1. Introduction
In recent years, the study of exact solutions in magneto-electro-elastic media (such as sensors, actuators and controllers) has attracted more and more attention from mathematicians and physicists [1–4]. As everyone knows, the governing equation that describes the propagation of solitons through magneto-electro-elastic media is modeled by a longitudinal wave equation of the form [1]1 ) has been studied by many researchers, with solutions such as the one-soliton and peaked solitary wave solutions [2], the bright and singular soliton solutions [3], the hyperbolic function solitary wave solutions and plane wave solutions [4]. In addition, equation (1 ) can be reduced to the Boussinesq equation in the case of traveling wave solutions, which have N-soliton solutions [5]. Further, equation (1 ) can be viewed as a special case of the (2+1)-dimensional Boussinesq equation [6].
$\begin{eqnarray}\displaystyle \frac{{\partial }^{2}u}{\partial {t}^{2}}-{c}_{0}\displaystyle \frac{{\partial }^{2}u}{\partial {x}^{2}}-\displaystyle \frac{1}{2}{c}_{0}^{2}\displaystyle \frac{{\partial }^{2}{u}^{2}}{\partial {x}^{2}}-N\displaystyle \frac{{\partial }^{4}u}{\partial {t}^{2}{x}^{2}}=0,\end{eqnarray}$
where c0 is the linear longitudinal velocity of a magneto-electro-elastic circular rod and N > 0 is the dispersion parameter [1]. It is well known that equation (Fractional derivatives have the characteristics of heredity and memory. They are widely used in the fields of natural science and engineering technology. In general, nonlinear evolution equations of fractional order can better express scientific phenomena than differential equations of integer order. Therefore, we have mainly discussed the fractional version of equation (1 ) with the conformable fractional derivative in a magneto-electro-elastic circular rod. That is
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\partial }^{2\varepsilon }{\rm{\Phi }}(\chi ,\tau )}{\partial {\tau }^{2}}-{c}_{0}\displaystyle \frac{{\partial }^{2\varepsilon }{\rm{\Phi }}(\chi ,\tau )}{\partial {\chi }^{2}}-\displaystyle \frac{1}{2}{c}_{0}^{2}\displaystyle \frac{{\partial }^{2}{\rm{\Phi }}{\left(\chi ,\tau \right)}^{2}}{\partial {\chi }^{2}}\\ -N\displaystyle \frac{{\partial }^{4}{\rm{\Phi }}(\chi ,\tau )}{\partial {\tau }^{2}{\chi }^{2}}=0,\end{array}\end{eqnarray}$
where 0 < ϵ ≤ 1 is the fractional derivative order.Here, we applied the soliton ansatz method and the direct integration approach to construct the soliton and solitary wave solutions of equation (2 ). The soliton and solitary wave solutions are interesting in soliton theory. They can be used to describe the velocity and amplitude of water wave propagation. The Hirota bilinear method and the Riemann–Hilbert method are systematic schemes to obtain the N-soliton solutions [7, 8]. Recently, the fractional Lie group method was used to solve fractional differential equations [9, 10] and the nonlocal group reduction was utilized to deal with nonlocal integrable equations [11, 12], attracting the attention of many scholars in the field of mathematical physics.
The plan of this article is as follows: in section 2 , we give some basic definitions and theorems on the conformable fractional derivative. Through the soliton ansatz method, the soliton solutions of this considered model are obtained in section 3 . In section 4 , the solitary wave solutions are derived with the help of the direct integration approach. Meanwhile, their nonlinear dynamic behaviors are presented in two-dimensional graphics. In the last section, some important conclusions and discussions are shown.
2. Basic definitions
In the first place, some basic definitions and theorems on the conformable fractional derivative that will be used in the below text can be given.
In addition, if f is differentiable, then ${T}_{\alpha }(f)(t)={t}^{1-\alpha }\tfrac{{\rm{d}}f}{{\rm{d}}t}$.
[13, 14] Let $f:(0,\infty )\to R$, then the conformable fractional derivative of f of order α is defined as:
$\begin{eqnarray}{T}_{\alpha }(f)(t)=\mathop{\mathrm{lim}}\limits_{\delta \to 0}\displaystyle \frac{f(t+\delta {t}^{1-\alpha })-f(t)}{\delta },0\lt \alpha \leqslant 1,\end{eqnarray}$
for all $t\gt 0$.[13] Let $\alpha \in (0,1]$ and $f,g$ be α-differentiable at a point t, then we have
| 1. | (1)${T}_{\alpha }({af}+{bg})={{aT}}_{\alpha }(f)+{{bT}}_{\alpha }(g),{\rm{for}}\,{\rm{all}}\,a,b\in R,$ |
| 2. | (2)${T}_{\alpha }({t}^{\lambda })=\lambda {t}^{\lambda -\alpha },{\rm{for}}\,{\rm{all}}\,\lambda \in R,$ |
| 3. | (3)${T}_{\alpha }({fg})={{fT}}_{\alpha }(g)+{{gT}}_{\alpha }(f),$ |
| 4. | (4)${T}_{\alpha }\left(\tfrac{f}{g}\right)=\tfrac{{{fT}}_{\alpha }(g)-{{gT}}_{\alpha }(f)}{{g}^{2}}.$ |
[14] Assume $f,g:[0,\infty )\to R$ are α-differentiable functions ($0\lt \alpha \leqslant 1$), then $f(g(t))$ is α-differentiable with $t\ne 0$ and $g(t)\ne 0$, we have
$\begin{eqnarray}{T}_{\alpha }(f\circ g)(t)=({T}_{\alpha }f)(g(t)){T}_{\alpha }(g)(t)g{\left(t\right)}^{\alpha -1}.\end{eqnarray}$
If t = 0, then we have $\begin{eqnarray}{T}_{\alpha }(f\circ g)(0)=\mathop{\mathrm{lim}}\limits_{t\to {0}^{+}}({T}_{\alpha }f)(g(t)){T}_{\alpha }(g)(t)g{\left(t\right)}^{\alpha -1}.\end{eqnarray}$
[13, 14] Assume $a\geqslant 0$ and $t\geqslant a$. If f is a function defined on $(a,t]$ and $\alpha \in f$, then the α-fractional integral of f is defined by
$\begin{eqnarray*}{I}_{\alpha }^{a}(f)(t)={\int }_{a}^{t}\displaystyle \frac{f(x)}{{x}^{1-\alpha }}{dx}.\end{eqnarray*}$
[13, 14] Suppose $a\geqslant 0$ and $\alpha \in (0,1)$. Also, let f be a continuous function such that ${I}_{\alpha }^{a}(f)(t)$ exists. Then
$\begin{eqnarray*}{T}_{\alpha }{I}_{\alpha }^{a}(f)(t)=f(t).\end{eqnarray*}$
Note 2. When α → 1, the above definitions and theorems become the classical cases.
In what follows, we will use the conformable fractional derivative and theorems to solve the nonlinear time–space fractional longitudinal wave equation (2 ).
3. Soliton solutions of equation (2 )
In order to get soliton solutions of equation (2 ), we first suppose that equation (2 ) has the form of solution as follows:
$\begin{eqnarray}\begin{array}{c}{\rm{\Pi }}(\xi )={\rm{\Phi }}(\chi ,\tau )=\alpha {{\rm{sech}} }^{p}(\xi ),\\ \xi =\frac{{\chi }^{\varepsilon }}{\varepsilon }-v\frac{{\tau }^{\varepsilon }}{\varepsilon },0\lt \varepsilon \leqslant 1,\end{array}\end{eqnarray}$
where α is the amplitude of the soliton, v is the velocity of the soliton and ϵ is the conformable fractional derivative.Substituting (6 ) into (2 ) yields6 ) and setting the coefficients of ${\rm{{\rm{sech}} }}{(\xi )}^{i},i\,=\,2,4$ to zero, one has that8 ), we have2 ) with (6 ) as follows:
$\begin{eqnarray}\begin{array}{l}{{\rm{sech}} }^{p}(\xi )\alpha \,{v}^{2}-{{\rm{sech}} }^{p}(\xi )\alpha \,{c}_{0}-\displaystyle \frac{1}{2}\,{{c}_{0}}^{2}{\alpha }^{2}{{\rm{sech}} }^{2p}(\xi )\\ -N{{\rm{sech}} }^{p}(\xi )(1-{{\rm{sech}} }^{2}(\xi ))\alpha \,{v}^{2}{p}^{2}\\ -N{{\rm{sech}} }^{p}(\xi )(1-{{\rm{sech}} }^{2}(\xi ))\alpha \,{c}^{2}p+N{{\rm{sech}} }^{p}(\xi )\alpha \,{v}^{2}p=0.\end{array}\end{eqnarray}$
Balancing the powers 2p and p + 2, gives $\begin{eqnarray*}p=2.\end{eqnarray*}$
Substituting p = 2 into ( $\begin{eqnarray}\left\{\begin{array}{l}-\displaystyle \frac{1}{2}\,{{c}_{0}}^{2}{\alpha }^{2}+6\,N\alpha \,{v}^{2}=0,\\ -4\,N\alpha \,{v}^{2}+\alpha \,{v}^{2}-\alpha \,{c}_{0}=0.\\ \end{array}\right.\end{eqnarray}$
Solving systems ( $\begin{eqnarray}\alpha =-\displaystyle \frac{212N}{{c}_{0}\left(4\,N-1\right)},\,v=\pm \sqrt{-\displaystyle \frac{{c}_{0}}{4\,N-1}},\end{eqnarray}$
where the existing conditions of soliton solutions are $\begin{eqnarray}\displaystyle \frac{{c}_{0}}{4\,N-1}\leqslant 0\,\,{\rm{and}}\,\,N\ne \displaystyle \frac{1}{4}.\end{eqnarray}$
Therefore, we can obtain two soliton solutions of equation ( $\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}(\chi ,\tau ) & = & -\displaystyle \frac{12N}{{c}_{0}\left(4\,N-1\right)}\\ & & \times \,{{\rm{sech}} }^{2}\left(-\displaystyle \frac{{\chi }^{\varepsilon }}{\varepsilon }\pm \displaystyle \frac{{\tau }^{\varepsilon }}{\varepsilon }\sqrt{-\displaystyle \frac{{c}_{0}}{4\,N-1}}\right).\end{array}\end{eqnarray}$
Similarly, if the following form of ansatz solution of equation (2 ) holds11 ) and (13 ) more directly, we show their three-dimensional and two-dimensional diagrams in figure 1.
$\begin{eqnarray}\begin{array}{rcl}{\rm{\Pi }}(\xi ) & = & {\rm{\Phi }}(\chi ,\tau )=\alpha \csc {{\rm{h}}}^{p}(\xi ),\\ \xi & = & \displaystyle \frac{{\chi }^{\varepsilon }}{\varepsilon }-v\displaystyle \frac{{\tau }^{\varepsilon }}{\varepsilon },0\lt \varepsilon \leqslant 1,\end{array}\end{eqnarray}$
then we can obtain the other two soliton solutions of the forms $\begin{eqnarray}\begin{array}{rcl}{\rm{\Phi }}(\chi ,\tau ) & = & \displaystyle \frac{12N}{{c}_{0}\left(4\,N-1\right)}\\ & & \times \,{\mathrm{csch}}^{2}\left(-\displaystyle \frac{{\chi }^{\varepsilon }}{\varepsilon }\pm \displaystyle \frac{{\tau }^{\varepsilon }}{\varepsilon }\sqrt{-\displaystyle \frac{{c}_{0}}{4\,N-1}}\right).\end{array}\end{eqnarray}$
In order to see the nonlinear evolution processes of solutions (
Figure 1. (a) Soliton solution ( |
From figure 1, we can see that the fractional order ϵ did not change the amplitude of the soliton waves, but slightly changes their widths.
4. Solitary wave solutions of equation (2 )
In this section, we discuss the solitary wave solutions of equation (2 ) with the symmetry condition. A wave transformation presents that
$\begin{eqnarray}{\rm{\Pi }}(\xi )={\rm{\Phi }}(\chi ,\tau ),\xi =\displaystyle \frac{{\chi }^{\varepsilon }}{\varepsilon }-c\displaystyle \frac{{\tau }^{\varepsilon }}{\varepsilon },0\lt \varepsilon \leqslant 1,\end{eqnarray}$
where c is the velocity of the solitary wave.Substituting (14 ) into (2 ) yields
$\begin{eqnarray}{c}^{2}{\rm{\Pi }}^{\prime\prime} -{c}_{0}{\rm{\Pi }}^{\prime\prime} -\displaystyle \frac{1}{2}{c}_{0}^{2}({{\rm{\Pi }}}^{2})^{\prime\prime} -{cN}{{\rm{\Pi }}}^{\unicode{x02057}}=0.\end{eqnarray}$
We can see that there exists the symmetry condition
$\begin{eqnarray}{\rm{\Pi }}(\xi )={\rm{\Pi }}(-\xi ),\xi \in (-\infty ,+\infty ).\end{eqnarray}$
Hence, we only need to consider the solutions in the interval ξ ≥ 0.
Integrating equation (15 ) twice, and taking the integration constants to be equal to zero, one obtains17 ) leads to
$\begin{eqnarray}({c}^{2}+{c}_{0}){\rm{\Pi }}-\displaystyle \frac{1}{2}{c}_{0}^{2}{{\rm{\Pi }}}^{2}-{c}^{2}N{\rm{\Pi }}^{\prime\prime} =0,\xi \geqslant 0.\end{eqnarray}$
Separating variables and integrating once more, equation ( $\begin{eqnarray}\pm \xi =\int \displaystyle \frac{{\rm{d}}{\rm{\Pi }}}{{\rm{\Pi }}\sqrt{\tfrac{{c}^{2}-{c}_{0}}{{c}^{2}N}-\tfrac{{c}_{0}^{2}}{3{c}^{2}N}{\rm{\Pi }}}},\xi \geqslant 0.\end{eqnarray}$
This implies the existing condition of solution $\begin{eqnarray}{\rm{\Pi }}\leqslant \displaystyle \frac{3{c}^{2}-3{c}_{0}}{{c}_{0}^{2}}.\end{eqnarray}$
Hence, we suppose c2 > c0.From equation (18 ), two kinds of solitary wave solutions can be obtained in the following:
Case 1:
When $0\leqslant {\rm{\Pi }}\leqslant \tfrac{3{c}^{2}-3{c}_{0}}{{c}_{0}^{2}}$, we have a solitary wave solution for equation (18 )18 ) with symmetry condition (16 ) as follows:
$\begin{eqnarray}\begin{array}{c}{\rm{\Pi }}(\xi )=\frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{{\rm{sech}} }^{2}\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\xi +\lambda \right\},\\ \xi \geqslant 0,\lambda \geqslant 0.\end{array}\end{eqnarray}$
Therefore, we have the solitary solution for equation ( $\begin{eqnarray}\begin{array}{c}{\rm{\Pi }}(\xi )=\frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{{\rm{sech}} }^{2}\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\left|\xi \right|+\lambda \right\},\\ \xi \in (-\infty ,+\infty ),\lambda \geqslant 0,\end{array}\end{eqnarray}$
in the whole interval.So, we have the following solitary wave solution
$\begin{eqnarray}\begin{array}{ccl}{\rm{\Phi }}(\chi ,\tau ) & = & \frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{{\rm{sech}} }^{2}\\ & & \times \,\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\left|\frac{{\chi }^{\varepsilon }}{\varepsilon }-c\frac{{\tau }^{\varepsilon }}{\varepsilon }\right|+\lambda \right\},\lambda \geqslant 0.\end{array}\end{eqnarray}$
Case 2:
When ${\rm{\Pi }}\leqslant 0\leqslant \tfrac{3{c}^{2}-3{c}_{0}}{{c}_{0}^{2}}$, we can get the other solitary wave solution for equation (18 )18 ) with symmetry condition (16 ) as follows:
$\begin{eqnarray}\begin{array}{c}{\rm{\Pi }}(\xi )=-\frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{\mathrm{csch}}^{2}\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\xi +\delta \right\},\\ \xi \geqslant 0,\delta \gt 0.\end{array}\end{eqnarray}$
Therefore, we have the solitary solution for equation ( $\begin{eqnarray}\begin{array}{c}{\rm{\Pi }}(\xi )=-\frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{\mathrm{csch}}^{2}\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\left|\xi \right|+\delta \right\},\\ \xi \in (-\infty ,+\infty ),\delta \gt 0,\end{array}\end{eqnarray}$
in the whole interval.So, we have the following solitary wave solution22 ) by figure 2.
$\begin{eqnarray}\begin{array}{c}{\rm{\Phi }}(\chi ,\tau )=-\frac{3({c}^{2}-{c}_{0})}{{c}_{0}^{2}}{\mathrm{csch}}^{2}\\ \times \,\left\{\frac{1}{2}\sqrt{\frac{{c}^{2}-{c}_{0}}{{c}^{2}N}}\left|\frac{{\chi }^{\varepsilon }}{\varepsilon }-c\frac{{\tau }^{\varepsilon }}{\varepsilon }\right|+\delta \right\},\delta \gt 0.\end{array}\end{eqnarray}$
We plotted the nonlinear evolution behaviors with different parameters of solitary wave solution (
Figure 2. Solitary wave solution ( |
From figure 2, we can see that the fractional order ϵ did not change the amplitude of the solitary waves. Besides, when λ = 0, solution (22 ) is a solitary wave solution, while solution (22 ) is a solitary peak wave solution if λ > 0. That is to say, it is not differentiable at the peak of the solitary wave solution.
5. Conclusions and discussion
By applying two different schemes, we derived some new soliton solutions and solitary wave solutions to the nonlinear time–space fractional longitudinal wave equation. After analyzing them, we found that these results became the classical cases of ϵ → 1. In addition, the fractional order ϵ did not change the amplitude of the soliton solutions and solitary wave solutions from figures 1 and 2. In the future, we will further study the rational solutions and integrability of this considered model.