Let us investigate if the system (9)-(10) accepts solitary wave solutions of the form:

$P=A_1\mathrm{sech}\left(\frac{t-az}{w}\right){\text{e}}^{i\left(k_{1}z-{\omega }_{1}t\right)}$ (15)

$Q=A_2\mathrm{sech}\left(\frac{t-az}{w}\right){\text{e}}^{i\left(k_{2}z-{\omega }_{2}t\right)}$ (16)

Substituting (15)-(16) into Equation (9) we obtain an equation containing five types of hyperbolic functions: ch⁻¹, ch⁻³, ch⁻⁵, sh/ch² and sh/ch⁴. For this equation to be satisfied, the coefficients of each of the hyperbolic functions must be equal to zero. Setting equal to zero the coefficients of sh/ch⁴ and sh/ch² we obtain that:

${\omega }_1=0$, (17)

$a=0$. (18)

Taking into account these results, the equation obtained when (15) and (16) are substituted into Equation (9) takes the form:

$\begin{array}{l}\left(-k_1+\frac{{\alpha }_1}{w^2}+\frac{{\beta }_1}{w^4}\right)c{h}^4\left(\theta \right)+\left[-\frac{2{\alpha }_1}{w^2}-\frac{20{\beta }_1}{w^4}+{\gamma }_1\left(A_1^2+{\sigma }_1{A}_2^2\right)\right]c{h}^2\left(\theta \right)\\ +\frac{24{\beta }_1}{w^4}-{\delta }_1{\left(A_1^2+{\sigma }_2{A}_2^2\right)}^2=0,\end{array}$ (19)

where we have defined $\theta =\left(t-az\right)/w$ For this equation to be satisfied, the coefficient of $c{h}^4$, $c{h}^2$ and the independent term (independent of ch) must be equal to zero. In this way we obtain the following three equations:

$k_1=\frac{{\alpha }_1}{w^2}+\frac{{\beta }_1}{w^4}$, (20)

$-\frac{2{\alpha }_1}{w^2}-\frac{20{\beta }_1}{w^4}+{\gamma }_1\left(A_1^2+{\sigma }_1{A}_2^2\right)=0$, (21)

$\frac{24{\beta }_1}{w^4}-{\delta }_1{\left(A_1^2+{\sigma }_2{A}_2^2\right)}^2=0$. (22)

Proceeding in a similar way, if we substitute (15) and (16) in Equation (10), we can obtain the following equations:

${\omega }_2=0$, (23)

$k_2=\frac{{\alpha }_2}{w^2}+\frac{{\beta }_2}{w^4}$, (24)

$-\frac{2{\alpha }_2}{w^2}-\frac{20{\beta }_2}{w^4}+{\gamma }_2\left(A_2^2+{\sigma }_1{A}_1^2\right)=0$, (25)

$\frac{24{\beta }_2}{w^4}-{\delta }_2{\left(A_2^2+{\sigma }_2{A}_1^2\right)}^2=0$. (26)

Now, if we solve Equation (22) for $w^4$, and substitute the resulting expression in (21), we obtain the following equation:

$-2{\alpha }_1{\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}\left(A_1^2+{\sigma }_2{A}_2^2\right)-\frac{5}{6}{\delta }_1{\left(A_1^2+{\sigma }_2{A}_2^2\right)}^2+{\gamma }_1\left(A_1^2+{\sigma }_1{A}_2^2\right)=0$. (27)

In a similar way, from Equations (25) and (26) we can obtain:

$-2{\alpha }_2{\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}\left(A_2^2+{\sigma }_2{A}_1^2\right)-\frac{5}{6}{\delta }_2{\left(A_2^2+{\sigma }_2{A}_1^2\right)}^2+{\gamma }_2\left(A_2^2+{\sigma }_1{A}_1^2\right)=0$. (28)

Moreover, from Equations (22) and (26) we can obtain two different expressions for $w^{-4}$. By equating these two expressions we obtain:

$\frac{{\delta }_1}{{\beta }_1}{\left(A_1^2+{\sigma }_2{A}_2^2\right)}^2=\frac{{\delta }_2}{{\beta }_2}{\left(A_2^2+{\sigma }_2{A}_1^2\right)}^2,$ (29)

and if we consider that the coefficients ${\beta }_1$, ${\beta }_2$, ${\delta }_1$, ${\delta }_2$ and ${\sigma }_2$ are all positive, Equation (29) implies that:

${\left(\frac{{\delta }_1}{{\beta }_1}\right)}^{1/2}\left(A_1^2+{\sigma }_2{A}_2^2\right)={\left(\frac{{\delta }_2}{{\beta }_2}\right)}^{1/2}\left(A_2^2+{\sigma }_2{A}_1^2\right).$ (30)

If we considered that the values of the 10 coefficients $\left\{{\alpha }_n,{\beta }_n,{\gamma }_n,{\delta }_n,{\sigma }_n|n=1,2\right\}$ had been chosen arbitrarily, then it would be impossible to find values of $A_1^2$ and $A_2^2$ which satisfy the three equations (27), (28) and (30), as we would have an overdetermined system of three equations with only two unknowns. However, if we consider that one of the coefficients is unknown (for example, ${\gamma }_2$), then the Equations (27), (28) and (30) would be a system of three equations with three unknowns ( $A_1^2$, $A_2^2$ and ${\gamma }_2$), and it would be possible to obtain the solution of this system. However, if ${\alpha }_1$ and ${\alpha }_2$ are arbitrary real numbers, and ${\sigma }_1\ne {\sigma }_2$, it will not be possible to obtain the solution of this system in analytical form. Therefore, let us consider the simpler case when:

${\sigma }_1={\sigma }_2\equiv \sigma$. (31)

In this case the Equations (27), (28) and (30) reduce to:

$-2{\alpha }_1{\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}-\frac{5}{6}{\delta }_1\left(A_1^2+\sigma A_2^2\right)+{\gamma }_1=0$, (32)

$-2{\alpha }_2{\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}-\frac{5}{6}{\delta }_2\left(A_2^2+\sigma A_1^2\right)+{\gamma }_2=0$, (33)

${\left(\frac{{\delta }_1}{{\beta }_1}\right)}^{1/2}\left(A_1^2+\sigma A_2^2\right)={\left(\frac{{\delta }_2}{{\beta }_2}\right)}^{1/2}\left(A_2^2+\sigma A_1^2\right)$, (34)

and if we obtain $\left(A_1^2+\sigma A_2^2\right)$ and $\left(A_2^2+\sigma A_1^2\right)$ from Equations (32) and (33), and substitute the resulting expressions into Equation (34), this equation transforms into:

${\left(\frac{{\delta }_1{\beta }_2}{{\delta }_2{\beta }_1}\right)}^{1/2}\frac{{\delta }_2}{{\delta }_1}\left[{\gamma }_1-2{\alpha }_1{\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}\right]={\gamma }_2-2{\alpha }_2{\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}.$ (35)

In this way, instead of the system (27), (28) and (30), now we just have a system of two Equations, (32) and (33), subject to the condition that the coefficients ${\alpha }_n$, ${\beta }_n$, ${\gamma }_n$ and ${\delta }_n$ must satisfy the condition (35). And this condition can be used to obtain ${\gamma }_2$ as a function of the remaining coefficients $\left\{{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2,{\gamma }_1,{\delta }_1,{\delta }_2\right\}$. If this function is now introduced in Equation (33), we can obtain $A_1^2$ and $A_2^2$ by solving the system (32)-(33), and the solution is:

$A_1^2=-\left[\frac{1}{5{\delta }_1\left({\sigma }^2-1\right)}\right]\left[\sqrt{6}{\alpha }_1\sqrt{{\delta }_1/{\beta }_1}-6{\gamma }_1\right]\left[-1+\sigma {\left(\frac{{\beta }_2{\delta }_1}{{\beta }_1{\delta }_2}\right)}^{1/2}\right],$ (36)

$A_2^2=\left[\frac{1}{5{\delta }_1\left({\sigma }^2-1\right)}\right]\left[\sqrt{6}{\alpha }_1\sqrt{{\delta }_1/{\beta }_1}-6{\gamma }_1\right]\left[-\sigma +{\left(\frac{{\beta }_2{\delta }_1}{{\beta }_1{\delta }_2}\right)}^{1/2}\right].$ (37)

Therefore, if ${\sigma }^2\ne 1$ and we have coefficients $\left\{{\alpha }_1,{\alpha }_2,{\beta }_1,{\beta }_2,{\gamma }_1,{\gamma }_2,{\delta }_1,{\delta }_2\right\}$ which produce positive values of $A_1^2$ and $A_2^2$ when they are substituted in the right-hand-sides of Equations (36) and (37), then the system (9)-(10) will have a vector soliton solution of the form (15)-(16), with ${\omega }_1={\omega }_2=0$. Once with the values of $A_1^2$ and $A_2^2$, we can calculate the value of w (the width of the solitons) with Equation (22) or (26), if we introduce in these equations the value of the parameter $\sigma$. And once with the value of w, we will be able to calculate the values of the wavenumbers $k_1$ and $k_2$ with the Equations (20) and (24). For example, let us consider the coefficients:

${\alpha }_1=0.5$ and ${\alpha }_2=0.25$, (38)

${\beta }_1=5$ and ${\beta }_2=4$, (39)

${\gamma }_1=5$ and ${\gamma }_2=4.44152$, (40)

${\delta }_1=1$ and ${\delta }_2=1$, (41)

$\sigma =2$. (42)

where the value of ${\gamma }_2$ was calculated with Equation (35). Substituting these coefficients in Equations (36)-(37) we obtain:

$A_1^2=1.5489$, (43)

$A_2^2=2.17078$. (44)

Then, substituting these values in (22) or (26), we obtain the value of w:

$w=1.36370$, (45)

and introducing this value in Equations (20) and (24) we obtain the values of the wavenumbers:

$k_1=1.71459$ and $k_2=1.29101$. (46)

As these wavenumbers are positive, they are contained in the range of the linear dispersion relations of Equations (9) and (10), as these two equations have dispersion relations of the form:

$k={\beta }_n{\omega }^4-{\alpha }_n{\omega }^2$. (47)

Therefore, the solitons (15) and (16), with the amplitudes, width and wavenumbers defined in Equations (43)-(46), are embedded solitons. This result proves that the NLS system (9)-(10) does indeed possess embedded vector solitons (EVSs).

In this way we have demonstrated that embedded vector solitons do indeed exist.

In this sub-section we will use the VK criterion to estimate the stability of the EVSs of the NLS system (9)-(10). Solitons which are stable according to the VK criterion will be referred to as VK-stable solitons.

To determine the VK stability of the soliton solutions (15)-(16) of the system (9)-(10) we will need the integrals:

$N_1={\displaystyle {\int }_{-\infty }^{+\infty }{\left|P\right|}^2\text{d}t}$ and $N_2={\displaystyle {\int }_{-\infty }^{+\infty }{\left|Q\right|}^2\text{d}t}$. (48)

Substituting the functions (15) and (16) in these integrals we obtain:

$N_1=2A_1^{2}w$ and $N_2=2A_2^{2}w$, (49)

and to determine the stability of the solitons (15) and (16) we have to calculate the signs of the derivatives:

$\frac{\text{d}{N}_1}{\text{d}{k}_1}$ and $\frac{\text{d}{N}_2}{\text{d}{k}_2}$. (50)

Therefore, in order to calculate these derivatives, we need to express $A_1^2$, $A_2^2$ and $w$ as functions of the wavenumbers $k_1$ and $k_2$. In the following we’ll see how to obtain these expressions.

To begin with, we should observe that if we consider the simpler case when (31) holds, and we consider that $\sigma >0$, then Equations (22) and (26) permit us to express $w^{-2}$ in the following two forms:

$\frac{1}{w^2}={\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}\left(A_1^2+\sigma A_2^2\right),$ (51)

$\frac{1}{w^2}={\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}\left(A_2^2+\sigma A_1^2\right).$ (52)

On the other hand, from (20) and (24) we can obtain an expression for $w^2$ as a function of $k_1$ and $k_2$:

$w^2=\frac{{\alpha }_1-\frac{{\beta }_1}{{\beta }_2}{\alpha }_2}{k_1-\frac{{\beta }_1}{{\beta }_2}{k}_2}.$ (53)

Then, the substitution of (53) in (51) leads to the following equation:

$\left(A_1^2+\sigma A_2^2\right){\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}\left({\alpha }_1-\frac{{\beta }_1}{{\beta }_2}{\alpha }_2\right)=k_1-\frac{{\beta }_1}{{\beta }_2}{k}_2.$ (54)

In a similar way, from (52) and (53) we can obtain:

$\left(A_2^2+\sigma A_1^2\right){\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}\left({\alpha }_2-\frac{{\beta }_2}{{\beta }_2}{\alpha }_2\right)=k_2-\frac{{\beta }_2}{{\beta }_1}{k}_1.$ (55)

And from Equations (54) and (55) we can obtain the following equations:

$\left({\sigma }^2-1\right)A_1^2=\left(\frac{\sigma }{F_2}+\frac{{\beta }_1}{F_1{\beta }_2}\right)k_2-\left(\frac{\sigma {\beta }_2}{F_2{\beta }_1}+\frac{1}{F_1}\right)k_1,$ (56)

$\left(\sigma -\frac{1}{\sigma }\right)A_2^2=\left(\frac{1}{F_1}+\frac{{\beta }_2}{\sigma F_2{\beta }_1}\right)k_1-\left(\frac{1}{\sigma F_2}+\frac{{\beta }_1}{F_1{\beta }_2}\right)k_2,$ (57)

where we have defined:

$F_1={\left(\frac{{\delta }_1}{24{\beta }_1}\right)}^{1/2}\left({\alpha }_1-\frac{{\beta }_1}{{\beta }_2}{\alpha }_2\right),$ (58)

$F_2={\left(\frac{{\delta }_2}{24{\beta }_2}\right)}^{1/2}\left({\alpha }_2-\frac{{\beta }_2}{{\beta }_1}{\alpha }_1\right).$ (59)

The Equations (53), (56) and (57) are particularly important. They reveal how the solitons’ parameters $w$, $A_1^2$ and $A_2^2$ depend on the wavenumbers $k_1$ and $k_2$. It is worth observing that the Equations (36)-(37), obtained in Sec. 3.1, permitted us to calculate the values of the solitons’ amplitudes $A_1$ and $A_2$, but those equations do not tell us how these amplitudes depend on the wavenumbers. Now, with the Equations (53), (56) and (57), we know how the integrals $N_1$ and $N_2$ [given in Equation (49)] depend on $k_1$ and $k_2$, and the knowledge of the functions $N_1\left(k_1,k_2\right)$ and $N_2\left(k_1,k_2\right)$ permits us to know the signs of the derivatives mentioned in (50).

Let us use the functions $N_1\left(k_1,k_2\right)$ and $N_2\left(k_1,k_2\right)$, obtained by substituting the Equations (53), (56) and (57) in (49), to determine the VK-stability of the solitons of the system (9)-(10) which were determined in Sec. 3.1. These solitons correspond to the coefficients shown in Equations (38)-(42). Therefore, introducing these coefficients in Equations (53), (56) and (57) we can obtain the form of the functions $N_1\left(k_1,k_2\right)$ and $N_2\left(k_1,k_2\right)$.

The VK-stability of the first soliton [the soliton P defined in Equation (15)] is defined by the slope of the curve $N_1\left(k_1,k_2=1.29101\right)$, where $k_2=1.29101$ is the wavenumber of the second soliton [the soliton Q defined in Equation (16)]. This curve is shown in Figure 1 .

This figure shows that $\text{d}{N}_1/\text{d}{k}_1>0$, thus implying that the P-soliton defined in Equation (15) is VK-stable.

On the other hand, Figure 2 shows the shape of the curve $N_2\left(k_1=1.71459,k_2\right)$, where $k_1=1.71459$ is the wavenumber of the first soliton [the soliton P shown in Equation (15)]. We can see that in this case $\text{d}{N}_2/\text{d}{k}_2<0$, thus implying that the Q-soliton, defined in Equation (16), is VK-unstable.

We have thus seen that with the coefficients shown in Equations (38)-(42) the P-soliton is VK-stable and the Q-soliton is VK-unstable. However, this situation may change if we use other coefficients. For example, let us consider the following coefficients:

${\alpha }_1=0.35$ and ${\alpha }_2=0.65$, (60)

${\beta }_1=5$ and ${\beta }_2=4$, (61)

${\gamma }_1=5$ and ${\gamma }_2=6.40008$, (62)

${\delta }_1=0.5$ and ${\delta }_2=1$, (63)

$\sigma =2$, (64)

where the value of ${\gamma }_2$ was calculated with Equation (35). With these coefficients the Equations (36)-(37) imply that:

$A_1^2=1.05007$, (65)

$A_2^2=5.42074$. (66)

Substituting these values in (22) we obtain the value of w:

$w=1.14139$, (67)

and introducing this value in Equations (20) and (24) we obtain the wavenumbers of the solitons P and Q defined in Equations (15)-(16):

$k_1=3.21468$ and $k_2=2.85575$. (68)

Introducing the coefficients (60)-(64) in in Equations (53), (56) and (57) we can obtain $w^2$, $A_1^2$ and $A_2^2$ in terms of $k_1$ and $k_2$, and with these expressions we can obtain the forms of the functions $N_1\left(k_1,k_2=2.85575\right)$ and $N_2\left(k_1=3.21468,k_2\right)$. In Figure 3 and Figure 4 we can see the forms of these functions.

We can see that in the two examples shown above [i.e., with the coefficients (38)-(42), and also with the coefficients (60)-(64)], the VK criterion predicts that one of the solitons shown in (15)-(16) is VK-stable, while the other is VK-unstable. This result might be a bit unexpected at first sight. However, the VK prediction that the vector solitons of the NLS system (9)-(10) are neither fully stable, nor completely unstable, might be related to the fact that the embedded solitons of the cubic-quintic NLS equation with fourth-order dispersion [i.e., Equation (9) with ${\sigma }_1={\sigma }_2=0$] are semistable objects, i.e., they are stable in a linear approximation, but are subject to a one-sided nonlinear instability.

Let us consider the following system of complex modified Korteweg-de Vries (cmKdV) equations:

$u_z-{\epsilon }_1{u}_{ttt}-{\gamma }_1\left({\left|u\right|}^2+\sigma {\left|v\right|}^2\right)u_t=0$, (69)

$v_z-{\epsilon }_2{v}_{ttt}-{\gamma }_2\left({\left|v\right|}^2+\sigma {\left|u\right|}^2\right)v_t=0$. (70)

And now let us investigate if this system accepts solutions of the form:

$u=A_1\mathrm{sech}\left(\frac{t-az}{w}\right){\text{e}}^{i\left(q_{1}z+r_{1}t\right)},$ (71)

$v=A_2\mathrm{sech}\left(\frac{t-az}{w}\right){\text{e}}^{i\left(q_{2}z+r_{2}t\right)}.$ (72)

Substituting (71)-(72) in the system (69)-(70) we find that the functions (71)-(72) are indeed solutions of (69)-(70) if the following conditions are satisfied (for $n=1$ and $n=2$):

$a=3{\epsilon }_n{r}_n^2-\frac{{\epsilon }_n}{w^2},$ (73)

$q_n=-{\epsilon }_n{r}_n^3+\frac{3{\epsilon }_n{r}_n}{w^2},$ (74)

${\gamma }_n\left(A_n^2+\sigma A_{3-n}^2\right)=\frac{6{\epsilon }_n}{w^2}.$ (75)

If we consider the two possible values, $n=1$ and $n=2$, in Equations (73)-(75), we obtain 6 equations. And these 6 equations involve the 8 parameters which appear in the soliton solutions (71)-(72), namely: a, w, $A_1$, $A_2$, $q_1$, $q_2$, $r_1$ and $r_2$. Therefore, we may choose the values of two of these 8 parameters, and the Equations (73)-(75) will permit us to obtain the values of the remaining 6 parameters. In the following we will see that it is convenient to choose the values of the parameters a and w [which are the parameters which appear in the two Equations (71)-(72)].

Once the values of the coefficients $\left\{\sigma ,{\epsilon }_n,{\gamma }_n\right\}$ have been specified, and the values of a and w have been chosen, Equation (73) will permit us to obtain the values of $r_1$ and $r_2$. And once the values of $r_1$ and $r_2$ are defined, the Equation (74) will permit us to calculate the values of $q_1$ and $q_2$. Finally, we only need to calculate the values of $A_1$ and $A_2$ in order to have the exact shape of the solitons (71)-(72).

We can see that Equation (75) defines a system of 2 linear equations for the variables $A_1^2$ and $A_2^2$. Solving this system we find that:

$A_n^2=\frac{6}{w^2\left(1-{\sigma }^2\right)}\left(\frac{{\epsilon }_n}{{\gamma }_n}-\sigma \frac{{\epsilon }_{3-n}}{{\gamma }_{3-n}}\right),$ (76)

and these equations will give us the values of the solitons’ amplitudes $A_1$ and $A_2$. As $A_1^2$ and $A_2^2$ must be positive, Equation (76) implies that the coefficients $\left\{\sigma ,{\epsilon }_n,{\gamma }_n\right\}$ must satisfy the condition:

$\frac{1}{1-{\sigma }^2}\left(\frac{{\epsilon }_n}{{\gamma }_n}-\sigma \frac{{\epsilon }_{3-n}}{{\gamma }_{3-n}}\right)>0$. (77)

Therefore, only of this condition is satisfied, the cmKdV system (69)-(70) will have soliton solutions of the form (71)-(72).

It should be emphasized that the soliton solutions (71)-(72) of the system (69)-(70) are indeed embedded solitons. This reason of this embedding can be easily understood if we observe that the linear dispersion relations of the Equations (69)-(70) have the following form:

$k={\epsilon }_n{\omega }^3$, (78)

and the range of wavenumbers permitted by this dispersion relation is the entire real axis. Therefore, the solitons’ wavenumbers $q_1$ and $q_2$ (irrespective of their particular values) will be necessarily contained in this range of wavenumbers, thus implying that the solitons (71)-(72) are indeed embedded, and consequently the Equations (71)-(72) constitute an embedded vector soliton of the system (69)-(70).

To close this sub-section, it is worth emphasizing that the cmKdV system (69)-(70) has an infinity of EVSs, since there exist EVSs of the form (71)-(72) for arbitrary values of the width, w, and the (inverse) velocity α.

Headings, or heads, are organizational devices that guide the reader through your paper. There are two types: component heads and text heads.

In order to determine the VK stability of the solitons (71)-(72) we need to calculate the signs of the derivatives:

$\frac{\text{d}{N}_n}{\text{d}{q}_n},$ (79)

where we have defined:

$N_1={\displaystyle {\int }_{-\infty }^{+\infty }{\left|u\right|}^2\text{d}t}$ and $N_2={\displaystyle {\int }_{-\infty }^{+\infty }{\left|v\right|}^2\text{d}t}$. (80)

Introducing the forms of the solitons (71)-(72) in these integrals we find that:

$N_n=2A_n^{2}w=\frac{2A_n^2}{M^{1/2}},$ (81)

where we have defined $M=1/w^2$, and $n=1$ or $n=2$.

Now, taking into account that Equation (76) implies that $A_1^2$ and $A_2^2$ depend on $M=1/w^2$, and Equations (73)-(74) imply that $q_1$ and $q_2$ also depend on M, we can calculate the derivative shown in (79) in the following form:

$\frac{\text{d}{N}_n}{\text{d}{q}_n}=\frac{\text{d}}{\text{d}M}\left(\frac{2A_n^2}{M^{1/2}}\right)\frac{1}{\text{d}{q}_n/\text{d}M}.$ (82)

The first derivative which appears in the rhs (right-hand side) of (82) can be evaluated using (76), and in this way we find:

$\frac{\text{d}{N}_n}{\text{d}{q}_n}=\frac{6}{\left({\sigma }^2-1\right)M^{1/2}}\left(\sigma \frac{{\epsilon }_{3-n}}{{\gamma }_{3-n}}-\frac{{\epsilon }_n}{{\gamma }_n}\right)\frac{1}{\text{d}{q}_n/\text{d}M}.$ (83)

Therefore, the sign of $\text{d}{N}_n/\text{d}{q}_n$ will be determined by the values of the coefficients $\sigma$, ${\epsilon }_n$ and ${\gamma }_n$, and the sign of the derivative $\text{d}{q}_n/\text{d}M$. And to calculate this last derivative we need an expression which gives us $q_n$ as a function of M. To obtain this function we get $r_n$ from (73), and introduce this expression of $r_n$ in (74). In this way we obtain:

$q_n^2={\displaystyle {\sum }_{k=1}^5{h}_k^{\left(n\right)}{M}^k},$ (84)

where the coefficients $h_k^{\left(n\right)}$ have been defined as follows:

$h_1^{\left(n\right)}=\frac{a^5}{243{\epsilon }_n},$ (85)

$h_2^{\left(n\right)}=-\frac{32a^4}{243},$ (86)

$h_3^{\left(n\right)}=\frac{128}{81}{a}^3{\epsilon }_n,$ (87)

$h_4^{\left(n\right)}=-\frac{2048}{243}{a}^2{\epsilon }_n^2,$ (88)

$h_5^{\left(n\right)}=\frac{4096}{243}a{\epsilon }_n^3.$ (89)

With Equations (84)-(89) we can calculate the derivative $\text{d}{q}_n/\text{d}M$, and replacing $M=1/w^2$ in the resulting expression, we obtain:

$\frac{\text{d}{q}_n}{\text{d}M}=\frac{Q_1^{\left(n\right)}\left(a,w,{\epsilon }_n\right)}{Q_2^{\left(n\right)}\left(a,w,{\epsilon }_n\right)},$ (90)

where we have defined:

$Q_1^{\left(n\right)}\left(a,w,{\epsilon }_n\right)=\frac{a^5}{243{\epsilon }_n}+\frac{20480a{\epsilon }_n^3}{243w^8}-\frac{8192a^2{\epsilon }_n^2}{243w^6}+\frac{128a^3{\epsilon }_n}{27w^4}-\frac{64a^4}{243w^2}$, (91)

$Q_2^{\left(n\right)}\left(a,w,{\epsilon }_n\right)=-2{\epsilon }_n{\left(\frac{a}{3{\epsilon }_n}+\frac{1}{3w^2}\right)}^{3/2}+\frac{6{\epsilon }_n}{w^2}{\left(\frac{a}{3{\epsilon }_n}+\frac{1}{3w^2}\right)}^{1/2}$. (92)

Therefore, the Equations (83) and (90)-(92) permit us to obtain the values of the derivatives $\text{d}{N}_n/\text{d}{q}_n$, and the signs of these derivatives will define the VK stability of the solitons of the cmKdV system (69)-(70). In Table 1 we show the signs of the derivatives $D1\equiv \text{d}{N}_1/\text{d}{q}_1$ and $D2\equiv \text{d}{N}_2/\text{d}{q}_2$, corresponding to four different choices of the set of parameters $\left\{\sigma ,a,w,{\epsilon }_1,{\epsilon }_2,{\gamma }_1,{\gamma }_2\right\}$.

Table 1 shows that in the four cases considered in the table, the two bright solitons (71)-(72) of the cmKdV system (69)-(70) turned out to be VK-stable. This is the result that we might have guessed, as the solitons of the standard cmKdV equation are always stable [4] .