2.1. Bilinear forms

Introducing the dependent variable transformation,

(2.1) $$\begin{eqnarray}\displaystyle u=2\text{e}^{-{\it\Gamma}t}\left[(\ln f)_{xy}+\frac{1}{2}\int \text{e}^{{\it\Gamma}t}g(t)\,\text{d}t\right]\!, & & \displaystyle\end{eqnarray}$$

where $f=f(x,y,t)$ is a real function, we have the following bilinear forms of (1.6):

(2.2a ) $$\begin{eqnarray}\displaystyle & D_{y}D_{t}\,f\cdot f+BD_{x}^{3}D_{y}\,f\cdot f+CD_{x}D_{y}^{3}\,f\cdot f=0, & \displaystyle\end{eqnarray}$$

(2.2b ) $$\begin{eqnarray}\displaystyle & AD_{x}D_{y}\,f\cdot f-6BD_{x}^{2}\,f\cdot f-6CD_{y}^{2}\,f\cdot f=0, & \displaystyle\end{eqnarray}$$

$D_{x}$

$D_{y}$

(2.3) $$\begin{eqnarray}\displaystyle D_{x}^{n_{1}}D_{y}^{n_{2}}D_{t}^{n_{3}}F\cdot G=[\partial (x-x^{\prime })]^{n_{1}}[\partial (y-y^{\prime })]^{n_{2}}[\partial (t-t^{\prime })]^{n_{3}}F(x,y,t)G(x^{\prime },y^{\prime },t^{\prime })|_{x^{\prime }=x,y^{\prime }=y,t^{\prime }=t}, & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

where $n_{1}$ , $n_{2}$ and $n_{3}$ are all the non-negative integers, $F(x,y,t)$ is the real function about the variables $x$ , $y$ and $t$ , and $G(x^{\prime },y^{\prime },t^{\prime })$ is the real function about the formal variables $x^{\prime }$ , $y^{\prime }$ and $t^{\prime }$ .

2.2. Soliton-like solutions of (1.6)

To construct the soliton-like solutions of (1.6), we expand $f$ in the power series of a formal parameter ${\it\epsilon}$ as

(2.4) $$\begin{eqnarray}\displaystyle f=1+{\it\epsilon}f_{1}+{\it\epsilon}^{2}f_{2}+\cdots \,, & & \displaystyle\end{eqnarray}$$

where $f_{j}$ $(j=1,2,\dots )$ are the real functions to be determined. Substituting expression (2.4) into bilinear forms (2.2) and equating the coefficients of the same powers of ${\it\epsilon}$ to zero, we can obtain the soliton-like solutions of (1.6).

2.2.1. One-soliton-like solutions

Truncating expression (2.4) as

(2.5) $$\begin{eqnarray}\displaystyle f=1+{\it\epsilon}f_{1}, & & \displaystyle\end{eqnarray}$$

and substituting it into bilinear forms (2.2), we get

(2.6) $$\begin{eqnarray}f_{1}=\text{e}^{{\it\theta}},\quad {\it\theta}=kx+py+{\it\omega}t,\end{eqnarray}$$

(2.7a,b ) $$\begin{eqnarray}{\it\omega}=-\frac{1}{72}\left(\frac{Ak^{3}\sqrt{A^{2}-144BC}}{C}+\frac{A^{2}k^{3}}{C}\right),\quad p=\frac{k\sqrt{A^{2}-144BC}+Ak}{12C},\end{eqnarray}$$

with $k$ as a real constant. Setting ${\it\epsilon}=1$ , we derive the one-soliton-like solutions of (1.6) as

(2.8) $$\begin{eqnarray}\displaystyle u=2\text{e}^{-{\it\Gamma}t}\left[\ln (1+f_{1})_{xy}+\frac{1}{2}\int \text{e}^{{\it\Gamma}t}g(t)\,\text{d}t\right]\!. & & \displaystyle\end{eqnarray}$$

Based on solutions (2.8), propagation of the one soliton is given in figure 1, indicating that the amplitude, velocity and width of the bell-shaped soliton remain invariant during such propagation.

Figure 1.

Effects of the parameters on the one-soliton propagation via solutions (2.8) with $y=1$ at $t=1$ . Parameters are $k=1$ , $g(t)=\sin t$ , and (a) $A=1$ , $B=1$ , $C=0.2$ , ${\it\Gamma}=1.5$ , 1 and 0.7; (b) ${\it\Gamma}=1$ , $B=1$ , $C=0.2$ , $A=1$ , 3 and 6; (c) ${\it\Gamma}=1$ , $A=1$ , $B=-0.5$ , $-2$ and $-5$ , $C=0.2$ ; (d) ${\it\Gamma}=1$ , $A=1$ , $B=-2$ , $C=0.1$ , 0.3 and 1. The solid, dotted and dashed lines in each plot mark the values of the parameter that is varied in each case.

It is further investigated that the soliton’s amplitude is positively related to ${\it\Gamma}$ when ${\it\Gamma}\leqslant 1$ and reaches the maximum value when ${\it\Gamma}=1$ , with ${\it\Gamma}$ as the coefficient of the damped term, as shown in figure 1(a). Effects of $A$ , $B$ and $C$ , i.e. the nonlinear, dispersive and disturbed coefficients, are given in figure 1(b–d): the soliton’s amplitude is positively related to $A$ , while inversely related to $B$ and $C$ . Thus, based on expression (1.4), it is found that the amplitude of the ES soliton in the forced magnetized e–p–i plasma is enhanced with ${\it\lambda}$ decreasing or ${\it\beta}$ increasing, where ${\it\lambda}$ and ${\it\beta}$ represent the phase velocity and ion-to-electron density ratio of the weakly nonlinear ion-acoustic waves in a magnetized e–p–i plasma, respectively.

2.2.2. Two-soliton-like solutions

Truncating expression (2.4) as

(2.9) $$\begin{eqnarray}\displaystyle f=1+{\it\epsilon}f_{1}+{\it\epsilon}^{2}f_{2}, & & \displaystyle\end{eqnarray}$$

and substituting it into bilinear forms (2.2), we have

(2.10a,b ) $$\begin{eqnarray}f_{1}=\text{e}^{{\it\theta}_{1}}+\text{e}^{{\it\theta}_{2}},\quad {\it\theta}_{j}=k_{j}x+p_{j}y+{\it\omega}_{j}t,\quad j=1,2,\end{eqnarray}$$

(2.11a,b ) $$\begin{eqnarray}\displaystyle {\it\omega}_{j}=-\frac{1}{72}\left(\frac{Ak_{j}^{3}\sqrt{A^{2}-144BC}}{C}+\frac{A^{2}k_{j}^{3}}{C}\right),\quad p_{j}=\frac{k_{j}\sqrt{A^{2}-144BC}+Ak_{j}}{12C},\end{eqnarray}$$

(2.12) $$\begin{eqnarray}\displaystyle f_{2}={\it\sigma}\text{e}^{{\it\theta}_{1}+{\it\theta}_{2}},\quad {\it\sigma}=\frac{(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}},\end{eqnarray}$$

with $k_{1}$ and $k_{2}$ as the real constants. Setting ${\it\epsilon}=1$ , we derive the two-soliton-like solutions of (1.6) as follows:

(2.13) $$\begin{eqnarray}\displaystyle u=2\text{e}^{-{\it\Gamma}t}\left[\ln (1+f_{1}+f_{2})_{xy}+\frac{1}{2}\int \text{e}^{{\it\Gamma}t}g(t)\,\text{d}t\right]. & & \displaystyle\end{eqnarray}$$

In accordance with solutions (2.13), interactions between the two solitons are exhibited in figures 2–7, including the effects of the damped coefficient ${\it\Gamma}$ and external force $g(t)$ .

Figure 2.

Interaction between the two solitons via solutions (2.13) at $t=1$ . Parameters are $k_{1}=0.5$ , $k_{2}=-0.95$ , $A=1$ , $B=2$ , $C=0.2$ , and $g(t)=\sin t$ : (a) ${\it\Gamma}=1.5$ ; (b) ${\it\Gamma}=0.6$ .

Figure 3.

The same as figure 2 except that $t$ is not fixed while $y=1$ and (a) ${\it\Gamma}=0.6$ ; (b) ${\it\Gamma}=0.4$ .

Figure 4.

The same as figure 3 except that $y$ is not fixed while $x=1$ .

Considering a periodically-fluctuating external force like $g(t)=\sin t$ , as shown in figures 2–4, we see that the two solitons are always parallel during the propagation on the $x{-}y$ , $x{-}t$ and $y{-}t$ planes. Amplitudes of the two solitons decrease with ${\it\Gamma}$ decreasing, while the background, which has been defined in expression (20) in Shally, Murugan & Rama (Reference Shally, Murugan and Rama2014), of the two solitons rises. Thus, it is found that the periodic external force leads to the parallel ES solitons in a forced magnetized e–p–i plasma, while the amplitudes and background of the ES solitons decrease with ${\it\Gamma}$ decreasing.

However, when there exists an exponentially-decreasing force like $g(t)=\text{e}^{-2t}$ , the two parallel solitons on the $x{-}y$ plane merge into a whole one during the propagation, while the background of the two solitons rises with ${\it\Gamma}$ increasing, as displayed in figure 5. Interaction between the two solitons with the exponentially-decreasing force on the $x{-}t$ and $y{-}t$ planes is displayed in figures 6 and 7, respectively. It is found that the two solitons are also parallel during the propagation, and the amplitudes of the two solitons decreases with ${\it\Gamma}$ increasing. Thus, the exponentially-decreasing force leads to a merged ES soliton on the $x{-}y$ plane or two parallel ES solitons on the $x{-}t$ and $y{-}t$ planes in a forced magnetized e–p–i plasma.

3.1. Phase-plane analysis on (1.7)

Substituting $u(x,y,t)={\it\phi}({\it\xi})$ into (1.7), with ${\it\xi}=r_{1}x+r_{2}y-r_{3}t$ and $r_{j}$ $(j=1,2,3)$ as real constants, we have

(3.1) $$\begin{eqnarray}\displaystyle {\it\phi}_{{\it\xi}{\it\xi}{\it\xi}}+\frac{Ar_{1}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}{\it\phi}{\it\phi}_{{\it\xi}}-\frac{r_{3}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}{\it\phi}_{{\it\xi}}+\frac{{\it\Gamma}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}{\it\phi}=\frac{{\it\alpha}_{1}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}\sin ({\it\alpha}_{2}t).\quad & & \displaystyle\end{eqnarray}$$

Assuming that ${\it\phi}=X$ , ${\it\phi}_{{\it\xi}}=Y$ and ${\it\phi}_{{\it\xi}{\it\xi}}=Z$ , we can rewrite (3.1) as

(3.2a,b ) $$\begin{eqnarray}X_{{\it\xi}}=Y,\quad Y_{{\it\xi}}=Z,\end{eqnarray}$$

(3.2c ) $$\begin{eqnarray}\displaystyle Z_{{\it\xi}} & = & \displaystyle -\frac{Ar_{1}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}XY+\frac{r_{3}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}Y\nonumber\\ \displaystyle & & \displaystyle -\,\frac{{\it\Gamma}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}X+\frac{{\it\alpha}_{1}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}\sin ({\it\alpha}_{2}t).\end{eqnarray}$$

We note that $|{\it\alpha}_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2})|$ and $|Ar_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2})|$ refer to the strengths of the external force and nonlinearity, respectively.

3.2. Chaotic motions of (1.7)

When the external force in a magnetized e–p–i plasma is ignored, (1.6) reduces to (Paul Reference Paul2015)

(3.3) $$\begin{eqnarray}\displaystyle u_{t}+Auu_{x}+Bu_{xxx}+Cu_{xyy}+{\it\Gamma}u=0. & & \displaystyle\end{eqnarray}$$

Then, (3.3) can be written as

(3.4a−c ) $$\begin{eqnarray}\displaystyle X_{{\it\xi}}=Y,\quad Y_{{\it\xi}}=Z,\quad Z_{{\it\xi}}=Z_{{\it\xi}}=-\frac{Ar_{1}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}XY+\frac{r_{3}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}Y-\frac{{\it\Gamma}}{Br_{1}^{3}+Cr_{1}r_{2}^{2}}X. & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

Next, we will study the effect of an external force through the comparison between systems (3.2a,b ) and (3.4a−c ), and by means of the spectral analysis, we will find out whether (1.7) can indeed give rise to the chaotic motions. With the varying strengths of external force and nonlinearity taken into consideration, the corresponding power spectra are given in figure 8, where the abscissa is the frequency of $Z$ , and the ordinate $\text{fft}(Z)$ is the fast Fourier transform (FFT) of $Z$ (Stoica & Moses Reference Stoica and Moses1997).

Power spectrum for $g(t)=0.25\sin ({\rm\pi}t)$ for a particular case when $|Ar_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2})|\gg |({\it\alpha}_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2}))\sin ({\it\alpha}_{2}t)|$ is shown in figure 8(a), in which we can see that the frequency of the external force $g(t)$ is superposed on that of system (3.4a−c ). However, as $g(t)$ is increased to $15\sin ({\rm\pi}t)$ , i.e. $|Ar_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2})|\ll |({\it\alpha}_{1}/(Br_{1}^{3}+Cr_{1}r_{2}^{2}))\sin ({\it\alpha}_{2}t)|$ , the frequency of system (3.4a−c ) is completely broken, as displayed in figure 8(b). Figure 8 exhibits that the different strengths of the external force can lead to different chaotic motions.

Figure 8.

Power spectra of the solutions produced by system (3.2a,b ). Parameters are: $Br_{1}^{3}+Cr_{1}r_{2}^{2}=1$ , $r_{1}=1$ , $r_{3}=0.425$ , ${\it\Gamma}=0.56$ , $Ar_{1}=8.5$ . (a) $g(t)=0.25\sin ({\rm\pi}t)$ ; (b) $g(t)=15\sin ({\rm\pi}t)$ .

Phase projections are investigated for us to exhibit the final chaotic states corresponding with the power spectra, as shown in figure 9.

Figure 9.

Phase projections of the chaotic motions of system (3.2a,b ). Parameters are the same as figure 4.

Phase projection, which is corresponding to the power spectrum in figure 8(a), is shown in figure 9(a), from which we find that the periodicity is broken gradually and (1.7) turns into the chaotic state. Since there only exists a small chaotic deviation from the periodic solutions, as defined in Laptyeva et al. (Reference Laptyeva, Bodyfelt, Krimer and Flach2010) and Mulansky et al. (Reference Mulansky, Ahnert, Pikovsky and Shepelyansky2011), we know that such a chaotic state as in figure 9(a) is the weak one, and the deviation is rooted in the superposition of the two frequencies. The corresponding Poincaré section is given in figure 10(a). There are more than three points in figure 10(a), indicating that the chaotic motion of (1.7) is a weak one after a sequence of quasi-period doubling.

Figure 10.

Poincaré section of system (3.2a,b ) corresponding to (a) figure 9(a); (b) figure 9(b).

When $g(t)$ is increased to $15\sin ({\rm\pi}t)$ , the phase projection is given in figure 9(b), which corresponds to the power spectra in figure 8(b). As defined in Zhang & Yang (Reference Zhang and Yang2011), Lalescu, Meneveau & Eyink (Reference Lalescu, Meneveau and Eyink2013) and Peletan, Baguet & Torkhani (Reference Peletan, Baguet and Torkhani2014), we know that such a chaotic motion is the developed one because the solutions ignore the driver periods and represent a random sequence of uncorrelated shocks. Evolution of the developed chaos can be seen in figure 9(b), and we cannot see any periodicity. Poincaré section for the developed chaos is a three-dimensional strange attractor with the stretching-and-folding structure, as given in figure 10(b), again indicating that the motions in figures 8(b) and 9(b) are indeed chaotic. As seen in figures 8–10, the chaotic motions can be weakened when the phase velocity ${\it\lambda}$ decreases, or the ion-to-electron density ratio ${\it\beta}$ increases.

3.3. Effect of the external force on the chaotic motions

Aiming to find out whether other factors can affect the chaotic motions of (1.7), we fix the strengths of the nonlinearity and external force $g(t)$ and choose the varying parameter as the frequency of $g(t)$ . Phase projections and the corresponding power spectra are shown in figures 11 and 12, respectively.

With the frequency of $g(t)$ varying from 0.5 to $2{\rm\pi}$ , transitions among the periodic motion, weak chaos and developed chaos are given, as shown in figure 11. As the frequency is increased from ${\rm\pi}$ to $2{\rm\pi}$ , the weak chaotic motion is transformed into the periodic one, as shown in figure 11(a). The corresponding power spectrum in figure 12(a) is a single frequency, which can be used to corroborate that the motion in figure 11(a) is indeed periodic.

Figure 11.

Phase projections for the states of system (3.2a,b ). Parameters are the same as those in figure 8 except for (a) $g(t)=0.25\sin (2{\rm\pi}t)$ ; (b) $g(t)=0.25\sin (0.5t)$ .

Figure 12.

Power spectra corresponding to the motions in (a) figure 11(a); (b) figure 11(b).

However, a transition from the weak chaos to developed one occurs when the frequency of $g(t)$ decreases from ${\rm\pi}$ to 0.5, as shown in figure 11(b). From the phase projection, we can see a strange attractor with the stretching-and-folding structure. The corresponding power spectrum is given in figure 12(b), in which there are more than two frequencies, again indicating that the motion in figure 11(b) is developed chaotic. As a consequence, increasing the frequency of the external force can weaken the chaotic motions of (1.7).