except soliton solutions, may also have explode decay mode solutions. This kind of soliton solutions is called a singular-soliton one that can be written by an analytical relation. All solutions Eqs. (18), (32), (46), and (59) are singular-soliton ones that are plotted in Figs. 3, 6, 9 and 12. Furthermore, parts (c) of all Figures demonstrate three plots of our solutions in 2D case for a special spatial domain and some values of time which are t = 1, 2, 3. As one can see from these figures all of them have soliton properties.

Figure 12 a) The singular-soliton solution u₄₃ in (59) for k = 1.2, b) the corresponding 2D plot for t = 0, c) graphical representation with -5 ≤ x ≤ 10; and t = 1, 2, 3. 

4. A concluding remark

In the present manuscript, the ansätz method is used for constructing singular-, dark-, and bright-soliton solutions for the distinct forms of Boussinesq-type equations. The graphs of soliton solutions with different selections of their parameters were plotted to demonstrate the localizations of the solutions that describe the nonlinear waves in coastal and ocean engineering, like tsunami waves. According to the best of our knowledge, these closed forms of solutions were not presented previously and all of them are new ones and novel. All presented solutions in this paper were tested for satisfy in their relevant equations. Also the obtained results have shown that an excellent performance of the ansätz scheme in using Boussinesq-type equations. Our results can be further extended in future research works by working on various classes of the investigated equation.