1 Introduction
The Zakharov–Kuznetsov equation (ZK, Reference Zakharov and Kuznetsov1974) is one of the most important nonlinear model equations of magnetized plasmas. It describes ion acoustic waves in a plasma whose magnetic pressure is much greater than the kinetic pressure (low-
$\unicode[STIX]{x1D6FD}$
plasma), so that the magnetic pressure constitutes a significant fraction of the ion rest energy density. In such a plasma slow magneto-acoustic waves can be considered as ion acoustic ones. Some time ago the problem of stability of waves described by the Zakharov–Kuznetsov equation was solved for an arbitrary shape of the wave. Its propagation, however, was limited to clinging to the magnetic field (Infeld Reference Infeld1985). As a check, the soliton case, further limited to perpendicular instabilities, was recovered as found by Laedke & Spatchek (Reference Laedke and Spatchek1982). Since then several authors have looked at the propagation and some at stability in various configurations (Infeld & Frycz Reference Infeld and Frycz1987, Reference Infeld and Frycz1989; Murawski & Edwin Reference Murawski and Edwin1992; Allen & Rowlands Reference Allen and Rowlands1993, Reference Allen and Rowlands1995, Reference Allen and Rowlands1997; Munro & Parkes Reference Munro and Parkes1999; Bas & Bulent Reference Bas and Bulent2010; Nawaz, Yildirim & Mohyud-Din Reference Nawaz, Yildirim and Mohyud-Din2013; Mothibi & Khalique Reference Mothibi and Khalique2015) and several others. Here we present a small
$K$
stability analysis of a nonlinear wave propagating at an angle to the magnetic field. Its shape is treated exactly and a cubic is obtained for the frequency of the perturbation (or growth rate). For zero angle, the cubic found in Infeld (Reference Infeld1985) is recovered (see also the book by Infeld & Rowlands Reference Infeld and Rowlands2000). The ZK equation is taken in the form
$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}t}+\frac{1}{3}(\unicode[STIX]{x2202}_{x}^{2}+\unicode[STIX]{x2202}_{y}^{2}+3n)\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}x}=0.\end{eqnarray}$$
Here
$x$
is along the uniform field
$\boldsymbol{B}$
. The wave is propagating at an angle
$\unicode[STIX]{x1D703}$
and it proves convenient to rotate the system by this angle. Thus
$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}x=\cos \unicode[STIX]{x1D703}\,\unicode[STIX]{x1D709}+\sin \unicode[STIX]{x1D703}\,\unicode[STIX]{x1D702}\\ y=\sin \unicode[STIX]{x1D703}\,\unicode[STIX]{x1D709}-\cos \unicode[STIX]{x1D703}\,\unicode[STIX]{x1D702},\end{array}\right\}\end{eqnarray}$$
and ZK is now
$$\begin{eqnarray}\frac{\unicode[STIX]{x2202}n}{\unicode[STIX]{x2202}t}+\left(n+\frac{\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D702}}^{2}}{3}\right)(\cos \unicode[STIX]{x1D703}\,n_{\unicode[STIX]{x1D709}}+\sin \unicode[STIX]{x1D703}\,n_{\unicode[STIX]{x1D702}})=0.\end{eqnarray}$$
2 Shape of the wave and possible perturbation
We assume the nonlinear wave or soliton to be a function of
$\unicode[STIX]{x1D709}-Ut$
.
We now look for stationary nonlinear solutions of the form
$$\begin{eqnarray}n_{0}^{\prime }(\unicode[STIX]{x1D709}^{\prime })=n_{0}+U/\cos \,\unicode[STIX]{x1D703},\quad \unicode[STIX]{x1D709}^{\prime }=\unicode[STIX]{x1D709}-Ut\end{eqnarray}$$
thus adding a constant to
$n_{0}$
which we simply include at zero order. Thus, in the new variables, dropping the primes and integrating twice
$$\begin{eqnarray}\left.\begin{array}{@{}c@{}}{\textstyle \frac{1}{3}}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}n_{0}+{\textstyle \frac{1}{2}}n_{0}^{2}=C\\ {\textstyle \frac{1}{6}}(\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}n_{0})^{2}=Cn_{0}-{\textstyle \frac{1}{6}}n_{0}^{3}+D.\end{array}\right\}\end{eqnarray}$$
By rescaling the variables and the constants we may always reduce the number of parameters putting
$$\begin{eqnarray}C=1/6,\quad D(C=1/6)=\unicode[STIX]{x1D701}/6.\end{eqnarray}$$
To obtain positive
$$\begin{eqnarray}n_{0\unicode[STIX]{x1D709}}^{2}=n_{0}-n_{0}^{3}+\unicode[STIX]{x1D701}\end{eqnarray}$$
in a finite interval of
$n_{0}$
, the parameter
$\unicode[STIX]{x1D701}$
has to satisfy
$$\begin{eqnarray}-2/\sqrt{27}<\unicode[STIX]{x1D701}<2/\sqrt{27}\end{eqnarray}$$
and the stationary solution
$n_{0}(\unicode[STIX]{x1D709})$
is periodic with a period
$\unicode[STIX]{x1D706}$
, which may be defined in terms of complete elliptic integrals.
Suppose a periodic wave solution is perturbed such that the wave vector
$\boldsymbol{K}$
of the perturbation forms an angle
$\unicode[STIX]{x1D713}$
with the direction
$\unicode[STIX]{x1D743}$
of the nonlinear wave. In the coordinate system of the basic wave we have
$$\begin{eqnarray}\displaystyle & n=n_{0}(\unicode[STIX]{x1D709})+\unicode[STIX]{x1D6FF}n=n_{0}(\unicode[STIX]{x1D709})+\tilde{\unicode[STIX]{x1D6FF}}n(\unicode[STIX]{x1D709})\text{e}^{\text{i}[(K\cos \unicode[STIX]{x1D713})\unicode[STIX]{x1D709}+(K\sin \unicode[STIX]{x1D713})\unicode[STIX]{x1D702}-\unicode[STIX]{x1D6FA}t]} & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \boldsymbol{K}=K(\cos \unicode[STIX]{x1D713},\sin \unicode[STIX]{x1D713}) & \displaystyle\end{eqnarray}$$
and
$\tilde{\unicode[STIX]{x1D6FF}}n(\unicode[STIX]{x1D709})$
is
$\unicode[STIX]{x1D706}$
periodic. We now assume
$K$
small and expand:
$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D6FA}=\unicode[STIX]{x1D6FA}_{1}K+\unicode[STIX]{x1D6FA}_{2}K^{2}+\cdots & \displaystyle\end{eqnarray}$$
$$\begin{eqnarray}\displaystyle & \tilde{\unicode[STIX]{x1D6FF}}n=\unicode[STIX]{x1D6FF}n_{0}+K\unicode[STIX]{x1D6FF}n_{1}+K^{2}\unicode[STIX]{x1D6FF}n_{2}+\cdots . & \displaystyle\end{eqnarray}$$
The second-order equation which arises from substitution of (1.3) with (2.6) and (2.8), together with the spatial non-secularity conditions (see the appendix A for details), result in a consistency condition of the form
$$\begin{eqnarray}G(\unicode[STIX]{x1D6FA},\boldsymbol{K},\unicode[STIX]{x1D701})=0,\end{eqnarray}$$
generalizing a dispersion relation theory. A short summary of the path leading to such a dispersion relation, cubic in
$\unicode[STIX]{x1D6FA}$
, is given below. The more detailed derivation is given in appendix A (see also the book by Infeld & Rowlands Reference Infeld and Rowlands2000).
3 Dispersion relation cubic for
$\unicode[STIX]{x1D6FA}/K$
Introducing
$$\begin{eqnarray}L={\textstyle \frac{1}{3}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{2}+n_{0}\end{eqnarray}$$
we find
$$\begin{eqnarray}\displaystyle \cos \unicode[STIX]{x1D703}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}L\tilde{\unicode[STIX]{x1D6FF}}n & = & \displaystyle {\textstyle \frac{1}{3}}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}^{3}\tilde{\unicode[STIX]{x1D6FF}}n+\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}n_{0}\tilde{\unicode[STIX]{x1D6FF}}n=\text{i}\unicode[STIX]{x1D6FA}^{\ast }\tilde{\unicode[STIX]{x1D6FF}}n\nonumber\\ \displaystyle & & \displaystyle -\,\text{i}K\left[\cos \unicode[STIX]{x1D703}\cos \unicode[STIX]{x1D713}(n_{0}\tilde{\unicode[STIX]{x1D6FF}}n+\tilde{\unicode[STIX]{x1D6FF}}n_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}})+\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D713}\left(n_{0}\tilde{\unicode[STIX]{x1D6FF}}n+{\textstyle \frac{1}{3}}\tilde{\unicode[STIX]{x1D6FF}}n_{\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}}\right)\right]\nonumber\\ \displaystyle & & \displaystyle +\,K^{2}\left[\cos \unicode[STIX]{x1D703}\left(\cos ^{2}\unicode[STIX]{x1D713}+{\textstyle \frac{1}{3}}\sin ^{2}\unicode[STIX]{x1D713}\right)+{\textstyle \frac{2}{3}}\sin \unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D713}\cos \unicode[STIX]{x1D713}\right]\tilde{\unicode[STIX]{x1D6FF}}n_{\unicode[STIX]{x1D709}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FA}^{\ast }=\unicode[STIX]{x1D6FA}-UK\sin \unicode[STIX]{x1D713}\tan \unicode[STIX]{x1D703}$
.
At second order in
$K$
, a dispersion relation is obtained under the assumption that the perturbation amplitude
$\tilde{\unicode[STIX]{x1D6FF}}n(\unicode[STIX]{x1D709})$
is
$\unicode[STIX]{x1D706}$
periodic (i.e. secular terms are absent).
$$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x1D6FA}^{\ast }/K)^{3}-\frac{2}{9}\frac{[4+3Y(Y+6\unicode[STIX]{x1D701})](3\cos \unicode[STIX]{x1D713}\cos \unicode[STIX]{x1D703}-\sin \unicode[STIX]{x1D713}\sin \unicode[STIX]{x1D703})}{Y^{3}-4Y-8\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6FA}^{\ast }/K)^{2}\nonumber\\ \displaystyle & & \displaystyle \quad +\,\frac{2}{15}\frac{(Y+3\unicode[STIX]{x1D701})(Y^{2}-4/3)(\cos ^{2}\unicode[STIX]{x1D703}-5\sin ^{2}\unicode[STIX]{x1D703})\sin ^{2}\unicode[STIX]{x1D713}+(5/2)[4\unicode[STIX]{x1D701}+Y(8/3+3Y\unicode[STIX]{x1D701})]\sin 2\unicode[STIX]{x1D713}\sin 2\unicode[STIX]{x1D703}}{Y^{3}-4Y-8\unicode[STIX]{x1D701}}(\unicode[STIX]{x1D6FA}^{\ast }/K)\nonumber\\ \displaystyle & & \displaystyle \quad -\,\frac{8}{135}\frac{[5(27\unicode[STIX]{x1D701}^{2}-4)\cos ^{2}\unicode[STIX]{x1D713}+3(Y+3\unicode[STIX]{x1D701})^{2}\sin ^{2}\unicode[STIX]{x1D713}]\cos ^{2}\unicode[STIX]{x1D703}\cos (\unicode[STIX]{x1D713}-\unicode[STIX]{x1D703})}{Y^{3}-4Y-8\unicode[STIX]{x1D701}}=0.\end{eqnarray}$$
Here
$n_{1}$
,
$n_{2}$
,
$n_{3}$
, are roots of
$n_{\unicode[STIX]{x1D709}}^{2}$
in increasing order. Also
$n_{0}$
is contained between
$n_{2}$
and
$n_{3}$
. Other definitions are
$$\begin{eqnarray}s=\frac{n_{3}-n_{2}}{n_{3}-n_{1}}\leqslant 1,\quad Y=2\langle n_{0}\rangle =2\left[n_{1}+(n_{3}-n_{1})\frac{E(s)}{K(s)}\right]\end{eqnarray}$$
$E(s)$
and
$K(s)$
are complete elliptic integrals of modulus
$s$
.
The cubic for
$\unicode[STIX]{x1D701}=0$
at three angles:
$\unicode[STIX]{x1D703}=0$
,
$\unicode[STIX]{x1D703}=45^{\circ }$
and
$\unicode[STIX]{x1D703}=46^{\circ }$
, while
$\unicode[STIX]{x1D713}=\unicode[STIX]{x03C0}/2$
. Single
$\unicode[STIX]{x1D703}$
-intercept means that two roots are complex conjugate; one of them corresponds to the unstable mode. When the cubic has 3 real roots, no instability occurs. The critical angle apparently lies between
$45^{\circ }$
and
$46^{\circ }$
.
Summary of the derivation of (3.3) may be found in the appendix A.
For
$\unicode[STIX]{x1D703}=0$
we regain the result of the Infeld and Rowlands book (equation (8.3.90), where
$\unicode[STIX]{x1D6FA}$
has conventionally the opposite sign).
We limit further analysis to the most unstable angle of perturbation
$\unicode[STIX]{x1D713}=\unicode[STIX]{x03C0}/2$
.
There is an instability for
$\unicode[STIX]{x1D703}=0$
, and
$\unicode[STIX]{x1D703}>0$
up to a critical angle. The angle may easily be obtained e.g. by an analysis of the discriminant of the cubic. For
$\unicode[STIX]{x1D701}=0$
(
$n_{1}=-1$
,
$n_{2}=0$
,
$n_{3}=1$
), the critical angle is equal
$0.792946~\text{rad}\approx 45.4325^{\circ }$
(figure 1).
The growth rate
$\unicode[STIX]{x1D6E4}/K$
increases from its value at
$\unicode[STIX]{x1D703}=0$
to a maximum, then falls to zero at the critical
$\unicode[STIX]{x1D703}$
(figure 2). For all acute angles
$\unicode[STIX]{x1D703}$
above the critical one the system is stable to first order in
$K$
.
For
$\unicode[STIX]{x1D703}>\unicode[STIX]{x03C0}/2$
the situation is symmetric with respect to
$\unicode[STIX]{x1D703}\rightleftarrows \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}$
.
The growth rate
$\unicode[STIX]{x1D6E4}/K$
as a function of
$\unicode[STIX]{x1D703}$
for
$\unicode[STIX]{x1D701}=0$
. The instability vanishes above the critical angle
$\unicode[STIX]{x1D703}\approx 0.793~\text{rad}\approx 45.43^{\circ }$
.
The growth rate
$\unicode[STIX]{x1D6E4}/K$
as a function of
$\unicode[STIX]{x1D703}$
for the soliton case
$\unicode[STIX]{x1D701}=2/\sqrt{27}$
. The growth rate decreases from 0.298 at
$\unicode[STIX]{x1D703}=0$
to zero at the critical angle
$\unicode[STIX]{x1D703}\approx 0.659~\text{rad}\approx 37.76^{\circ }$
.
Another limit of interest is for a soliton propagating at an angle to
$\boldsymbol{B}$
. For it
$\unicode[STIX]{x1D701}=2/\sqrt{27}$
and the cubic dispersion relation reads
$$\begin{eqnarray}(\unicode[STIX]{x1D6FA}^{\ast }/K)^{3}-\frac{\sin \unicode[STIX]{x1D703}}{3\sqrt{3}}(\unicode[STIX]{x1D6FA}^{\ast }/K)^{2}+\frac{4}{45}(\cos ^{2}\unicode[STIX]{x1D703}-5\sin ^{2}\unicode[STIX]{x1D703})(\unicode[STIX]{x1D6FA}^{\ast }/K)+\frac{4}{45\sqrt{3}}\cos ^{2}\unicode[STIX]{x1D703}\sin \unicode[STIX]{x1D703}=0.\end{eqnarray}$$
Here the growth rate is greatest for
$\unicode[STIX]{x1D703}=0$
, where it is equal
$2/(3\sqrt{5})\approx 0.298142$
. Then it decreases to zero at the critical angle, which is
$\text{arctan}\sqrt{3/5}\approx 0.659058~\text{rad}\approx 37.7612^{\circ }$
(figure 3), in agreement with Allen & Rowlands (Reference Allen and Rowlands1995). To first order in
$K$
the system is stable for acute angles greater than this angle. The behaviour for angles above
$\unicode[STIX]{x03C0}/2$
again follows from the symmetry
$\unicode[STIX]{x1D703}\rightleftarrows \unicode[STIX]{x03C0}-\unicode[STIX]{x1D703}$
.
4 A bit of history
Forty years ago Infeld and Rowlands pointed out flaws in the way people were calculating the stability of solitons. The perturbations introduced failed to vanish at infinity (Reference Infeld and Rowlands1977). Many scientists took the problem seriously. The two authors looked at the problem differently. Rowlands pointed out that the soliton problem involves four different kinds of secularity and removed all by introducing multivariables. Infeld pointed out that removing secular terms is simpler for periodic structures, so let us treat a soliton train and take it to the
$\unicode[STIX]{x1D706}\rightarrow \infty$
limit. (This effort is in that spirit.) Lowest-order results of the two methods so far agree, but the wave train
$\rightarrow$
soliton transition is not understood. All this notwithstanding the nonlinear wave problems’ importance.
Appendix A. On the derivation of the cubic dispersion relation (3.3)
To get a consistent expansion in
$K$
we substitute (2.8) into equation (3.2).
At zero order, (3.2) has a solution of the same form as in Infeld & Rowlands (Reference Infeld and Rowlands2000), namely
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}n_{0} & = & \displaystyle n_{0\unicode[STIX]{x1D709}}+A\,n_{0\unicode[STIX]{x1D709}}\int ^{\unicode[STIX]{x1D709}}n_{0\unicode[STIX]{x1D709}^{\prime }}^{-2}\,\text{d}\unicode[STIX]{x1D709}^{\prime }+3M\,n_{0\unicode[STIX]{x1D709}}\int ^{\unicode[STIX]{x1D709}}n_{0}\,n_{0\unicode[STIX]{x1D709}^{\prime }}^{-2}\,\text{d}\unicode[STIX]{x1D709}^{\prime }\nonumber\\ \displaystyle & = & \displaystyle n_{0\unicode[STIX]{x1D709}}+(A\unicode[STIX]{x1D6FD}+3M\unicode[STIX]{x1D6FE})\,\unicode[STIX]{x1D709}\,n_{0\unicode[STIX]{x1D709}}+A\,Q_{0}(\unicode[STIX]{x1D709})+3M\,Q_{1}(\unicode[STIX]{x1D709}),\end{eqnarray}$$
where the integrals on the right-hand side of (A 1) have been split into the secular and non-secular parts; the latter are
$Q_{0}$
and
$Q_{1}$
respectively, while
$\unicode[STIX]{x1D6FD}$
and
$\unicode[STIX]{x1D6FE}$
are constant gradients of the secular parts of the respective integrals. The assumed periodicity of
$\unicode[STIX]{x1D6FF}n_{0}$
requires that the first integrals of zero-order (3.2),
$A$
and
$M$
, have to satisfy
$A\unicode[STIX]{x1D6FD}+3M\unicode[STIX]{x1D6FE}=0$
. Further on, the non-secularity at the first order in
$K$
will infer that both these constants vanish, thus reducing the solution to simple
$\unicode[STIX]{x1D6FF}n_{0}=n_{0\unicode[STIX]{x1D709}}$
(this conclusion follows from two conditions obtained by integration of the first-order equation over a period: one by direct integration of the equation, the other one by integration of the equation multiplied by
$n_{0}$
). For such
$\unicode[STIX]{x1D6FF}n_{0}$
, the first-order equation reads
$$\begin{eqnarray}\cos \unicode[STIX]{x1D703}\,\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}\,L\,\unicode[STIX]{x1D6FF}n_{1}=\text{i}\unicode[STIX]{x1D6FA}_{1}^{\ast }-\text{i}\cos (\unicode[STIX]{x1D703}-\unicode[STIX]{x1D713})n_{0}n_{0\unicode[STIX]{x1D709}}-i(\cos \unicode[STIX]{x1D703}\,\cos \unicode[STIX]{x1D713}+{\textstyle \frac{1}{3}}\sin \unicode[STIX]{x1D703}\,\sin \unicode[STIX]{x1D713})\,n_{0\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}\unicode[STIX]{x1D709}},\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FA}_{1}^{\ast }=\unicode[STIX]{x1D6FA}_{1}-U\sin \unicode[STIX]{x1D713}\tan \unicode[STIX]{x1D703}$
. Equation (A 2) may be solved similarly to the zero-order equation to yield, after removal of the secular part
$$\begin{eqnarray}\unicode[STIX]{x1D6FF}n_{1}=\text{i}\unicode[STIX]{x1D6FA}_{1}^{\ast }/\cos \unicode[STIX]{x1D703}+3\cos \unicode[STIX]{x1D713}(EQ_{2}+\text{i}Q_{0}/\unicode[STIX]{x1D6FD}),\end{eqnarray}$$
where
$Q_{2}=Q_{1}-\unicode[STIX]{x1D6FE}Q_{0}/\unicode[STIX]{x1D6FD}$
.
$E$
is an integration constant, whose value will be determined by the non-secularity condition at the next order. Namely, integration of the second-order equation over a period yields a non-secular expression, provided that
$$\begin{eqnarray}E=-\frac{\text{i}\unicode[STIX]{x1D6FA}_{1}^{\ast }(\unicode[STIX]{x1D6FA}_{1}^{\ast }-\langle n_{0}\rangle \sin \unicode[STIX]{x1D703})}{(3\unicode[STIX]{x1D6FA}_{1}^{\ast }\langle Q_{2}\rangle -\sin \unicode[STIX]{x1D703})\cos \unicode[STIX]{x1D703}\,\cos \unicode[STIX]{x1D713}},\end{eqnarray}$$
where the averaging of
$\unicode[STIX]{x1D709}$
-dependent quantities over their period
$\unicode[STIX]{x1D706}$
is denoted by
$\langle ~\rangle$
.
What remains is to calculate the averaged quantities from the solution
$n_{0}$
of (2.4) and substituting them to the second order of (3.2). Fortunately not only
$\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}L\unicode[STIX]{x1D6FF}n_{2}$
but also
$n_{0}\unicode[STIX]{x2202}_{\unicode[STIX]{x1D709}}L\unicode[STIX]{x1D6FF}n_{2}$
vanish on averaging, thus making the averaged second-order equation free of
$\unicode[STIX]{x1D6FF}n_{2}$
. The remaining quantities which require averaging are
$n_{0}$
,
$\unicode[STIX]{x1D6FF}n_{1}$
,
$n_{0}\unicode[STIX]{x1D6FF}n_{1}$
,
$n_{0}^{2}\unicode[STIX]{x1D6FF}n_{1}$
,
$n_{0\unicode[STIX]{x1D709}}^{2}$
,
$Q_{0}$
and
$Q_{2}$
. By using (A 3) and (2.4), these averages may be calculated in terms of
$Y=2\langle n_{0}\rangle$
without substituting the specific solution for
$n_{0}$
. A straightforward though cumbersome calculation yields (3.3).
The solution of (2.4) for
$n_{0}$
is necessary if we want to calculate the soliton limit (3.5) of the dispersion relation (3.3). For this purpose we use the averaged solution given by (3.3). Its right-hand limit is taken at
$n_{02}\rightarrow -1/\sqrt{3}$
, while the remaining roots of
$n_{\unicode[STIX]{x1D709}}^{2}$
have constant ‘soliton’ values
$n_{01}=-1/\sqrt{3}$
,
$n_{03}=2/\sqrt{3}$
.
