New Exact Solutions for Generalized (3+1) Shallow Water-Like (SWL) Equation

In various fields of physical sciences, nonlinear evolution equations (NLEEs) and their exact solutions are important for non-linear phenomena. In this paper, generalized (3 + 1) shallow water-like (SWL) equation [1, 2] which is one of these equations will be discussed and new solutions will be examined.

uxxxy+3uxxuy+3uxuxy−<!-- - -->uyt−<!-- - -->uxz=0. $$\begin{array}{} \displaystyle u_{xxxy}+3u_{xx} u_{y}+3u_{x} u_{xy}-u_{yt}-u_{xz}=0. \end{array}$$(1)

There are some studies on this equation. Rational solutions and lump solutions are obtained for equation(1) by Zhang et al. [1] and Grammian and Pfaffian solutions are obtained by Tang et al. [2]. Also, this equation solved by Tian and Gao [3] via the tanh method, by Zayed [4] via the (G′/G) expansion method. Lump-type solutions and their interaction solutions are generated by Sadat[5]. In this context, various papers were presented to the literature [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23]. The organization of this paper is as follows: firstly, we give the methodology of the improved Bernoulli sub-equation function method. Then we apply this method to the SWL equation for finding new exact solutions. At last, we give a conclusion.

In this part, we use the improved Bernoulli sub-equation function method [24, 25, 26, 27, 28] for solutions eq. (1).

Step 1

Let’s consider the following partial differential equation;

P(u,ux,uy,uz,ut,uxx,uxy,...)=0. $$\begin{array}{} \displaystyle P(u,u_{x},u_{y},u_{z},u_{t},u_{x x},u_{x y},... )=0. \end{array}$$(2)

and take the wave transformation

u(x,y,z,t)=U(ξ<!-- ? -->),ξ<!-- ? -->=x+ky+mz−<!-- - -->wt, $$\begin{array}{} \displaystyle u(x,y,z,t)=U(\xi),\xi=x+ky+mz-wt, \end{array}$$(3)

where k, m and w are nonzero constants.Substituting Eq. (2) into Eq. (3), we obtain the following nonlinear ordinary differential equation;

N=(U,U′,U″,U‴,...)=0. $$\begin{array}{} \displaystyle N=(U, U^{'} ,U^{''} ,U^{'''} ,...)=0. \end{array}$$(4)

In this section, application of the improved Bernoulli sub-equation function method to SWL equation is presented. Using the wave transformation on Eq. (1)

u(x,y,z,t)=U(ξ<!-- ? -->),ξ<!-- ? -->=x+ky+mz−<!-- - -->wt, $$\begin{array}{} \displaystyle u(x,y,z,t)=U(\xi), \xi=x+ky+mz-wt, \end{array}$$(11)

we get the following nonlinear ordinary differential equation:

kU(4)+6kU′U″+(kw−<!-- - -->m)U″=0. $$\begin{array}{} \displaystyle kU^{(4)}+6kU^{'}U^{''}+(kw-m)U^{''}=0. \end{array}$$(12)

Integrating the equation in (12), we get

kU‴+3k[U′]2+(kw−<!-- - -->m)U′=0. $$\begin{array}{} \displaystyle kU^{'''}+3k[U^{'}]^{2}+(kw-m)U^{'}=0. \end{array}$$(13)

Finally, if we write V instead of U^′, the equation (13) becomes a second order nonlinear ordinary differential equation:

kV″+3kV2+(kw−<!-- - -->m)V=0. $$\begin{array}{} \displaystyle kV^{''}+3kV^{2}+(kw-m)V=0. \end{array}$$(14)

Balancing Eq. (14) by considering the highest derivative V^″ and the highest power V², we obtain

n+2=2M+m. $$\begin{array}{} \displaystyle n+2=2M+m. \end{array}$$

Choosing M = 2, m = 1, gives n = 3. Thus, the trial solution to Eq. (1) takes the following form:

U(ξ<!-- ? -->)=a0+a1F(ξ<!-- ? -->)+a2F2(ξ<!-- ? -->)+a3F3(ξ<!-- ? -->)b0+b1F(ξ<!-- ? -->). $$\begin{array}{} \displaystyle U(\xi)=\frac{a_{0}+a_{1} F(\xi)+a_{2} F^{2} (\xi)+a_{3} F^{3} (\xi)}{b_{0}+b_{1} F(\xi)}. \end{array}$$(15)

where F^′ = αF + βF², α, β ≠ 0. Substituting Eq. (15), its second derivative and power along with F^′ = αF + βF², α, β ≠ 0, into Eq. (14), yields a polynomial in F. Solving the system of the algebraic equations, yields the values of the parameter involved. Substituting the obtained values of the parameters into Eq. (15), yields the solutions to Eq. (1). We can find following coefficients:

Case 1

a0=−<!-- - -->(1+k)wb03k,a1=2w3/2b0k1+k−<!-- - -->(1+k)wb13k,a2=−<!-- - -->2w2b0+2w3/2b1k1+k,a3=−<!-- - -->2w2b1,σ<!-- s -->=−<!-- - -->wk1+k; $$\begin{array}{} \displaystyle a_0=-\frac{(1+k)w b_0}{3k} ,a_1=\frac{2w^{3/2} b_0 }{\sqrt{\frac{k}{1+k}}}-\frac{(1+k) w b_1} {3k}, a_2=-2w^{2} b_0 + \frac{2w^{3/2} b_1 }{\sqrt{\frac{k}{1+k}}} ,a_3=-2w^{2} b_1, \sigma=-\frac{\sqrt{w}} {\sqrt{\frac{k}{1+k}}}; \end{array}$$(16)

Case 4

a0=0,a1=(−<!-- - -->kw)b13k,a2=−<!-- - -->2iwm−<!-- - -->kwb1k,a3=−<!-- - -->2w2b1,b0=0,σ<!-- s -->=im−<!-- - -->kwk; $$\begin{array}{} \displaystyle a_0=0 ,a_1=\frac{(-kw) b_1}{3k} ,a_2=-\frac{2iw \sqrt{m-kw} b_1 }{\sqrt{k}},a_3=-2w^{2} b_1, b_0=0, \sigma=\frac{i\sqrt{m-kw}} {\sqrt{k}}; \end{array}$$(19)

Choosing the suitable values of parameters, we performed the numerical simulations of the obtained solutions for (16, 17) case by plotting their 2D and 3D.

Fig. 1

Fig. 2

In this article, new solutions are obtained for the SWL equation using the IBSEFM method. We have seen that the results we obtained are new solutions when we compare them with previous ones. The results may be useful to explain the physical effects of various nonlinear models in non-linear sciences. IBSEFM is a powerful and efficient mathematical tool that can be used to process various nonlinear mathematical models.