2 Reductive perturbation formalism and evolution equations

There is a plethora of plasma compositions treated in the literature, allowing critical densities leading to the mKdV equation. These plasmas are usually described by a number of cold and/or warm fluid species, of quite different characteristics, in the presence of some inertialess Boltzmann or non-thermal distributions for the hot species.

RPT rests on two pillars: a suitable stretching of space and time and the expansion of the dependent variables. For the family of KdV-type equations the stretching can be chosen as

(2.1a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D709}=\unicode[STIX]{x1D700}(X-VT),\quad \unicode[STIX]{x1D70F}=\unicode[STIX]{x1D700}^{3}T,\end{eqnarray}$$

or equivalent choices, with $V$ the linear acoustic phase velocity, $\unicode[STIX]{x1D700}$ a small parameter and $X$ and $T$ the physical space and time coordinates, respectively. This is inspired by the dispersion law for linear waves with frequency $\unicode[STIX]{x1D714}$ and wavenumber $k$ , by taking the limit $k\rightarrow 0$ but keeping the phase velocity $\unicode[STIX]{x1D714}/k$ finite, based on the idea that acoustic modes have $\unicode[STIX]{x1D714}$ proportional $k$ to lowest order. For the expansion of the electrostatic potential, $\unicode[STIX]{x1D711},$ we take

(2.2) $$\begin{eqnarray}\unicode[STIX]{x1D711}=\unicode[STIX]{x1D700}\unicode[STIX]{x1D711}_{1}+\unicode[STIX]{x1D700}^{2}\unicode[STIX]{x1D711}_{2}+\unicode[STIX]{x1D700}^{3}\unicode[STIX]{x1D711}_{3}+\cdots \,,\end{eqnarray}$$

with similar expansions for the plasma variables like densities, pressures and fluid velocities of the different species.

Applying (2.1) and (2.2) to the basic fluid equations and Poisson’s equation produces order by order in $\unicode[STIX]{x1D700}$ sets of equations for the successive $\unicode[STIX]{x1D711}_{i}$ . In the generic case, $\unicode[STIX]{x1D711}_{1}\equiv 0$ leads to the KdV equation,

(2.3) $$\begin{eqnarray}A\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+B\unicode[STIX]{x1D711}_{2}\,\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{1}{2}\,\frac{\unicode[STIX]{x2202}^{3}\unicode[STIX]{x1D711}_{2}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}=0.\end{eqnarray}$$

The coefficients $A$ and $B$ include the compositional plasma parameters and $V$ ; the latter as a solution of the linear dispersion law is a function of those same parameters. No matter how complicated the plasma model is, one gets the KdV equation (2.3) which has soliton solutions and other special properties, provided the plasma species obey barotropic pressure–density relations (Verheest Reference Verheest2000). Given its ubiquity in physics, this is the best known and most studied nonlinear soliton.

We assume that the plasma composition is critical ( $B=0$ ). Continuing with $\unicode[STIX]{x1D711}_{1}\not =0$ then leads to the mKdV equation,

(2.4) $$\begin{eqnarray}A\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D70F}}+C\unicode[STIX]{x1D711}_{1}^{2}\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}}+\frac{1}{2}\frac{\unicode[STIX]{x2202}^{3}\unicode[STIX]{x1D711}_{1}}{\unicode[STIX]{x2202}\unicode[STIX]{x1D709}^{3}}=0,\end{eqnarray}$$

where $A$ and $C$ again depend on the plasma parameters and $V$ . Perusing the literature, it becomes clear that almost all plasmas need to have at least three constituents before $B=0$ can be imposed under the condition that $V$ annuls the linear dispersion relation.

The earliest examples are plasmas with two cold ion species (one positive, one negative) in the presence of Boltzmann electrons (Das Reference Das1975; Das & Tagare Reference Das and Tagare1975; Watanabe Reference Watanabe1984; Tagare Reference Tagare1986). Later, this was extended to include more cold or warm ion species (Verheest Reference Verheest1988), and the picture remains unchanged when various non-thermal distributions are introduced for the hot electrons (Verheest Reference Verheest2015), such as the kappa distribution (Vasyliunas Reference Vasyliunas1968; Summers & Thorne Reference Summers and Thorne1991) and Tsallis non-extensive distribution (Tsallis Reference Tsallis1988; Lima, Silva Jr & Santos Reference Lima, Silva and Santos2000). The only exception to this three or more species rule is the Cairns distribution (Cairns et al. Reference Cairns, Mamun, Bingham, Boström, Dendy, Nairn and Shukla1995) because it admits a critical density in the presence of only one cold ion species. Regardless of the fine tuning of the parameters needed for criticality, mKdV solitons were even observed experimentally in multi-ion plasmas and their collision properties were investigated by numerically solving the mKdV equation (Nakamura & Tsukabayashi Reference Nakamura and Tsukabayashi2009).

Note that the mKdV equation (2.4) is invariant under the change $\unicode[STIX]{x1D711}$ to $-\unicode[STIX]{x1D711}$ . As a consequence of this uncommon property, every positive-polarity soliton has an equivalent negative one. The one-soliton solution of (2.4) is well known,

(2.5) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{1}=\pm \sqrt{\frac{6UA}{C}}\text{sech}[\sqrt{2UA}(\unicode[STIX]{x1D709}-U\unicode[STIX]{x1D70F})],\end{eqnarray}$$

where $U$ is an arbitrary velocity, in the frame moving with velocity $V$ with respect to an inertial observer. On the other hand, the two-soliton solution is much less known in the context of plasma physics, and will therefore be the main focus of our efforts in the present paper.

In what follows, we will start from a generic form with coefficients chosen such that the application of Hirota’s formalism (Hirota Reference Hirota1971, Reference Hirota1972, Reference Hirota2004) is as simple as possible (Drazin & Johnson Reference Drazin and Johnson1989). A change of variables,

(2.6a-c ) $$\begin{eqnarray}\unicode[STIX]{x1D709}=2x\sqrt{\frac{3}{C}},\quad \unicode[STIX]{x1D70F}=48t\frac{A}{C}\sqrt{\frac{3}{C}},\quad \unicode[STIX]{x1D711}_{1}=u,\end{eqnarray}$$

transforms the mKdV equation (2.4) into

(2.7) $$\begin{eqnarray}u_{t}+24u^{2}u_{x}+u_{xxx}=0.\end{eqnarray}$$

This transformation requires that $C>0$ (and $B=0$ ). The coefficient 24 in (2.7) is chosen to minimize the numerical factors when applying Hirota’s method, and derivatives are denoted by lower-case subscripts $x$ and $t$ , where, for example, $u_{xxx}$ denotes the third derivative of $u$ with respect to $x$ . Note that we have transformed the space and time variables, but $u$ remains a normalized electrostatic potential at the lowest non-zero order.

3 Hirota’s bilinear method and soliton solutions

Hirota (Reference Hirota1971) developed an ingenious method to find exact $N$ -soliton solutions for the KdV equation. The method uses bilinear operators, hence the name Hirota’s bilinear method. It was later shown that the method can be applied to large classes of nonlinear evolution equations, including the mKdV equation.

The key idea is to change the dependent variable so that the given nonlinear equation becomes bilinear in one or more new dependent variables. Once the appropriate bilinear forms have been found, a formal series expansion is used to generate its solutions in an iterative way. If pure solitons exist the iterative process terminates at a certain level and the finite series leads to an exact solution.

Hirota’s method proceeds in three steps: (i) a judicious guess for the transformation of the dependent variable, (ii) writing the transformed equation as a single bilinear equation or a coupled system of bilinear equations and (iii) using a formal expansion scheme to solve these bilinear equation(s).

Inspiration for the initial step has sometimes come from performing the Painlevé integrability test (Drazin & Johnson Reference Drazin and Johnson1989; Ablowitz & Clarkson Reference Ablowitz and Clarkson1991) of an evolution equation or knowing its $N$ -soliton solution from application of the inverse scattering method.

Let us now recall some of the steps for the mKdV equation. Hirota (Reference Hirota1971) introduced bilinear differential operators, $D_{x}$ and $D_{t}$ , defined for ordered pairs of arbitrary functions $f(x,t)$ and $g(x,t)$ , as follows,

(3.1) $$\begin{eqnarray}D_{x}^{m}D_{t}^{n}\{\,f\cdot g\}=\left.\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x^{\prime }}\right)^{m}\left(\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t}-\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}t^{\prime }}\right)^{n}f(x,t)g(x^{\prime },t^{\prime })\right|_{x^{\prime }=x,t^{\prime }=t},\end{eqnarray}$$

where $m$ and $n$ are non-negative integers.

Operators like $D_{x}$ and $D_{t}$ are bilinear because of their evident linearity in both arguments $f$ and  $g$ .

Whereas the KdV equation can be replaced by one bilinear equation (Hirota Reference Hirota1971; Drazin & Johnson Reference Drazin and Johnson1989), the mKdV equation (2.7) requires a coupled system of bilinear equations (Hirota Reference Hirota1972) because the change of the dependent variable

(3.2) $$\begin{eqnarray}u=\frac{\unicode[STIX]{x2202}}{\unicode[STIX]{x2202}x}\left(\arctan \left(\frac{f}{g}\right)\right)=\frac{f_{x}g-fg_{x}}{f^{2}+g^{2}}\end{eqnarray}$$

involves two functions $f(x,t)$ and $g(x,t)$ . Substitution of (3.2) into (2.7) yields, after one integration with respect to $x$ ,

(3.3) $$\begin{eqnarray}\displaystyle & & \displaystyle (f^{2}+g^{2})(f_{t}g-fg_{t}+f_{xxx}g-3f_{xx}g_{x}+3f_{x}g_{xx}-fg_{xxx})\nonumber\\ \displaystyle & & \displaystyle \quad -6(fg_{x}-f_{x}g)(ff_{xx}-f_{x}^{2}+gg_{xx}-g_{x}^{2})=0.\end{eqnarray}$$

Setting each term equal to zero results in the following pair of bilinear equations,

(3.4) $$\begin{eqnarray}\displaystyle & (D_{t}+D_{x}^{3})\{\,f\cdot g\}=0, & \displaystyle\end{eqnarray}$$

(3.5) $$\begin{eqnarray}\displaystyle & D_{x}^{2}\{\,f\cdot f+g\cdot g\}=0. & \displaystyle\end{eqnarray}$$

The expansions are

(3.6) $$\begin{eqnarray}\left.\begin{array}{@{}c@{}}f=f_{0}+\unicode[STIX]{x1D700}f_{1}+\unicode[STIX]{x1D700}^{2}f_{2}+\cdots \,,\\ g=g_{0}+\unicode[STIX]{x1D700}g_{1}+\unicode[STIX]{x1D700}^{2}g_{2}+\cdots \,,\end{array}\right\}\end{eqnarray}$$

where $f_{0}$ and $g_{0}$ are constants (not both zero to avoid a trivial solution) and $\unicode[STIX]{x1D700}$ is a bookkeeping parameter to disentangle the different orders. Once the successive $f_{i}$ and $g_{i}$ have been computed, one sets $\unicode[STIX]{x1D700}=1$ . Doing the computations, one finds from the lowest non-zero order of (3.5) that either $f_{0}=g_{1}=0$ or $g_{0}=f_{1}=0$ . Both choices are equivalent for they amount to a sign change in the polarity of the nonlinear wave.

To derive the one and two-soliton solutions, we continue with $f_{0}=g_{1}=0$ . Then, without loss of generality, we normalize $g_{0}=1$ and recover from $f_{1}=\text{e}^{\unicode[STIX]{x1D703}_{1}}$ (in the one-soliton case) the well-known sech solution (2.5). Here and below one uses the notation $\unicode[STIX]{x1D703}_{i}=k_{i}x-k_{i}^{3}t+\unicode[STIX]{x1D6FF}_{i}$ , which incorporates the linear dispersion $\unicode[STIX]{x1D714}_{i}=k_{i}^{3}$ characterizing the KdV family. In the two-soliton case, $f_{1}=\text{e}^{\unicode[STIX]{x1D703}_{1}}+\text{e}^{\unicode[STIX]{x1D703}_{2}}$ and one can set $f_{2}=0$ . For one and two-soliton interactions one can shift the origins of $x$ and $t$ so as to absorb $\unicode[STIX]{x1D6FF}_{i}$ , which will henceforth be omitted. The same argument no longer works for three-soliton interactions (and higher). Suppressing the $\unicode[STIX]{x1D6FF}_{i}$ then requires non-unit amplitudes $a_{i}=\text{e}^{\unicode[STIX]{x1D6FF}_{i}}$ .

After computing $g_{2}$ and verifying that $g_{i}=0$ for $i\geqslant 3$ and $f_{i}=0$ for $i\geqslant 2$ , one obtains

(3.7) $$\begin{eqnarray}\displaystyle \left.\begin{array}{@{}c@{}}f=\text{e}^{\unicode[STIX]{x1D703}_{1}}+\text{e}^{\unicode[STIX]{x1D703}_{2}},\\ g=1-{\displaystyle \frac{(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}}\text{e}^{\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}},\end{array}\right\} & & \displaystyle\end{eqnarray}$$

and, from (3.2),

(3.8) $$\begin{eqnarray}u=\frac{k_{1}\text{e}^{\unicode[STIX]{x1D703}_{1}}+k_{2}\text{e}^{\unicode[STIX]{x1D703}_{2}}+\displaystyle \frac{(k_{1}-k_{2})^{2}}{(k_{1}+k_{2})^{2}}(k_{1}\text{e}^{\unicode[STIX]{x1D703}_{2}}+k_{2}\text{e}^{\unicode[STIX]{x1D703}_{1}})\text{e}^{\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}}}{1+\text{e}^{2\unicode[STIX]{x1D703}_{1}}+\text{e}^{2\unicode[STIX]{x1D703}_{2}}+\displaystyle \frac{8k_{1}k_{2}}{(k_{1}+k_{2})^{2}}\text{e}^{\unicode[STIX]{x1D703}_{1}+\unicode[STIX]{x1D703}_{2}}+\displaystyle \frac{(k_{1}-k_{2})^{4}}{(k_{1}+k_{2})^{4}}\text{e}^{2\unicode[STIX]{x1D703}_{1}+2\unicode[STIX]{x1D703}_{2}}}.\end{eqnarray}$$

This is a family of two-soliton solutions of (3.2) with arbitrary constants $k_{1}$ and $k_{2}$ . Written as such, it is not readily imagined what the solution looks like. To get an idea of the shape one has to resort to visualizations which will be the focus of the next section. We have also checked that (3.8) solves the mKdV equation (2.7), as other (equivalent) expressions of the solution are possible (Anco, Ngatat & Willoughby Reference Anco, Ngatat and Willoughby2011).

4 Interaction dynamics of two solitons for the mKdV equation

In this section we will visualize the dynamics of the two-soliton solution (3.8). The wavenumbers $k_{1}$ and $k_{2}$ determine the amplitude and polarity of the two solitons comprised inside (3.8), and through the linear dispersion, also the phase speed $k_{i}^{2}$ . Far away from the collision region, the two solitons each assume the shape of a single soliton (2.5). The larger the wavenumber $k_{i}$ , the taller the soliton will be and the faster it will travel since the phase velocity equals $k_{i}^{2}$ . A positive wavenumber produces a hump (or bump); a negative wavenumber produces a hole (sometimes referred to as dip or depression), but both travel to the right.

To visualize the asymptotic behaviour (in particular, separation into individual solitons) we use long time intervals in figures 1 and 3, typically for $t$ varying between $-40$ and 40, with large (equal) time steps, e.g. $\unicode[STIX]{x0394}t=40$ . To zoom into the collision regions, shown in figures 2 and 4, we let $t$ vary between, say, $-5$ and 5, and use small (equal) time steps of length 5. All pictures show snapshots of both solitons at various moments in time to locate them on the $x$ -axis before, during, and after their collision, going from left to right. The large soliton peaks are easily identified, far away from the interaction region.

Figure 1.

Typical overtaking of a smaller by a larger hump, for (a) $k_{1}=0.5$ and $k_{2}=1$ and (b) $k_{1}=0.3$ and $k_{2}=1$ . The curves correspond to $t=-40$ (blue dashed), 0 (red) and 40 (black dotted), where the peaks move to the right as time increases.

4.1 Overtaking of same-polarity solitons

Figure 1 illustrates the dynamics of solitons with the same polarity (two humps). The parameters determining the respective shape of the solitons are (a) $k_{1}=0.5$ or (b) $k_{1}=0.3$ , and $k_{2}=1$ . The curves are shown for $t=-40$ (blue dashed), 0 (red) and 40 (black dotted), where the peaks move to the right as time increases. The pictures for two holes would be the same apart from a vertical flip over the $x$ -axis. The distinguishing feature between figure 1(a,b) is what happens in the interaction zone, details of which are shown in the corresponding graphs in figure 2.

Although we have graphed the solitons at equal time intervals, it can be seen that when the larger hump has overtaken the smaller one, the net effect is a slight forward shift of the larger soliton. Such shifts do not persist in time, and are only to be compared to what the trajectory of the larger hump would have been, had it been a true one-soliton solution, alone in the physical system. After a while the solitons separate and resume their original shapes. Asymptotically, e.g. for times $t<-40$ or $t>40$ , the profile of each soliton becomes undistinguishable from the one-soliton shape.

To start, figure 1 illustrates how a faster, larger but narrower hump with positive polarity overtakes a slower, smaller but wider one, also with positive polarity, but for different ratios $k_{1}/k_{2}$ . By reducing the time step near the overtaking region we show in figure 2 how the interaction evolves between the larger and the smaller soliton, for the cases shown in figure 1. The parameters match those in figure 1, but with smaller time steps, $t=-5$ (blue dashed), 0 (red) and 5 (black dotted).

Figure 2.

Details of the overtaking interaction of a smaller by a larger hump, for the parameters used in figure 1. The curves correspond to $t=-5$ (blue dashed), 0 (red) and 5 (black dotted).

Figure 3.

(a) Shows the typical overtaking of a smaller hole by a larger hump, for $k_{1}=-0.5$ and $k_{2}=1$ , whereas (b) concerns the opposite, where a larger hole overtakes a smaller hump, for $k_{1}=0.5$ and $k_{2}=-1$ . The curve coding is as in figure 1.

Figure 4.

Details of the centre regions for the overtaking interaction, in (a) of a smaller hole by a larger hump, for the parameters given in figure 3(a), and in (b) of a smaller hump by a larger hole, for the parameters given in figure 3(b). The curve coding is as in figure 2.

There are two distinct possibilities. For the first possibility the parameters are chosen such that during the interaction there are at all times two distinct peaks, as shown in figure 2(a). Thus, as the first peak increases in amplitude and the second peak decreases, it looks as if the larger and the smaller peaks have swapped places. In other words, after the collision the larger soliton is ahead, having undergone a forward phase shift, and mutatis mutandis for the smaller soliton.

The other possibility is that, as we decrease $k_{1}$ , keeping $k_{2}$ fixed, the figure changes qualitatively and the two solitons temporarily merge into a single but distorted peak, illustrated in figure 2(b). This occurs for $k_{1}/k_{2}<0.39$ . Indeed, only the ratio is important, as we can, without loss of generality, take $k_{2}=1$ for the largest of the peaks, and $k_{1}<1$ for the smaller one. Analogous results have been found in the mathematics literature (Anco et al. Reference Anco, Ngatat and Willoughby2011).

Switching the polarities from positive to negative produces the same type of figures, except that figures 1 and 2 are vertically flipped (not shown). All this is reminiscent of what happens for the interaction of KdV solitons, which are always of the same polarity, depending on the sign of $B$ (not shown here, but presented in many papers in the literature).

4.2 Overtaking of opposite-polarity solitons

The case of two solitons having opposite polarities leads to graphs which have not been systematically studied before in a plasma physics context. Figure 3 illustrates the dynamics of solitons of the opposite polarity (hump and hole). Figure 4 gives a clearer (zoomed) view of what happens in the interaction zones.

We first illustrate in figure 3(a) how a faster, larger but narrower hump of positive polarity overtakes a slower, smaller but wider hole of negative polarity. Larger and smaller refer to the absolute values of the amplitudes of the respective solitons. Figure 3(b) shows a larger hole overtaking a smaller hump. The parameters are $k_{1}=-0.5$ , $k_{2}=1$ in (a), and $k_{1}=0.5$ , $k_{2}=-1$ in (b). In (a), increasing $k_{2}$ makes the hump larger but narrower; decreasing $|k_{1}|$ makes the hole shallower but wider. For (b) the converse holds, increasing $k_{2}$ makes the hole deeper but narrower; decreasing $|k_{1}|$ makes the hump smaller but wider.

By reducing the time step near the overtaking region of the two solitons, we show in figure 4 how the interaction evolves between the larger and the smaller soliton. The parameters are as in figure 3, but with smaller time steps, for $t=-5$ , 0, 5.

In particular, note that in (a) the hole is split in two when the hump passes through, and the holes are less deep, rendering the hump larger. The net result is that the hole is transferred from being ahead of the hump to trailing it. On the other hand, in case (b) the hump is split when the hole passes through, with similar remarks about smaller humps and a deeper hole.

This mechanism has been observed in a variety of other parameter ranges we have investigated, but omitted here to avoid repetitive figures that are qualitatively similar. And, as before, the mechanism occurs for all ratios $|k_{1}/k_{2}|$ .

The analogy between panels (a) and (b) of figure 3 is due to an underlying antisymmetry:

(4.1) $$\begin{eqnarray}u(-x,-t)|_{k_{1}\rightarrow -k_{1},k_{2}\rightarrow -k_{2}}=-u(x,t).\end{eqnarray}$$

A consequence is that one can prove mathematically from (3.8) that $k_{1}+k_{2}\rightarrow 0$ yields $u(x,t)\rightarrow 0$ . This is to be anticipated: when the hole is as deep as the hump is tall, both will cancel and travel at the same speed, hence one is reduced to the trivial solution of the mKdV equation.

5 Electric field profiles

We recall that $u=\unicode[STIX]{x1D711}_{1}$ represents the electrostatic potential. In the theoretical analysis of acoustic solitons in multispecies plasmas this is usually the more important quantity. However, its derivatives have a physical meaning: up to a change of sign, the first derivative gives the electric field, and the second, the global charge density of the plasma. This is in contrast to the original application of the KdV equation – describing solitary waves on the surface of shallow water – where the height of the wave is important but the derivatives of that elevation are not of physical interest.

In plasma applications, the electric field of the wave is given by

(5.1) $$\begin{eqnarray}E=-\frac{\unicode[STIX]{x2202}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}x}.\end{eqnarray}$$

With respect to solitary waves having a potential hump or hole, it follows that the associated electric fields show a typical bipolar signature, of which a generic picture is given in figure 5 for a positive single-soliton solution (2.5). For a potential hump, where the electrostatic potential rises from zero to a maximum, before decreasing again to zero at infinity, its derivative is initially positive but then switches to negative. Because of the minus sign in the definition of the electric field, the bipolar structure starts with a negative part and ends with a positive part. For potential holes the opposite occurs. Thus, the electric field is also an indicator for the polarity of the mode.

Figure 5.

Generic profiles for mKdV single-soliton solution and associated electric field.

The interest in such bipolar electric field structures stems from the fact that they have been observed in a plethora of space observations by ionospheric and auroral satellite missions as diverse as GEOTAIL (Matsumoto et al. Reference Matsumoto, Kojima, Miyatake, Omura, Okada, Nagano and Tsutsui1994), POLAR (Franz, Kintner & Pickett Reference Franz, Kintner and Pickett1998; Franz et al. Reference Franz, Kintner, Pickett and Chen2005), FAST (McFadden et al. Reference McFadden, Carlson, Ergun, Mozer, Muschietti, Roth and Moebius2003) and CLUSTER (Pickett et al. Reference Pickett, Chen, Kahler, Santolík, Gurnett, Tsurutani and Balogh2004, Reference Pickett, Chen, Mutel, Christopher, Santolík, Lakhina, Singh, Reddy, Gurnett and Tsurutani2008; Norgren et al. Reference Norgren, André, Vaivads and Khotyaintsev2015) where the electric field (rather than the electrostatic potential) is observed. In some of the satellite observations both the electrostatic potential and the electric field are observed by different instruments (McFadden et al. Reference McFadden, Carlson, Ergun, Mozer, Muschietti, Roth and Moebius2003), which can thus serve as a way to check the consistency of the observations.

However, there are many different ways of modelling such structures, which in the case of electron holes refer to a localized plasma region where the electron density is lower than the surrounding plasma (Hutchinson Reference Hutchinson2017). The decreased electron density causes a local maximum in charge density and electrostatic potential. They might be viewed as a Bernstein–Greene–Kruskal (BGK) mode (Bernstein, Greene & Kruskal Reference Bernstein, Greene and Kruskal1957), but can also be described in terms of solitons. For further details we refer to an excellent overview by Hutchinson (Reference Hutchinson2017) and to recent extensions of BGK theory by Harikrishnan, Kakad & Kakad (Reference Harikrishnan, Kakad and Kakad2018a ,Reference Harikrishnan, Kakad and Kakad b ). Among the many interpretations, the one that interests us here is that of electrostatic solitary waves and structures, but there remains ambiguity in identifying the nature of observed phenomena.

Furthermore, there are many observations that do not fit the simple patterns and cannot be explained by simple solitons and their electric fields. A specific example is found in figure 1(b) in Pickett et al. (Reference Pickett, Chen, Kahler, Santolík, Gurnett, Tsurutani and Balogh2004), where besides the typical bipolar structures there are tripolar structures or modifications of the bipolar electric field structure by wiggles on the sides. For the latter we advanced a possible explanation in terms of supersolitons (Dubinov & Kolotkov Reference Dubinov and Kolotkov2012; Verheest, Hellberg & Kourakis Reference Verheest, Hellberg and Kourakis2013; Verheest & Hellberg Reference Verheest and Hellberg2015; Kakad, Lotekar & Kakad Reference Kakad, Lotekar and Kakad2016; Olivier, Verheest & Hereman Reference Olivier, Verheest and Hereman2018), but in personal correspondence with the lead author of Pickett et al. (Reference Pickett, Chen, Kahler, Santolík, Gurnett, Tsurutani and Balogh2004) these were thought to indicate an overtaking or even merging of solitons (J.S. Pickett, 2013, private communication).

Moreover, in most of the observed electric fields, the bipolar profiles have larger or smaller amplitudes, which – if they are due to propagating solitons – implies that the smaller ones will be overtaken by the larger ones. The property that larger amplitude structures are faster than smaller ones also holds for large amplitude solitary waves (Verheest Reference Verheest2010) which are commonly described by a Sagdeev pseudopotential formalism (Sagdeev Reference Sagdeev and Leontovich1966). The drawback of this method is that it usually does not give analytical expressions for the profiles nor for their interactions. Outside the KdV range, results will have to come from numerical simulations (see, e.g. Kakad et al. Reference Kakad, Kakad and Omura2017).

Of course, observing the precise overtaking of two bipolar structures would be serendipitous, but signals close to or after an overtaking collision have been recorded. Regardless of their theoretical explanation, these observations have been interpreted in terms of propagating structures, even though that has rarely been unambiguously confirmed. There are some CLUSTER observations where a fortuitous configuration of two of the four satellites was able to capture propagation of the major structures, while others were already quite distorted (Pickett et al. Reference Pickett, Chen, Mutel, Christopher, Santolík, Lakhina, Singh, Reddy, Gurnett and Tsurutani2008; Norgren et al. Reference Norgren, André, Vaivads and Khotyaintsev2015).

Hence, in what follows we focus on the interaction properties of the electric fields. The graphs to follow show the negative of the derivative of (3.8) with respect to $x$ . We omit the cumbersome mathematical expression of the derivative because it offers no physical insight. Using the same parameter conditions as in figures 1–4, we start with the positive electrostatic polarities in figure 6. As in figure 1, the parameters determining the respective shapes of the solitons are (a) $k_{1}=0.5$ and (b) $k_{1}=0.3$ , both with $k_{2}=1.$ The curves are shown for $t=-40$ , 0, 40 with the usual curve coding. Far away from the interaction region, the electric field shows two characteristic bipolar signatures, a stronger one for the larger soliton and a weaker one for the smaller soliton.

Figure 6.

Electric field profiles of the soliton interactions for (a) $k_{1}=0.5$ or (b) $k_{1}=0.3$ and $k_{2}=1$ , as shown in figure 1. The curves correspond to $t=-40$ , 0, 40, with a similar curve coding.

As can be seen on these graphs, during the interaction the curves become quite muddled. This is shown in more detail in figure 7, for both sets of parameters. Figures 1 and 2(a) showed two distinct peaks. The (b) panels of the same figures illustrated that the peaks briefly merged into a single distorted one. Consequently, in figures 6 and 7 there are always two distinct bipolar signatures in (a), sometimes very close together, whereas in (b) there is a stage when there is only one bipolar structure, but with wiggles on the wings. Some of the latter characteristics have also been seen on supersolitons, but in larger amplitude descriptions outside the reductive perturbation approach (Dubinov & Kolotkov Reference Dubinov and Kolotkov2012; Verheest et al. Reference Verheest, Hellberg and Kourakis2013) or in simulations (Kakad et al. Reference Kakad, Lotekar and Kakad2016).

Figure 7.

Details of the electric field profiles for the parameters used in figure 6. The curves correspond to $t=-5$ , 0, 5.

Next, in figures 8 and 9 we graph the overtaking of a smaller hole by a larger hump. Here again, the electric field bipolar signatures can be recognized far from the interaction zone, except that now the signatures differ not only in amplitude but also in sign. This is less often seen in space observations but the complicated interaction region itself might perhaps help with identifying new features.

Figure 8.

Electric field profiles for $k_{1}=-0.5$ and $k_{2}=1$ for the soliton interactions shown in figure 3.

Figure 9.

Details of the electric field profiles for the parameters used in figure 8. The curves correspond to $t=-5$ , 0, 5.

The figures for the converse case, i.e. the overtaking of a smaller hump by a larger hole, would be obtained by flipping the graphs shown in figures 8 and 9 over the $x$ axis (not shown).

The second derivative of $u=\unicode[STIX]{x1D711}$ also has a physical interpretation. Indeed, it follows from Poisson’s equation

(5.2) $$\begin{eqnarray}\unicode[STIX]{x1D70E}=-\frac{\unicode[STIX]{x2202}^{2}\unicode[STIX]{x1D711}}{\unicode[STIX]{x2202}x^{2}}\end{eqnarray}$$

that, up to a sign, it corresponds to the global charge density in the solitary wave(s). However, this has rarely been plotted in theoretical papers. Viewed more as a curiosity than for its additional physical insight, the available plots are only for single solitons or double layers (Verheest & Hellberg Reference Verheest and Hellberg2015). Due to the additional local extrema, we have not attempted to graph these density curves for two interacting mKdV solitons.