2. Basic set of equations

A magnetized EPI plasma is under consideration, which is comprised of $q$ -distributed electrons and positrons, as well as inertial warm ions. An external magnetic field of strength $B_{0}$ is applied along the $x$ -axis i.e.  $\boldsymbol{B}_{0}=B_{0}\hat{x}$ . The charge neutrality condition at equilibrium is $n_{e0}=n_{i0}+n_{p0},$ where $n_{j0}$ is the equilibrium number density of $j$ th species ( $j=\text{e}$ for electrons, $\text{i}$ for ions and $\text{p}$ for positrons).

The nonlinear dynamics of the IA waves in the proposed magnetized plasma can be described by the following equation of ion continuity,

the equation of ion momentum,

and the Poisson equation,

The one-dimensional equilibrium $q$ -distribution function determining the non-extensivity behaviours of electrons and positrons ( $j=\text{e},\text{p}$ ) is given by (Silva, Plastino & Lima Reference Silva, Plastino and Lima1998; Verheest Reference Verheest2013)

where

and

Here, the parameter $q_{j}$ stands for the strength of non-extensivity. It is useful to note that for $q_{j}<-1$ , the $q$ -distribution is not normalizable. In the extensive limiting case ( $q_{j}=1$ ), the $q$ -distribution reduces to the well-known Maxwell–Boltzmann distribution. Note that for $q_{j}>1$ , the $q$ -distribution function exhibits a thermal cutoff on the maximum value allowed for the velocity of the $j$ th-particles, which is given by (Pakzad & Tribeche Reference Pakzad and Tribeche2011)

The densities of $q$ -distributed non-thermal electrons and positrons are obtained from the zeroth order moment of the distribution function and integrating over $v_{j}$ i.e.  $-\infty$ to $+\infty$ (for $q_{j}<1$ ) and $v_{j}$ from $-v_{_{j}\mathit{max}}$ to $+v_{j\,\mathit{max}}$ (for $q_{j}>1$ ) case, which is described as follows,

The parameters $q_{e}$ and $q_{p}$ stand for the strengths of electron and positron non-extensivity. Unlike the description of Maxwellian distribution, such that most of the particles are centred around the thermal speed, a power law distribution characterizes the systems containing an ample supply of superthermal particles with the constraint $q_{e}$ , $q_{p}<1$ , or by including a large number of low velocity particles with the restriction $q_{e}$ and $q_{p}>1$ (Liyan & Jiulin Reference Liyan and Jiulin2008; El-Taibany & Tribeche Reference El-Taibany and Tribeche2012).

The energy of the $q$ -distributed particles can be obtained from the second moment of the distribution function i.e. involving the integral of $v_{j}^{2}f(v_{j})$ over $v_{j}$ , which comes out to be

for both $-1<q_{j}<1$ and $q_{j}>1$ cases. The range of the non-extensive parameter $q_{j}$ (for the case $q_{j}<1$ ) for which the energy of the system is positive should be $1/3<q_{j}<1$ . This is in contrast to kappa distribution theory, where the argument that the energy should remain finite has led to the standard requirement that ${\it\kappa}>3/2$ (Vasyliunas Reference Vasyliunas1968; Hellberg et al. Reference Hellberg2009). There is no maximum limit for $q_{j}>1$ case, but its values are not taken to be very large in the literature (Silva et al. Reference Silva, Plastino and Lima1998; Pakzad & Tribeche Reference Pakzad and Tribeche2011; Verheest Reference Verheest2013) for a non-thermal plasma system.

Here $\boldsymbol{{\rm\nabla}}=(\partial /\partial x,\partial /\partial y,0)$ and $\boldsymbol{V}_{i}=(V_{ix},V_{iy},V_{iz})$ and the quantities $n_{j}$ and ${\it\varphi}$ are the number density of $j$ th species and the electrostatic potential, respectively. The ion gyro-frequency is denoted by ${\it\omega}_{ci}=eB_{0}/m_{i}c$ , $e$ is the electronic charge, $m_{i}$ is the mass of ions and $c$ is the speed of light in a vacuum. For the warm ion pressure we have $P_{i}={\it\gamma}n_{i}T_{i}$ (Tiwari, Jain & Chawla Reference Tiwari, Jain and Chawla2007), where ${\it\gamma}$ is the adiabatic constant and for a magnetized plasma we have ${\it\gamma}=5/3$ and $T_{e}(T_{p})$ is the electron (positron) temperature.

Using the already mentioned dimensional equations (2.1)–(2.9), the dispersion relation in normalized form is obtained as follows

where

Here $k_{x}$ and $k_{y}$ are the wave vectors in the $x$ - and $y$ -directions joined by the relation $k^{2}=k_{x}^{2}+k_{y}^{2}$ . Also, ${\it\lambda}_{D}=\sqrt{T_{e}/4{\rm\pi}n_{i0}e^{2}}$ (Debye length), ${\it\omega}_{pi}=\sqrt{4{\rm\pi}n_{i0}e^{2}/m_{i}}$ (ion plasma frequency), $C_{s}={\it\omega}_{pi}{\it\lambda}_{D}=\sqrt{T_{e}/m_{i}}$ (IA speed) and ${\it\Omega}={\it\omega}_{ci}/{\it\omega}_{pi}$ , ${\it\sigma}={\it\gamma}T_{i}/T_{e}$ , ${\it\beta}=T_{e}/T_{p}$ , ${\it\mu}_{e}=n_{e0}/n_{i0}$ and $p=n_{p0}/n_{i0}$ . The roots of the quartic equation are given by

The limiting cases for obliquely propagating electrostatic waves in the parallel and perpendicular directions to the magnetic field $\boldsymbol{B}_{0}$ are discussed below.

In order to study wave propagation in a direction parallel to the magnetic field $\boldsymbol{B}_{0}$ , we put $k_{y}=0$ in (2.13) to obtain the following two roots:

The positive root is a frequency and can therefore be neglected; the second root is the dispersion relation for an IA wave propagating along the magnetic field and the phase velocity of IA wave depends on the non-extensive parameters of the electrons and positrons, temperature and their density ratios.

On the other side, to find the electrostatic wave propagating in a direction perpendicular to the magnetic field $\boldsymbol{B}_{0}$ , we put $k_{x}=0$ in (2.13) and obtain

The positive root gives the dispersion relation for an ion cyclotron wave (ICW), whereas the slow wave vanishes. Here, the phase velocity of the ICW is not independent of $k_{y}$ and also depends on the non-extensive parameters of the electrons and positrons, temperature and their density ratios.

We can express (2.1)–(2.9) in the form of scalar components as follows,

where we have used the normalizations as $n_{e,i,p}=n_{e,i,p}/n_{e0,i0,p0},$ $\boldsymbol{V}_{i}=\boldsymbol{V}_{i}/C_{s}$ , $\boldsymbol{{\rm\nabla}}=\hat{\boldsymbol{x}}\partial /\partial x+\hat{\boldsymbol{y}}\partial /\partial y\equiv \boldsymbol{{\rm\nabla}}{\it\lambda}_{D}$ , $t=t{\it\omega}_{pi}$ , ${\it\Omega}={\it\omega}_{ci}/{\it\omega}_{pi},$ and ${\it\Phi}=e{\it\varphi}/T_{e}.$ Here ${\it\mu}_{e}=(1+p),$ with, $c_{1}=(q_{e}+1)/2$ , $c_{2}=(q_{e}-3)/4$ , $p_{1}=(q_{p}+1){\it\beta}/2$ , $p_{2}=(q_{p}-3){\it\beta}/4$ . The non-extensive effects due to the electrons and positrons are represented by the indices $q_{e}$ and $q_{p}$ respectively.

3. Nonlinear analysis

In order to derive a ZK equation for the proposed magnetized EPI plasmas, we define the stretching of independent variables, from (Washimi & Taniuti Reference Washimi and Taniuti1966; Kourakis et al. Reference Kourakis2009)

where ${\it\epsilon}$ is a small $(0<{\it\epsilon}\leqslant 1)$ expansion parameter characterizing the strength of the nonlinearity and ${\it\lambda}$ (normalized by $C_{s}$ ) is the phase speed of the wave, which will be determined later. Using the RPM, Kourakis et al. (Reference Kourakis2009), we can expand the perturbed quantities about their equilibrium values as follows:

Substituting (3.1) and (3.2) into (2.16)–(2.22) and collecting terms of lowest order $({\sim}{\it\epsilon}^{3/2})$ terms of continuity and $x$ -, $y$ -, $z$ -components of the momentum equation for the ions, we obtain

where $N=1-({\it\sigma}/{\it\lambda}^{2})$ .

The lowest order $({\sim}{\it\epsilon})$ terms for the non-extensive electrons and positrons from (2.21)–(2.22) give

The Poisson equation gives the lowest order $({\sim}{\it\epsilon})$ terms,

The phase speed ${\it\lambda}$ of the wave is obtained using (3.3), (3.5) and (3.6), i.e.

which is the normalized phase velocity of the IA wave which can be obtained from (2.14) for a dispersionless medium.

Proceeding through the perturbation theory and using the relations provided in (3.3)–(3.7), we can finally obtain the ZK equation for an electrostatic wave in magnetized EPI plasmas in terms of ${\it\Phi}_{1}$ as follows,

where the nonlinear coefficient $A$ and the dispersive coefficients $B$ and $C$ are defined as,

The ZK equation (3.8) has a well-known soliton solution (Kourakis et al. Reference Kourakis2009), which is given by

where ${\it\phi}_{m}=3u_{0}/A$ and $W=\sqrt{4(B+C)/u_{0}}$ are the amplitude and the width of the soliton, and ${\it\eta}$ is the transformed coordinate in the co-moving frame with speed $u_{0}$ i.e.  ${\it\zeta}=\ell _{x}{\it\xi}+\ell _{y}{\it\eta}-u_{0}{\it\tau}$ . Here $l_{x}^{2}+l_{y}^{2}=1$ . The coefficients of the ZK equation, which depend on the entropic indices i.e.  $q_{e}$  and $q_{p}$ , directly affect the basic features of the produced soliton.

From (3.10) and the relation $\boldsymbol{E}_{1}=-\boldsymbol{{\rm\nabla}}{\it\Phi}_{1}$ , the normalized electric field of the obliquely propagating two-dimensional IAW becomes

Stability analysis. To study the stability of the two-dimensional solitons satisfying (3.8), we apply a small- $k$ expansion perturbation method (Moslem et al. Reference Moslem2007; Ghosh & Chakrabart Reference Ghosh and Chakrabart2013). Using the following transformations

we can rewrite (3.8) as

Supposing ${\it\chi}=X-{\it\nu}_{0}T,$  (3.13) becomes

where we assume that (3.13) has a planar solution ${\it\varphi}_{0}$ , that is given by

with ${\it\varphi}_{0m}=3{\it\nu}_{0}$ and ${\it\Delta}=2/\sqrt{{\it\nu}_{0}}$ . Now, let us write the full solution of (3.13) as

Substituting (3.16) into (3.13) and linearizing with respect to ${\it\psi}$ , we obtain

Now expanding ${\it\psi}$ and ${\it\Upsilon}$ as a power series of $k$

then substituting into (3.17) and equating the coefficients of equal powers of $k$ , the zeroth order yields

which can be written as

where $\hat{L}=((\text{d}^{2}/\text{d}{\it\chi}^{2})+{\it\varphi}_{0}-{\it\nu}_{0})$ . Though, (3.14) can be given in terms of the operator $\hat{L}$ as

Differentiating with respect to ${\it\chi}$ , we obtain

Comparing (3.20) and (3.22), we can write

So,

where ${\it\psi}_{0m}=-3{\it\nu}_{0}^{3/2}$ . Considering the next order of $k$ , one obtains

Integrating with respect to ${\it\chi}$ , we obtain

where $S_{1}$ is the integration constant. To solve (3.26), it is useful to write ${\it\psi}_{1}=V({\it\chi}){\it\psi}_{0}({\it\chi})$ , where $V({\it\chi})$ is an arbitrary function, to be determined. Substituting into (3.26) and looking for the localized solution under the condition ${\it\psi}_{1}\rightarrow 0$ as $|{\it\chi}|\rightarrow \infty$ , we obtain

So,

where ${\it\psi}_{1m}=-(3/4)\text{i}{\it\Upsilon}_{1}{\it\nu}_{0}{\it\Delta}$ . Considering the next order of $k$ , we obtain

To have a solution for (3.29), its left-hand side must be orthogonal to the kernal ${\it\varphi}_{0}$ , i.e.

This orthogonality condition yields the following consistency condition

from which we can determine ${\it\Upsilon}_{1}$ . After performing these integrals, we obtain

For the present problem, ${\it\nu}_{0}$ is always positive, so it is clear that the positive sign in the above expression leads to instability of the one-dimensional soliton solution in the transverse direction. Also, the instability growth rate is directly proportional to ${\it\nu}_{0}$ . The above result, (3.32), agrees exactly with that obtained in Moslem et al. (Reference Moslem2007) and Ghosh & Chakrabart (Reference Ghosh and Chakrabart2013) when ${\it\nu}_{o}=4;|{\it\Upsilon}_{1}|=8/\sqrt{15}$ .

Figure 1.

The low frequency mode of the electrostatic wave i.e.  ${\it\omega}_{-}$ versus $k$ is plotted for different values of entropic indices i.e.  $q_{e}$ , $q_{p}$ ; $q_{e}=q_{p}=0.35$ (solid curve), $q_{e}=$ $q_{p}=0.9$ (dotted curve), $q_{e}=q_{p}=2$ (dashed dotted curve). For the plasma ${\it\mu}=5$ , $p=4$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $T_{i}=0.2T_{e}$ , $k_{y}=0.1k_{x}$ and $B_{0}\sim 10^{2}~\text{G}$ .

Figure 2.

The high frequency mode of the electrostatic wave i.e.  ${\it\omega}_{+}$ versus $k$ is plotted for different values of entropic indices i.e.  $q_{e}$ , $q_{p}$ ; $q_{e}=q_{p}=0.35$ (solid curve), $q_{e}=$ $q_{p}=0.9$ (dotted curve), $q_{e}=q_{p}=2$ (dashed dotted curve). For the plasma ${\it\mu}=5$ , $p=4$ , $T_{e}=$ $T_{p}=10^{3}~\text{K}$ , $T_{i}=0.2T_{e}$ , $k_{y}=0.1k_{x}$ and $B_{0}\sim 10^{2}~\text{G}$ .

Figure 3.

The low frequency mode of the electrostatic wave i.e.  ${\it\omega}_{-}$ versus $k$ is plotted for different values of magnetic field. The plasma parameters are ${\it\mu}=5$ , $p=4$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $T_{i}=0.2T_{e}$ , $k_{y}=0.1k_{x}$ , $q_{e}=q_{p}=0.9$ for $B_{0}=10^{2}~\text{G}$ (solid curve), $B_{0}=4\times 10^{2}~\text{G}$ (dotted curve), $B_{0}=8\times 10^{2}~\text{G}$ (dashed dotted curve).

Figure 4.

The high frequency mode of the electrostatic wave i.e. ${\it\omega}_{+}$ vs $k$ is plotted for different values of magnetic field. The plasma parameters are ${\it\mu}=5$ , $p=4$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $T_{i}=0.2T_{e}$ , $k_{y}=0.1k_{x}$ , $q_{e}=$ $q_{p}=0.9$ for $B_{0}=10^{2}~\text{G}$ (solid curve), $B_{0}=4\times 10^{2}~\text{G}$ (dotted curve), $B_{0}=8\times 10^{2}~\text{G}$ (dashed dotted curve).

Figure 5.

The phase velocity ${\it\lambda}$ is plotted for IA wave in EPI plasmas at different ion temperature ( $T_{i}$ ) against entropic indices $q_{e}$ , $q_{p}$ for the non-thermal electrons and positrons. The plasma parameters are ${\it\mu}=1.2$ , $p=0.2$ , $T_{e}=T_{p}=10^{3}~\text{K}$ for $T_{i}=0.2T_{e}$ (red solid curve), $T_{i}=0$ (blue dotted curve).

Figure 6.

The phase velocity ${\it\lambda}$ is plotted for IA wave in EPI plasmas at different positrons density against entropic indices $q_{e}$ , $q_{p}$ for the non-thermal electrons and positrons. The plasma parameters are $T_{i}=0.2T_{e}$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $p=0.9$ (red solid curve), $p=0$ (blue dotted curve).

Figure 7.

The IA soliton amplitude ${\it\phi}_{m}=\left(3u_{0}/A\right)$ is plotted for different values of positron density concentration and entropic indices $q_{e}$ , $q_{p}$ for the non-thermal electrons and positrons. The plasma parameters are $u_{0}=0.3$ , $T_{i}=0.2T_{e}$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $B_{0}\sim 10~\text{G}$ , $p=0.4$ (red solid curve), $p=0.2$ (blue dotted curve) and $p=0$ (green dashed dotted curve).

Figure 8.

The width of the soliton $W=\sqrt{4(B+C)/u_{0}}$ is plotted for different values of positron density concentration and entropic indices $q_{e}$ , $q_{p}$ for the non-thermal electrons and positrons. The plasma parameters are $u_{0}=0.3$ , $T_{i}=0.2T_{e}$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $B_{0}\sim 10~\text{G}$ , $p=0.4$ (red solid curve), $p=0.2$ (blue dotted curve) and $p=0$ (green dashed dotted curve).

Figure 9.

The amplitude of the soliton ${\it\phi}_{m}=(3u_{0}/A)$ is plotted against entropic indices $q_{e}$ , $q_{p}$ for the electrons and positrons for different values of ${\it\sigma}$ . The plasma parameters are $u_{0}=0.3$ , ${\it\mu}=1.2$ , $p=0.2$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $B_{0}\sim 10~\text{G}$ , $T_{i}=0.28T_{e}$ (red solid curve), $T_{i}=0.1T_{e}$ (blue dotted curve) and $T_{i}=0$ (green dashed dotted curve).

Figure 10.

The soliton width $W=\sqrt{4(B+C)/u_{0}}$ is plotted against entropic indices $q_{e}$ , $q_{p}$ for the electrons and positrons for different values of ${\it\sigma}$ . The plasma parameters are $u_{0}=0.3$ , ${\it\mu}=1.2$ , $p=0.2$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $B_{0}\sim 10~\text{G}$ , $T_{i}=0.28T_{e}$ (red solid curve), $T_{i}=0.1T_{e}$ (blue dotted curve) and $T_{i}=0$ (green dashed dotted curve).

Figure 11.

The width $W=\sqrt{4(B+C)/u_{0}}$ of the soliton is plotted for different values of magnetic field intensity $B_{0}$ against entropic indices $q_{e}$ , $q_{p}$ for the electrons and positrons. The plasma parameters are $u_{0}=0.3$ , ${\it\mu}=1.2$ , $p=0.2$ , $T_{e}=T_{p}=10^{3}~\text{K}$ , $T_{i}=0.2T_{e}$ , $B_{0}=0.1~\text{G}$ (red solid curve), $B_{0}=20~\text{G}$ (blue dotted curve) and $B_{0}=10^{2}~\text{G}$ (green dashed dotted curve).

4. Numerical illustrations and discussion

In this section we illustrate the characteristics and instability of two-dimensional IASW in magnetized EPI plasmas based on the results presented in §§ 3 and 4.

Figure 12.

The variations of the planar solution, ${\it\varphi}_{0}$ , given by (3.15), in (a) and ${\it\psi}={\it\psi}_{0}+k{\it\psi}_{1}+\cdots \,,$ given by (3.18), in (b) versus ${\it\chi}$ where ${\it\nu}_{0}=2$ (black solid curve), ${\it\nu}_{0}=5$ (red dashed curve) and ${\it\nu}_{0}=8$ (blue dashed dotted curve). Here, $k=0.75$ .

The high and low frequency modes of the dispersion relation ${\it\omega}_{+,-}$ versus $k$ are plotted with variations of non-extensive parameter for electrons $q_{e}$ (same value for $q_{p}$ ) and the magnetic field $B_{0}$ . In figures 1 and 2 the low (IA wave) and high (ion cyclotron wave) frequency modes of the dispersion relation are shown at different values of the non-extensive parameter i.e.  $q_{e}$ . It can be seen from figure 1 that as we increase the values of $q_{e}$ , the saturation of the curve for the low frequency mode occurs at larger values of $k$ , while for the high frequency mode, saturation of the curves are created at small wavelengths or at large $k$ values as shown in figure 2. Magnetic field variation on the low and high frequency modes of the dispersion relation are presented in figures 3 and 4, respectively. The phase velocity of the low wave increases by enhancing the magnetic field strength while the saturation of the curve occurs at large values of $k$ shown in figure 3. The phase velocity of the high wave also increases with the increase in the magnetic field strength and the wave dispersion occurs at large $k$ values described in figure 4. The phase velocity of low or IAW ( ${\it\lambda}$ ) defined in (3.7) is plotted in figure 5 in EPI plasmas for different cases of ion temperature i.e.  ${\it\sigma}$ (ratio of ion-to-electron temperature) and variations of entropic indices $q_{e}$ , $q_{p}$ for the non-thermal electrons and positrons. It is found that the phase velocity of IA wave is decreased with an increase of entropic indices i.e.  $q_{e}$ , $q_{p}$ for the electrons and positrons. The phase velocity of IA wave increases in the presence of an ion temperature in comparison with the cold ions case (i.e.  ${\it\sigma}=0$ ) for $q_{e}$ , $q_{p}>0$ . In figure 6 phase velocity, ${\it\lambda}$ , is plotted for the different positron concentrations versus entropic indices i.e.  $q_{e}$ , $q_{p}$ for the electrons and positrons, respectively. The phase velocity, ${\it\lambda}$ , is larger in the presence of positrons in comparison with the absence of a positron concentration as shown in the figure. It is noticed that the phase velocity of the IA wave decreases with the increase in the value of entropic indices i.e.  $q_{e}$ , $q_{p}$ for the electrons and positrons keeping positron concentration fixed. The amplitude of the soliton ${\it\phi}_{m}=(3u_{0}/A)$ is plotted in figure 7 for different values of positron concentration against the entropic indices of electrons and positrons. The amplitude of the IA soliton increases as we decrease the positron concentration, keeping all others parameters fixed. The width of the soliton $W=\sqrt{4(B+C)/u_{0}}$ is plotted in figure 8 for all the same parameters used in figure 7, and we found a similar trend. The width of the soliton decreases at larger values of positron concentration and non-extensive parameter. The amplitude of the soliton is plotted in figure 9 for different values of ion temperature and entropic indices $q_{e}$ , $q_{p}$ . The amplitude of the soliton decreases for the larger values of ${\it\sigma}$ and entropic indices $q_{e}$ , $q_{p}$ . The soliton width $W$ is plotted for the temperature coefficient ${\it\sigma}$ in figure 10. The width of the soliton increases for large values of ${\it\sigma}$ and vice versa for the non-extensive parameters. The width for the ion acoustic wave in EPI is plotted for the magnetic field intensity. It can be seen that there is no change in the amplitude, whereas the width of the soliton reduces significantly with the increase in magnetic field strength i.e. $B_{0}$ (0.1 G, 20 G, 100 G) as shown in figure 11. It is also evident from (3.10) that the intensity of the external magnetic field $B_{0}$ has no direct effect on the amplitude of the solitary waves. The contribution of the ion gyro-frequency ${\it\Omega}$ comes into play on the width of the soliton through the coefficient $C$ in (3.9). The amplitude of the soliton remains the same while the width decreases with increasing ${\it\Omega}$ i.e. a stronger magnetic field leads to steeper and thus narrower soliton profiles, which agrees with the results of Sultana et al. (Reference Sultana2010). Figure 12 shows the variation of the planar solution, ${\it\varphi}_{0}$ , that is given by (3.15), and the function ${\it\psi}={\it\psi}_{0}+k{\it\psi}_{1}+\cdots \,$ appears in the perturbed solution; (3.18) against ${\it\chi}$ and different values of ${\it\nu}_{0}.$ It is clear that ${\it\varphi}_{0}$ ( ${\it\psi}$ ) decreases (increases) as ${\it\chi}$ increases (decreases). For ${\it\chi}<1,{\it\varphi}_{0}$ increases as ${\it\nu}_{0}$ increases. The opposite effect is observed where ${\it\chi}>1$ . Since, ${\it\varphi}_{0}$ changes its behaviour with ${\it\nu}_{0}$ at ${\it\chi}=1,$ this ${\it\psi}$ -corresponding characteristic point occurs at ${\it\chi}\simeq 1.75$ . Moreover, the variation curves of ${\it\psi}$ versus ${\it\chi}$ shows the existence of minimum points appearing at smaller ${\it\chi}$ by increasing ${\it\nu}_{0}$ . Figure 12 proves that there are some points at which $|{\it\psi}|>|{\it\varphi}_{0}|$ that may lead to instability of the produced solitons in the transverse direction (comparing the dash dotted curves in the two panels that correspond to higher values of ${\it\nu}_{0}$ ).