Soliton solutions to generalized (3 + 1)-dimensional shallow water-like equation using the (ϕ'/ϕ,1/ϕ)-expansion method

Abstract

The (3 + 1)-dimensional generalized shallow water equation is a significant mathematical framework for analyzing the dynamic behavior of waves in ocean physics. The purpose of this article is to investigate some more generic soliton solutions of the generalized shallow water-like model in three dimensions. The investigation is conducted utilizing the sophisticated mathematical methodology known as the double variables $(\varphi \prime /\varphi ,1/\varphi )$ expansion technique. With this approach, we produce new propagating wave solutions for this model in the form of hyperbolic, trigonometric, and rational functions. In addition, we offer two- and three-dimensional graphical representations to help visualize the intricate physical phenomena of the system. We have constructed many soliton solutions, such as kink shape soliton solutions, anti-bell shape solutions, single periodic solutions, singular soliton solutions, and anti-kink shape solutions for different values of the free parameters involved in the obtained solutions. These graphical representations are predicated on certain parameter selections, which facilitate the understanding of the complicated general behavior for this model. Through the presentation of new findings in the field of soliton solutions for the aforementioned equation, this paper offers fresh perspectives and highlights hitherto overlooked aspects of this fascinating mathematical challenge. The paper illuminates new results on soliton solutions with different geometrical structures for the given equation, revealing hitherto overlooked facets of this intriguing mathematical challenge.

Keywords:

• The (3 + 1)-dimensional generalized shallow water equation
• the double variables $(\varphi \prime /\varphi ,1/\varphi )$ expansion method
• analytical wave solutions

1. Introduction

In a wide range of scientific, engineering, and technological areas, mathematical solutions are superior to experimental results in terms of understanding physical phenomena (Fang, Nadeem, Habib, Karim, & Wahash, 2022; He, Nadeem, Habib, Sedighi, & Huang, 2022; Nadeem & He, 2021). As a result, physical phenomenon emerging in scientific and technological areas are mathematically represented using differential equations. Nonlinear evolution equations, often deployed in modeling phenomena, have emerged as a particularly captivating domain of inquiry for researchers. By formulating and solving these equations, we can gain a deeper understanding of complex systems and develop more accurate predictions about their behavior.

The (3 + 1)-dimensional shallow water equation is a useful model for studying waves in a variety of ocean science. It is used to explain the motion of water under the assumption that the depth of the water is small in comparison to the wavelength of the waves. The shallow water equation is a nonlinear evolution equation with traveling wave solutions, which are significant for different reasons. Soliton solutions to the aforementioned equation can reveal information about the behavior of waves in various physical systems, such as the formation and propagation of tsunamis, storm swells, and tidal waves. The shallow water equation and its traveling wave solutions are critical for the study of wave energy, which is a quickly expanding field.

There are some studies on this equation. Some academics obtained the solutions for equation (3.1)(3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1) by the improved Bernouli’s sub-equation method (Dusunceli, 2019), the generalized bilinear operator (Zhang, Dong, Zhang, & Yang, 2017), the Hirota bilinear form (Tang, Ma, & Xu, 2012; Tian & Gao, 1996), the tanh method (Zayed, 2010), and the $(G\prime /G)$ expansion method (Sadat, Kassem, & Ma, 2018).

Solving nonlinear evolution equation is not an easy task. There are many useful methods have been represented such as the generalized method (Boakye, Hosseini, Hinçal, Sirisubtawee, & Osman, 2024), the exp-function method (Bekir, 2009; He & Wu, 2006; Yusufoglu, 2008), the (G′/(G′+ G + A))-expansion method (Ganie et al., 2023), the generalized Kudryashov method (Kumar, Park, Tamanna, Paul, & Osman, 2020), the sine cosine method (Wazwaz, 2004), the Jacobi elliptic method (Chen & Wang, 2005; Liu, Fu, Liu, & Zhao, 2001; Lü, 2005), the homogeneous balance method (Radha & Duraisamy, 2021), the tanh function method (Abdou, 2007; Fan, 2000), the Sardar sub-equation method (Rehman, Akber, Wazwaz, Alshehri, & Osman, 2023), the first integral method (Martínez, Aguilar, & Atangana, 2018), the Backlund transformation method (Rogers & Shadwick, 1982), (G′/G)-expansion method (Abazari & Abazari, 2011; Borhanifar & Moghanlu, 2011; Elagan, Sayed, & Hamed, 2011; Feng, Li, & Wan, 2011; Islam, Al-Amin, Akbar, Wazwaz, & Osman, 2023; Jabbari, Kheiri, & Bekir, 2011; Naher, Abdullah, & Akbar, 2011; Öziş & Aslan, 2010; Wang, Li, & Zhang, 2008), the central finite difference approximations (Khalid et al., 2022), the two variables $(\mathrm{G}\prime /\mathrm{G},1/\mathrm{G})$-expansion method (Demiray, Ünsal, & Bekir, 2014, 2015; Mamun, Shahadat, Akbar, & Wazwaz, 2017; Miah, Iqbal, & Osman, 2023; Shakeel & Mohyud-Din, 2014), and many more (Abdel-Gawad & Osman, 2013; 2014; Baskonus et al., 2021; El-Sherif & Shoukry, 2007; Fahim, Kundu, Islam, Akbar, & Osman, 2022; Fetoh, Asla, El-Sherif, El-Didamony, & El-Reash, 2019; He, Hou, He, Saeed, & Hayat, 2021; He et al., 2023; He & Liu, 2023; Soliman, Amin, El-Sherif, Sahin, & Varlikli, 2017).

In this article, we shall endeavor to explore soliton solution of the generalized (3 + 1) shallow water equation utilizing the two variables $(\varphi \prime /\varphi ,1/\varphi )-$ expansion method, which is an advancement of the $(G\prime /G$)-expansion method, was originally introduced by Li et al. in 2010 to derive precise solutions for the Zakharov equation. Subsequently, this technique has exhibited remarkable efficacy in resolving a wide range of nonlinear partial differential equations. However, to the best of our knowledge, the $(\varphi \prime /\varphi ,1/\varphi )$-expansion method has not yet been employed to derive the soliton solution of the generalized (3 + 1) shallow water equation. Thus, this study has the potential to illuminate the behavior of this equation and the proficiency of the $(\varphi \prime /\varphi ,1/\varphi )$-expansion method in solving other nonlinear evolution equations.

2. Methodology

The $(\varphi \prime /\varphi ,1/\varphi )$-expansion technique, which provides a precise method of solving nonlinear equations, is the first thing we shall look at. Consider about a second-order LODE (linear ordinary differential equation). (2.1) $$\varphi ″\left(\xi \right)+\lambda \varphi \left(\xi \right)=\mu ,$$(2.1)

Afterward, consider the relationships that follows (2.2) $$\varphi =\frac{{\varphi }^{\prime }}{\varphi }\text{\hspace{0.5em}and\hspace{0.5em}}\psi =1/\varphi .$$(2.2)

The above relationships allow for its derivation. (2.3) $${\varphi }^{\prime }=-{\varphi }^2+\mu \psi -\lambda ,\psi \prime =-\varphi \mathrm{ }\psi .$$(2.3)

Three different sorts of solutions can be discovered for the Equation (2.1)(2.1) $$\varphi ″\left(\xi \right)+\lambda \varphi \left(\xi \right)=\mu ,$$(2.1) for various values of λ.

Remark 1:

When the value of $\lambda$ > 0, the Equation (2.1)(2.1) $$\varphi ″\left(\xi \right)+\lambda \varphi \left(\xi \right)=\mu ,$$(2.1) has a generic solution, which is expressed as follows. (2.4) $$\varphi \left(\xi \right)=m_1\mathrm{ }\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\mathrm{ }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\mathrm{ }\xi \right)+\frac{\mu }{\lambda }.$$(2.4) (2.5) $$\mathrm{\text{Thus}},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained\hspace{0.5em}}{\psi }^2=\frac{-\lambda ({\varphi }^2-2\mu \psi +\lambda )}{{\lambda }^2\beta +{\mu }^2},$$(2.5)

Wherein $\beta ={m_1}^2-{m_2}^2$ and $m_1,\hspace{0.5em}{m}_2$ are arbitrary constants.

Remark 2:

The generic solution of Equation (2.1)(2.1) $$\varphi ″\left(\xi \right)+\lambda \varphi \left(\xi \right)=\mu ,$$(2.1) has the following form for $\lambda$ > 0, (2.6) $$\varphi \left(\xi \right)=m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda }\mathrm{ }\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda }\mathrm{ }\xi \right)+\frac{\mu }{\lambda },$$(2.6) (2.7) $$\text{And\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained\hspace{0.5em}}{\psi }^2=\frac{\lambda ({\varphi }^2-2\mu \psi +\lambda )}{{\lambda }^2\beta -{\mu }^2},$$(2.7)

Wherein, $\mathrm{ }\beta ={m_1}^2+{m_2}^2.$

Remark 3:

Again, assume λ = 0, then the solution of Equation (2.1)(2.1) $$\varphi ″\left(\xi \right)+\lambda \varphi \left(\xi \right)=\mu ,$$(2.1) is of the form, (2.8) $$\varphi \left(\xi \right)=\frac{\mu }{2}{\xi }^2+m_1\xi +m_2,$$(2.8) (2.9) $$\text{And\hspace{0.5em}thus},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained},\hspace{0.5em}{\psi }^2=\frac{({\varphi }^2-2\mu \psi )}{{m_1}^2-2\mathrm{ }\mu m_2}.$$(2.9)

$\mathit{\text{Step}} 1:$ Let us consider a nonlinear evolution equation involving two independent variables, namely x and t, in the following context. (2.10) $$Q\left(u,u_t,u_x,u_{\mathit{\text{tt}}},u_{\mathit{\text{xt}}},u_{\mathit{\text{xx}}},\dots .\right)=0.$$(2.10)

Here $F$ is a polynomial in $u(x,t)$ and its partial derivatives. We present the independent variables $x$ and $t,$ along with the wave variable $\xi ,$ which is related to them as follows: (2.11) $$u\left(\xi \right)=u\left(x,t\right),\xi =x-\mathit{\text{vt}},$$(2.11)

Where, $v$ is the traveling wave velocity. Applying (2.10)(2.10) $$Q\left(u,u_t,u_x,u_{\mathit{\text{tt}}},u_{\mathit{\text{xt}}},u_{\mathit{\text{xx}}},\dots .\right)=0.$$(2.10) into (2.9)(2.9) $$\text{And\hspace{0.5em}thus},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained},\hspace{0.5em}{\psi }^2=\frac{({\varphi }^2-2\mu \psi )}{{m_1}^2-2\mathrm{ }\mu m_2}.$$(2.9) produces an ODE for $u(\xi )$ (2.12) $$F(u,u\mathrm{\prime },u\mathrm{″},\dots )=0.$$(2.12) $\mathit{\text{Step}} 2:$ Assume that the solution of the ODE (2.12)(2.12) $$F(u,u\mathrm{\prime },u\mathrm{″},\dots )=0.$$(2.12) can be described as a polynomial in $\varphi$ and $\psi$ (2.13) $$u\left(\xi \right)=\sum _{i=0}^n{a}_i{\varphi }^i\left(\xi \right)+\sum _{i=1}^n{b}_i{\varphi }^{i-1}\left(\xi \right){\psi }^i\left(\xi \right),$$(2.13)

Where $\varphi =\varphi (\xi )$ satisfies a second-order linear differential equation, and $\varphi$ and $\psi$ are defined in Equation (2.2)(2.2) $$\varphi =\frac{{\varphi }^{\prime }}{\varphi }\text{\hspace{0.5em}and\hspace{0.5em}}\psi =1/\varphi .$$(2.2) . In this context, the constants $a_i(i=0,1,2,\dots ,N),$ $b_i\left(i=0,1,2,\dots ,N\right),$λ, and μ are arbitrary, while N is a positive integer that can be regarded as the homogeneous balance number. The condition of homogeneous balance is applied by equating the highest-order derivatives with the nonlinear terms in the ODE (2.12)(2.12) $$F(u,u\mathrm{\prime },u\mathrm{″},\dots )=0.$$(2.12) . Utilizing the double $(\varphi \prime /\varphi ,1/\varphi )-$ expansion approach, the values of these arbitrary constants can be determined.

$\mathit{\text{Step}} 3:$ By substituting Equation (2.13)(2.13) $$u\left(\xi \right)=\sum _{i=0}^n{a}_i{\varphi }^i\left(\xi \right)+\sum _{i=1}^n{b}_i{\varphi }^{i-1}\left(\xi \right){\psi }^i\left(\xi \right),$$(2.13) into Equation (2.12)(2.12) $$F(u,u\mathrm{\prime },u\mathrm{″},\dots )=0.$$(2.12) while utilizing Equations (2.3)(2.3) $${\varphi }^{\prime }=-{\varphi }^2+\mu \psi -\lambda ,\psi \prime =-\varphi \mathrm{ }\psi .$$(2.3) and (2.5)(2.5) $$\mathrm{\text{Thus}},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained\hspace{0.5em}}{\psi }^2=\frac{-\lambda ({\varphi }^2-2\mu \psi +\lambda )}{{\lambda }^2\beta +{\mu }^2},$$(2.5) , we derive a polynomial that involves both $\varphi$ and $\psi .$ Notably, the degree of ψ in this polynomial does not exceed one. By equating the coefficients of ϕ and ψ to zero, we establish a system of algebraic equations encompassing variables such $a_i(i=0,1,2,\dots ,N),$ $b_i\left(i=0,1,2,\dots ,N\right),$ v, λ, μ, $m_1,$ and $m_2.$

$\mathit{\text{Step}} 4:$ The algebraic equations given in Step 3 can be solved using Maple, a mathematical program. The final accumulation of the exact solution for Equation (2.12)(2.12) $$F(u,u\mathrm{\prime },u\mathrm{″},\dots )=0.$$(2.12) can then be achieved by determining the consequent values for ${ a}_i,b_i,c,\lambda ,\mu ,m_1,m_2.$

3. Application to SWL equation

In this specific section, our objective is to elucidate the precise soliton solutions for the generalized (3 + 1)-dimensional shallow water-like equation, employing the two variables $(\varphi \prime /\varphi ,1/\varphi$)-expansion method. Here the (3 + 1) shallow water-like equation is (3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1)

Let us assume that the wave function has the following arrangement (3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),$$(3.2)

With the transformation $\xi =x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}},$ where $v$ is the traveling wave speed.

Using the above transformation in the Equation (3.1)(3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1) , we get the nonlinear ordinary differential equation as follows, (3.3) $${\mathit{\text{ku}}}^{″″}+6k{u}^{\prime }{u}^{″}+\left(\mathit{\text{kv}}-m\right)u^{″}=0.$$(3.3)

Now, integrating the above Equation (3.3.3), we have the nonlinear ODE in the form, (3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4)

To obtain the traveling wave solution, we resolve the Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) . By the method of homogeneous balance principle, here we get the balance number $N+2=3N.$ Hence $N=1.$

So, the solutions of the (3 + 1)-dimensional generalized shallow water like equation with the scheme of the two variables $(\varphi \prime /\varphi ,1/\varphi )$-expansion method takes form, (3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5)

Where $a_0,a_1,b_1$ are arbitrary constants are to be determined. Here there are three cases to be discussed to examine the nonlinear evolution equation for the sign of $\lambda .$ Putting the above solutions (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) , in the Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) , we illustrated the solutions, that are given below

Case 1: For hyperbolic function solutions is (when $\lambda <0$)

Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) ’s left side transforms into a polynomial when (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) , (2.3)(2.3) $${\varphi }^{\prime }=-{\varphi }^2+\mu \psi -\lambda ,\psi \prime =-\varphi \mathrm{ }\psi .$$(2.3) , and (2.5)(2.5) $$\mathrm{\text{Thus}},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained\hspace{0.5em}}{\psi }^2=\frac{-\lambda ({\varphi }^2-2\mu \psi +\lambda )}{{\lambda }^2\beta +{\mu }^2},$$(2.5) are added. Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) then becomes a system of algebraic equations when the coefficients of this polynomial are put to zero.

Using the mathematical application Maple, we were able to solve the aforementioned algebraic problems and arrive at $$a_0=a_0,{\mathrm{ }a}_1=1,b_1=\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda },v=\frac{m+k\lambda }{k}.$$

Putting these values into Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) and using Equation (3.2)(3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),$$(3.2) , we obtain the exact solution of the shallow water like Equation (3.2)(3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),$$(3.2) as follows (3.6) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{-\lambda }\left(m_1\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sinh}}\hspace{0.17em}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.6) where $\xi =x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt and}}\mathrm{ }\beta ={m_1}^2-{m_2}^2.$

In particular if $m_1\ne 0 \mathit{\text{and}} m_2=0 \mathit{\text{and}} \mu =0,$ the above solution takes form (3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7)

And if $m_1=0 \mathit{\text{and}} m_2\ne 0 \mathit{\text{and}} \mu =0,$ then the above solution turns out, (3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8)

Case 2: For Trigonometric solution (when $\lambda >0$)

When the value of λ is greater than zero, if we substitute Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) into Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) while making use of Equations (2.3)(2.3) $${\varphi }^{\prime }=-{\varphi }^2+\mu \psi -\lambda ,\psi \prime =-\varphi \mathrm{ }\psi .$$(2.3) and (2.7)(2.7) $$\text{And\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained\hspace{0.5em}}{\psi }^2=\frac{\lambda ({\varphi }^2-2\mu \psi +\lambda )}{{\lambda }^2\beta -{\mu }^2},$$(2.7) , the left-hand side of Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) undergoes a transformation resulting in a polynomial that involves the variables $\varphi$ and $\psi .$ By setting the coefficients of this polynomial to zero, we derive a system of algebraic equations.

Using the mathematical program Maple, it is obtained by solving those mentioned algebraic equations. $$a_0=a_0,{\mathrm{ }a}_1=1,b_1=\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda },v=\frac{m+k\lambda }{k}.$$

By substituting these specified values into Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) and combining it with Equation (3.2)(3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),$$(3.2) , we are able to derive the exact solution of the shallow water-like Equation (3.1)(3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1) in the following manner. (3.9) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{\lambda \mathrm{ }}\left(m_1\mathrm{\text{cos}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{s}\mathrm{\text{in}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.9)

Where $\beta ={m_1}^2+{m_2}^2,$ $\lambda >0$ and $\xi =x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}.$

For $m_1\ne 0\mathrm{ }\mathit{\text{and}}\mathrm{ }{m}_2=0\mathrm{ }\mathit{\text{and}}\mathrm{ }\mu =0,$ the attain solution becomes (3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10)

Thus for $m_1=0 \mathit{\text{and}} m_2\ne 0 \mathit{\text{and}} \mu =0,$ the solution (3.9)(3.9) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{\lambda \mathrm{ }}\left(m_1\mathrm{\text{cos}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{s}\mathrm{\text{in}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.9) takes the following form, (3.11) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }\text{sec}\hspace{0.17em}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.11)

Case 3: For rational function solution (when $\lambda =0$)

The left-hand side of Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) changes to a polynomial in $\varphi$ and $\psi$ and if $\lambda =0,$ putting Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) into Equation (3.4)(3.4) $${\mathit{\text{ku}}}^{″\prime }+3k{\left(u^{\prime }\right)}^2+\left(\mathit{\text{kv}}-m\right)u^{\prime }=0.$$(3.4) , using (2.3)(2.3) $${\varphi }^{\prime }=-{\varphi }^2+\mu \psi -\lambda ,\psi \prime =-\varphi \mathrm{ }\psi .$$(2.3) and (2.9)(2.9) $$\text{And\hspace{0.5em}thus},\text{\hspace{0.5em}it\hspace{0.5em}is\hspace{0.5em}obtained},\hspace{0.5em}{\psi }^2=\frac{({\varphi }^2-2\mu \psi )}{{m_1}^2-2\mathrm{ }\mu m_2}.$$(2.9) , and so on. This polynomial’s values can be set to zero to produce an algebraic system of equations.

Using Maple, the aforementioned algebraic equations are solved, and the result is received $$a_0=a_0,{\mathrm{ }a}_1=1,b_1=\pm \sqrt{({m_1}^2-2\mu m_2)},v=\frac{m}{k}.$$

Substituting these values into Equation (3.5)(3.5) $$u\left(\xi \right)=a_0+a_1\varphi \left(\xi \right)+b_1\psi \left(\xi \right),$$(3.5) , in addition to using Equation (3.2)(3.2) $$u\left(x,y,z,t\right)=u\left(\xi \right),$$(3.2) , we obtain the exact solution of the shallow water like Equation (3.1)(3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1) as follows (3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12)

Where $m_1$ and $m_2$ are arbitrary constants and $\xi =(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}).$

4. Graphical representation and result discussion

In this section, we present the graphical representation of the traveling wave solutions obtained for the generalized (3 + 1)-dimensional shallow water-like (SWL) equation. The graph, generated using MATLAB, offers a clear illustration of the soliton solutions obtained in the previous section. Due to the influence of graphical morphology on the dynamics of traveling wave solutions, we have depicted various types of soliton solutions such as kink-shaped soliton solution, anti-bell-shaped solution, singular periodic solution, singular soliton solution, anti-kink-shaped solution, etc., in both three-dimensional and two-dimensional formats.

We mention that a singular wave solution is very important for investigating many physical phenomena, for example when a sudden force is applied, such as an earthquake that might cause a devastating tsunami wave, a singular wave is created. Furthermore, for a porous media, an abrupt temperature shock may potentially cause a thermal tsunami (He & Liu, 2023).

Here, the solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) represents a kink shape soliton solution for the suitable parameters of values $\mu =0,\lambda =-0.5,v=0.2,y=2.5,z=0.5,-10\le x\le 10,0\le t\le 10$ and equivalents 2D plots depicts for $t=1,t=4,t=7$ which shown in the Figure 1. Again we observe from the Figure 2, the solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) represents a kink shape soliton solution for the suitable parameters choose values $\mathrm{ }\mu =0,\lambda =-0.25,v=0.3,y=3,z=2.75,-10\le x\le 10,0\le t\le 5$ and in a 2D representation, there are three spatial progressions associated with time points 1, 3, and 5. Next, we plot the Figure 3 of solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) from which we obtain the kink shape soliton solution for the values of parameters described as $\mu =0,\lambda =-0.1,v=0.3,y=4,z=2.9,-10\le x\le 10,0\le t\le 5$ and for $t=1,t=3,t=5,$ the corresponding 2D graphs are shown.

Figure 1. 3-D And 2-D graphical illustration of the solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) for values $\mu =0,\lambda =-0.5,v=0.2,y=2.5,z=2,-10\le x\le 10,0\le t\le 10.$

Figure 2. 3-D And 2-D graphical illustration the solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) for values $\mathrm{ }\mu =0,\lambda =-1,v=1,y=3,z=3,-5\le x\le 5,0\le t\le 5.$

Figure 3. 3-D And 2-D graphical illustration the solution (3.7)(3.7) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.7) for values $\mu =0,\lambda =-0.5,v=2,y=1,z=1,-5\le x\le 5,0\le t\le 5.$

Now, the Figure 4 of the solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) for the values of described parameter $\mu =0,\lambda =-2,v=2,y=2,z=2,-5\le x\le 5,0\le t\le 5$ with analogous plots can be seen in 2D graphical representations for $t=1,t=3,t=5$ and it represents anti-bell shape soliton solution. The solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) represents a kink shape soliton solution for the suitable parameters of values $\mu =0,\lambda =-1,v=1,y=3,z=3,-5\le x\le 5,0\le t\le 5$ and in 2D graphical representations, we can observe analogous plots for time instances 3, 4, and 5, which depicted in the Figure 5. Solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) describes a kink shape soliton solution which illustrated in the Figure 6 and the graph depicted for the values $\mu =0,\lambda =-0.5,v=1,y=1,z=1,-5\le x\le 5,0\le t\le 5$ and two-dimensional charts illustrate corresponding representations at $t=3,t=4,t=5.$

Figure 4. 3-D And 2-D graphical illustration the solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) for values $\mu =0,\lambda =-2,v=2,y=2,z=2,-5\le x\le 5,0\le t\le 5.$

Figure 5. 3-D And 2-D graphical illustration the solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) for values $\mu =0,\lambda =-1,v=1,y=3,z=3,-5\le x\le 5,0\le t\le 5.$

Figure 6. 3-D And 2-D graphical illustration the solution (3.8)(3.8) $$u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda } \mathit{\text{sech}}\sqrt{-\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.8) for values $\mathrm{ }\mu =0,\lambda =-0.5,v=1,y=1,z=1,-5\le x\le 5,0\le t\le 5.$

The solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) represents a singular periodic soliton solution for the suitable parameters of values $\mu =0,\lambda =2,v=1,y=3,z=3,-5\le x\le 5,0\le t\le 5$ and for $t=1,3,5$ in 2D images and displayed in the Figure 7. The Figure 8 of the solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) represents singular periodic soliton solution for the values of parameter $\mu =0,\lambda =1,v=1,y=4,z=4,-5\le x\le 5,0\le t\le 5$ and for the time points $t=1,t=3,\mathit{\text{and}} t=5,$a 2D graph displaying the same information is constructed. Next, we illustrated the Figure 9 of solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) from which we obtain the singular soliton solution for the values of parameters described as $\mu =0,\lambda =0.5,v=2,y=4,z=4,-5\le x\le 5,0\le t\le 5$ and in 2D we choose $t=1,t=3,\mathit{\text{and}} t=5.$

Figure 7. 3-D And 2-D graphical illustration the solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) for values $\mathrm{ }\mu =0,\lambda =2,v=1,y=3,z=3,-5\le x\le 5,0\le t\le 5.$

Figure 8. 3-D And 2-D graphical illustration the solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) for values $\mu =0,\lambda =1,v=1,y=4,z=4,-5\le x\le 5,0\le t\le 5.$

Figure 9. 3-D And 2-D graphical illustration the solution (3.10)(3.10) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }c\mathit{\text{osec}}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.10) for values $\mu =0,\lambda =0.5,v=2,y=4,z=4,-5\le x\le 5,0\le t\le 5.$

Figures 10–12 gives the singular periodic soliton solutions for the solution (3.11)(3.11) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }\text{sec}\hspace{0.17em}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.11) where the values of the parameters are $\mu =0,\lambda =1.5,v=2,y=3,z=3,-10\le x\le 10,0\le t\le 5$ and the 2D propagations are shown for $t=1,t=3,\mathit{\text{and}} t=5,$ in that order and $\lambda =1,v=2,y=2,z=2,-10\le x\le 10,0\le t\le 5$ and for a 2D graph exhibiting the same information is presented for time units $t=3,t=4,t=5$ and $\mu =0,\lambda =0.5,v=1,y=1,z=1,-10\le x\le 10,0\le t\le 5$ also The 2D visualizations represent at $t=1,t=3,\mathit{\text{and}} t=5$ time intervals respectively.

Figure 10. 3-D And 2-D graphical illustration the solution (3.11)(3.11) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }\text{sec}\hspace{0.17em}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.11) for $\mu =0,\lambda =1.5,v=2,y=3,z=3,-10\le x\le 10,0\le t\le 5.$

Figure 11. 3-D And 2-D graphical illustration of the solution (3.11)(3.11) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }\text{sec}\hspace{0.17em}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.11) for values $\mu =0,\lambda =1,v=2,y=2,z=2,-10\le x\le 10,0\le t\le 5.$

Figure 12. 3-D And 2-D graphical illustration the solution (3.11)(3.11) $$u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}\right)\right)\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda }\text{sec}\hspace{0.17em}\sqrt{\lambda }(x+\mathit{\text{ky}}+\mathit{\text{mz}}-\mathit{\text{vt}}))$$(3.11) for values $\mu =0,\lambda =0.5,v=1,y=1,z=1,-10\le x\le 10,0\le t\le 5.$

The Figure 13 of the solution (3.12)(3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12) for the values of described parameter $m_1=4,m_2=1 \mathrm{\text{and}}\mathrm{ }\mu =1,\lambda =0,v=2,y=2,z=2,-5\le x\le 5,0\le t\le 5$ and The same information is shown on a 2D graph for $t=3,t=4,t=5 \mathit{\text{and}}$ it represents anti-kink shape soliton solution and lastly the solution (3.12)(3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12) represents a parabolic shape soliton solution for the suitable parameters of values $m_1=4,m_2=2 \mathrm{\text{and}}\mathrm{ }\mu =3,\lambda =0,v=1,y=3,z=4,-10\le x\le 10,0\le t\le 10$ and an analogous two-dimensional diagram is drawn to represent data at $t=7,t=8,t=9$ as both are shown in Figure 14.

Figure 13. 3-D And 2-D graphical illustration the solution (3.12)(3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12) for values $m_1=4,m_2=1 \mathrm{\text{and}}\mathrm{ }\mu =1,\lambda =0,v=2,y=2,z=2,-5\le x\le 5,0\le t\le 5.$

Figure 14. 3-D And 2-D graphical illustration the solution (3.12)(3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12) for values $m_1=1,m_2=1 \mathrm{\text{and}}\mathrm{ }\mu =3,\lambda =0,v=1,y=3,z=2,-10\le x\le 10,0\le t\le 10.$

5. Comparison

Now, in this part, we will try to compare the solutions obtained by Dusunceli et al. (Nadeem & He, 2021) with the aid of improved Bernouli’s sub-equation method and result which we got by using the double variables $(\varphi \prime /\varphi ,1/\varphi )$-expansion method of the generalized (3 + 1) shallow water like equation (SWL). The $(\varphi \prime /\varphi ,1/\varphi )$-expansion method has used to describe in this article to find thesoliton solutions of the generalized (3 + 1)-dimensional shallow water like equation (SWL). Different values of full of nonlinearity parameter gives us different types of solution, Dusunceli has derived total four solutions of the generalized (3 + 1)-dimensional shallow water like equation by of improved Bernouli’s sub-equation method which are given in the table and all the solutions are in tan function. In this article we derived five soliton solutions $\left(3.7\right)-\left(3.8\right),\left(3.10\right)-(3.12)$ of the mentioned Equation (3.1)(3.1) $$u_{\mathit{\text{xxy}}}+{3u}_{\mathit{\text{xx}}}{u}_y+{3u}_x{u}_{\mathit{\text{xy}}}-u_{\mathit{\text{yt}}}-u_{\mathit{\text{xz}}}=0.$$(3.1) where each of the solutions are unique from solution Dusunceli’s solution and here all the solution are derived in terms of hyperbolic, trigonometric and rational functions. In particular, if we set $m_1=0,$ $m_2\ne 0$ and $\mu =0,$ then we get the solution in terms of hyperbolic functions of tanh and sech and in terms of trigonometric function we get secant and tangent functions. Similarly, if we set $m_1\ne 0,$ $m_2=0$ and $\mu =0,$ then we get other types of solution in terms hyperbolic functions in coth and cosech and cosec or cot in terms of trigonometric functions.

Table

Solutions obtained by Dusunceli (2019)

The obtained solutions in this article

(i) ${\mathrm{u}}_1\left(\mathrm{\xi }\right)=\frac{1}{{{\mathrm{b}}_0+\mathrm{b}}_1\mathrm{F}\left(\mathrm{\xi }\right)}\left(A_1+B_1\mathrm{F}\left(\mathrm{\xi }\right)+C_1{\mathrm{F}}^2\left(\mathrm{\xi }\right)-D_1{\mathrm{F}(\mathrm{\xi })}^3\right)$ Where $A_1,B_1,C_1\text{and}{D}_1$ are arbitrary constants. (ii) ${\mathrm{u}}_2\left(\mathrm{\xi }\right)=\frac{1}{{\mathrm{b}}_1\mathrm{F}\left(\mathrm{\xi }\right)}\left(A_2{\mathrm{F}}^2\left(\mathrm{\xi }\right)-B_1{\mathrm{F}(\mathrm{\xi })}^3\right)$ Where $A_2\text{and}{B}_2$ are arbitrary constants. (iii) ${\mathrm{u}}_3\left(\mathrm{\xi }\right)=\frac{1}{{\mathrm{b}}_1\mathrm{F}\left(\mathrm{\xi }\right)}(A_3\mathrm{F}(\mathrm{\xi })+B_3{\mathrm{F}}^2(\mathrm{\xi })-D_3{\mathrm{F}\left(\mathrm{\xi }\right)}^3)$ Where $A_3,B_3,\text{and} C_3$ are arbitrary constants. (iv) ${\mathrm{u}}_4\left(\mathrm{\xi }\right)=\frac{1}{{\mathrm{b}}_1\mathrm{F}\left(\mathrm{\xi }\right)}\left(A_4\mathrm{F}\left(\mathrm{\xi }\right)-{B_4\mathrm{F}}^2\left(\mathrm{\xi }\right)-C_4{\mathrm{F}(\mathrm{\xi })}^3\right)$ Where $A_4,B_4,\text{and} C_4$ are arbitrary constants. Also here $F\left(\xi \right),$ $b_{0 }\text{and}{b}_1$ are arbitrary parameters.

(i) If $m_1\ne 0 \text{and} m_2=0 \text{and} \mu =0,$ obtained solution (3.6)(3.6) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{-\lambda }\left(m_1\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sinh}}\hspace{0.17em}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.6) becomes $u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }c\mathit{\text{oth}}\left(\sqrt{-\lambda }\left(\xi \right)\right)\pm \mathrm{\Pi }\mathit{\text{cosech}}\sqrt{-\lambda }(\xi )),$ where $\mathrm{\Pi }$ is an arbitrary parameter and $\lambda <0.$ (ii) If $m_1=0 \text{and} m_2\ne 0 \text{and} \mu =0,$ obtained solution (3.6)(3.6) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{-\lambda }\left(m_1\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sinh}}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda ({\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sinh}}\hspace{0.17em}\left(\sqrt{-\lambda }\xi \right)+m_2\mathrm{\text{cosh}}\left(\sqrt{-\lambda }\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.6) becomes $u\left(x,y,z,t\right)=a_0+\sqrt{-\lambda }t\mathit{\text{anh}}\left(\sqrt{-\lambda }\left(\xi \right)\right)\mathrm{\Pi } \mathit{\text{sech}}\sqrt{-\lambda }\left(\xi \right),$ Where $\mathrm{\Pi }$ is an arbitrary parameter and $\lambda <0.$ (iii) If $m_1\ne 0 \text{and} m_2=0 \mathit{\text{and}} \mu =0,$ obtained solution (3.9)(3.9) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{\lambda \mathrm{ }}\left(m_1\mathrm{\text{cos}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{s}\mathrm{\text{in}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.9) becomes $u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\mathit{\text{cot}}\left(\sqrt{\lambda }\left(\xi \right)\right)\pm \mathrm{\Pi }\mathit{\text{cosec}}\sqrt{\lambda }(\xi ),$ Where $\mathrm{\Pi }$ is an arbitrary parameter and $\lambda >0.$ (iv) If $m_1=0 \text{and} m_2\ne 0 \text{and} \mu =0,$ obtained solution (3.9)(3.9) $$u\left(\xi \right)=a_0+\frac{\left\{\sqrt{\lambda \mathrm{ }}\left(m_1\mathrm{\text{cos}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{s}\mathrm{\text{in}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)\right\}}{\left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)\right)+\frac{\mu }{\lambda }}\pm \frac{\sqrt{-\lambda (-{\lambda }^2\beta +{\mu }^2)}}{\lambda \left(m_1\mathrm{\text{sin}}\hspace{0.17em}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+m_2\mathrm{c}\mathrm{\text{os}}\left(\sqrt{\lambda \mathrm{ }}\xi \right)+\frac{\mu }{\lambda }\right)}$$(3.9) becomes $u\left(x,y,z,t\right)=a_0+\sqrt{\lambda }\hspace{0.17em}\mathrm{\text{tan}}\hspace{0.17em}\left(\sqrt{\lambda }\left(\xi \right)\right)\pm \mathrm{\Pi }\mathit{\text{sec}}\sqrt{\lambda }(\xi )),$ Where $\mathrm{\Pi }$ is an arbitrary parameter and $\lambda >0.$ (v) If $\lambda =0,$ we attain the solution (3.12)(3.12) $$u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2},$$(3.12) $u\left(\xi \right)=a_0+\frac{\mu \xi +m_1}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}\pm \frac{\sqrt{({m_1}^2-2\mu m_2)}}{\frac{\mu }{2}{\xi }^2+m_1\xi +m_2}.$

6. Conclusion

In this article we have found the soliton solutions of the (3 + 1)-dimensional shallow water equation by using the two variables $(\varphi \prime /\varphi ,1/\varphi )$-expansion method which will be able to understand the physical phenomenon in ocean physics in a better way. Different types of soliton solutions have been obtained in this study like kink shape soliton solution, anti-bell shape solution, singular periodic solution, singular soliton solution, anti-kink shape solution etc. The solutions which we got in this study are capable of discussing the system’s complex behavior than the mentioned article. The computer program MATLAB is employed to visually demonstrate the three-dimensional (3-D) and two-dimensional (2-D) graphical depictions of the derived solutions. Moreover, this research has the capacity to have significant implications for the future of nonlinear partial differential equations (NPDEs). By demonstrating the efficacy of the two variables $(\varphi \prime /\varphi ,1/\varphi )$-expansion method for solving the mentioned equation, in near future our work could serve as an impetus for other researchers to apply this method to other NPDEs, potentially leading to groundbreaking discoveries in the field. Additionally, when the aforementioned model’s coefficients are not constants, we will use this method to analyze it in a future work.

Disclosure statement

No potential conflict of interest was reported by the author(s).