To find the relation between SSE, Hirota (NLS), and mKdV hierarchies, we need a consistent normalisation and use $b=\sqrt{2}$ in the results found above; we note that this is different from that used in the earlier article [29]. We denote our previously found [9] functionals for the Hirota (NLS) hierarchy by *K*_{j}, so that the equations of the infinite hierarchy are

$$i{u}_{t}+\sum _{n=1}^{\mathrm{\infty}}\left({\alpha}_{2n}{K}_{2n}-i{\alpha}_{2n+1}{K}_{2n+1}\right)=0.$$(25)

The NLSE is obtained from (25) when all real coefficients except *α*_{2} are zero. Thus,

$$i{u}_{t}+{\alpha}_{2}{K}_{2}=0,$$(26)

where the functional ${K}_{2}={u}_{xx}+2u{\left|u\right|}^{2}$.

If *u*(*x*, *t*) in (26) is real, this gives the real form *D*_{2} of the functional *K*_{2},

$${D}_{2}\equiv {u}_{xx}+2{u}^{3}.$$(27)

We can take the derivative of *D*_{2} with respect to *x*

$${M}_{3}=\frac{\partial {D}_{2}}{\partial x}={u}_{xxx}+6{u}_{x}{u}^{2}$$(28)

to get the mKdV equation,

$$i{u}_{t}-i{\alpha}_{3}{M}_{3}=0.$$

From [9], we have

$${K}_{3}={u}_{xxx}+6{u}_{x}{\left|u\right|}^{2}.$$(29)

Plainly, *S*_{3} and *K*_{3} are related through

$${S}_{3}-{K}_{3}=\left(\frac{3}{2}u\right)\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill {u}_{x}\hfill & \hfill {u}_{x}^{\ast}\hfill \end{array}\right|=\frac{3}{2}u\left(u{u}_{x}^{\ast}-{u}^{\ast}{u}_{x}\right)=\frac{3}{2}u{W}_{1},$$

where the determinant *W*_{1} is defined as

$${W}_{1}=\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill {u}_{x}\hfill & \hfill {u}_{x}^{\ast}\hfill \end{array}\right|.$$

If *u* is real, this determinant is zero.

For *u* real, the SSE

$${u}_{t}-{\alpha}_{3}\left[{u}_{xxx}+\frac{3}{2}{\left({\left|u\right|}^{2}\right)}_{x}u+3{\left|u\right|}^{2}{u}_{x}\right]=0$$

reduces to the basic mKdV equation

$${u}_{t}-{\alpha}_{3}\left({u}_{3x}+6{u}^{2}{u}_{x}\right)=0,$$

i.e. ${u}_{t}-{\alpha}_{3}{M}_{3}=0$, where ${M}_{3}={u}_{3x}+6{u}^{2}{u}_{x}$. The Hirota functional, ${K}_{3}={u}_{xxx}+6{\left|u\right|}^{2}{u}_{x}$, also reduces to *M*_{3} for *u* real.

Converting (13) to its *b*^{2} = 2 form, we get

$$\begin{array}{c}{S}_{4}=3{u}_{xx}^{\ast}{u}^{2}+6u{\left|u\right|}^{4}+6{\left|{u}_{x}\right|}^{2}u\hfill \\ +7{\left|u\right|}^{2}{u}_{xx}+4{u}^{\ast}{u}_{x}^{2}+{u}_{xxxx}.\hfill \end{array}$$

It is clear that (30) has the form of the fourth member of the Hirota (NLS) hierarchy, which is known as the LPD equation, but has different coefficients. Now, from [9]:

$$\begin{array}{c}{K}_{4}=2{u}_{xx}^{\ast}{u}^{2}+6u{\left|u\right|}^{4}+4{\left|{u}_{x}\right|}^{2}u\hfill \\ +8{\left|u\right|}^{2}{u}_{xx}+6{u}^{\ast}{u}_{x}^{2}+{u}_{xxxx}.\hfill \end{array}$$(30)

The difference between *S*_{4} and *K*_{4} is

$${S}_{4}-{K}_{4}=2{\left|{u}_{x}\right|}^{2}u-2{u}^{\ast}{u}_{x}^{2}+{u}_{xx}^{\ast}{u}^{2}-{\left|u\right|}^{2}{u}_{xx},$$

so the two fourth-order equations differ only by two terms:

$$\begin{array}{c}u\left({S}_{4}-{K}_{4}\right)=\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill \frac{\partial}{\partial x}\left({u}_{x}{u}^{2}\right)\hfill & \hfill \frac{\partial}{\partial x}\left({u}_{x}^{\ast}{u}^{2}\right)\hfill \end{array}\right|\hfill \\ =u\frac{\partial}{\partial x}\left({u}_{x}^{\ast}{u}^{2}\right)-{u}^{\ast}\frac{\partial}{\partial x}\left({u}_{x}{u}^{2}\right),\hfill \end{array}$$(31)

so

$${S}_{4}-{K}_{4}=u{W}_{2}+2{u}_{x}{W}_{1},$$

where

$${W}_{2}=\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill {u}_{2x}\hfill & \hfill {u}_{2x}^{\ast}\hfill \end{array}\right|.$$

If *u* is real, then *S*_{4} − *K*_{4} is also zero.

If we take *u* to be real, then we clearly have *S*_{4} − *K*_{4} = 0, and we get

$${S}_{4}={K}_{4}=10{u}_{xx}{u}^{2}+6{u}^{5}+10u{u}_{x}^{2}+{u}_{xxxx}\equiv {D}_{4}.$$(32)

Now *D*_{4} is the real form, which can be used to obtain the fifth-order mKdV equation [40],

$${u}_{t}-{\alpha}_{5}{M}_{5}=0.$$

Thus,

$$\begin{array}{c}\frac{\partial {D}_{4}}{\partial x}=30{u}_{x}{u}^{4}+10{u}_{xxx}{u}^{2}+40{u}_{x}{u}_{xx}u+10{u}_{x}^{3}+{u}_{5x}\hfill \\ ={M}_{5}.\hfill \end{array}$$(33)

This is the same as the functional *M*_{5} found by other means. It is also the same as (22) of the real-valued SSE hierarchy, *S*_{5}, found here, and it is the same as the real-valued form of *K*_{5} found in [9].

Indeed, from [9], we have

$$\begin{array}{c}{K}_{5}={u}_{5x}+10{\left|u\right|}^{2}{u}_{xxx}+10{\left(u{\left|{u}_{x}\right|}^{2}\right)}_{x}\hfill \\ +20{u}^{\ast}{u}_{x}{u}_{xx}+30{\left|u\right|}^{4}{u}_{x}.\hfill \end{array}$$(34)

We furthermore find that *S*_{5} and *K*_{5} are related as follows:

$$\begin{array}{c}{S}_{5}-{K}_{5}=\frac{5}{2}u{W}_{3}+\frac{5}{2}{u}_{x}{W}_{2}\hfill \\ +\left(\frac{5}{2}{u}_{2x}+10u{\left|u\right|}^{2}\right){W}_{1},\hfill \end{array}$$

where

$${W}_{3}=\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill {u}_{3x}\hfill & \hfill {u}_{3x}^{\ast}\hfill \end{array}\right|,$$

and *W*_{1} and *W*_{2} defined above. We can generalise these definitions as follows:

$${W}_{j}=\left|\begin{array}{cc}\hfill u\hfill & \hfill {u}^{\ast}\hfill \\ \hfill {u}_{jx}\hfill & \hfill {u}_{jx}^{\ast}\hfill \end{array}\right|,$$

so that the *W*_{j} determinant involves the *j*^{th} derivatives of *u* and its complex conjugate, *u*^{∗}. When *u* is real, then each *W*_{j} is zero. If *u* is real, then *S*_{5} − *K*_{5} is also zero, so *S*_{5} = *K*_{5} = *M*_{5}, as noted above.

The real scaled form of *S*_{6} from this present work is

$$\begin{array}{c}{S}_{6}=20{u}^{7}+70{u}^{4}{u}_{xx}+140{u}^{3}{u}_{x}^{2}+14{u}^{2}{u}_{xxxx}\hfill \\ +14u\left(4{u}_{x}{u}_{xxx}+3{u}_{xx}^{2}\right)+70{u}_{x}^{2}{u}_{xx}+{u}_{6x}.\hfill \end{array}$$(35)

The real form of *K*_{6}, given in [9], is exactly the same as (35). These two functionals are clearly identical, and we label them *D*_{6}. Now, we can obtain the seventh-order mKdV equation, ${u}_{t}-{\alpha}_{7}{M}_{7}=0$, where

$$\begin{array}{c}\frac{\partial {D}_{6}}{\partial x}={u}_{7x}+14{u}^{2}\left({u}_{5x}+30{u}_{x}^{3}\right)\hfill \\ +140{u}^{6}{u}_{x}+70{u}^{4}{u}_{xxx}+560{u}^{3}{u}_{x}{u}_{xx}+\hfill \\ +28u\left(3{u}_{x}{u}_{xxxx}+5{u}_{xx}{u}_{xxx}\right)\hfill \\ +182{u}_{x}{u}_{xx}^{2}+126{u}_{x}^{2}{u}_{xxx}\hfill \\ ={M}_{7}.\hfill \end{array}$$

This is the seventh-order mKdV functional. It is the same as the real form of *K*_{7}. The forms of the mKdV functionals, *M*_{2j+1}, agree with those found from the recursion operator.

Also, from [9], we have

$$\begin{array}{c}{K}_{6}={u}_{6x}+2\left[30{u}^{\ast}{\left|{u}_{x}\right|}^{2}+25{\left({u}^{\ast}\right)}^{2}{u}_{xx}+{u}_{xxxx}^{\ast}\right]{u}^{2}\hfill \\ +u\left[12{u}^{\ast}{u}_{xxxx}+8{u}_{x}{u}_{xxx}^{\ast}+22{\left|{u}_{xx}\right|}^{2}\right]\hfill \\ +u\left[18{u}_{xxx}{u}_{x}^{\ast}+70{\left({u}^{\ast}\right)}^{2}{u}_{x}^{2}\right]\hfill \\ +20{\left({u}_{x}\right)}^{2}{u}_{xx}^{\ast}+10{u}_{x}\left[5{u}_{xx}{u}_{x}^{\ast}+3{u}^{\ast}{u}_{xxx}\right]\hfill \\ +20{u}^{\ast}{u}_{xx}^{2}+10{u}^{3}\left[{\left({u}_{x}^{\ast}\right)}^{2}+2{u}^{\ast}{u}_{xx}^{\ast}\right]\hfill \\ +20u{\left|u\right|}^{6}.\hfill \end{array}$$(36)

We furthermore find that *S*_{6} and *K*_{6} are related as follows:

$$\begin{array}{c}{S}_{6}-{K}_{6}=2u{W}_{4}+8{u}_{x}{W}_{3}\hfill \\ +\frac{1}{2}\left(9{u}_{2x}+15u{\left|u\right|}^{2}+\frac{5}{u}{u}_{x}^{2}\right){W}_{2}\hfill \\ +\frac{1}{4}[15{u}^{2}{u}_{x}^{\ast}+105|u{|}^{2}{u}_{x}\hfill \\ -\frac{10}{u}{u}_{x}{u}_{xx}+2{u}_{xxx}]W{}_{1},\hfill \end{array}$$(37)

If *u* is real, then each *W*_{j} is zero, and thus, *S*_{6} = *K*_{6}.

In the expressions *S*_{j} − *K*_{j}, the first term is proportional to *uW*_{j−2}, while the second term is proportional to *u*_{x} W_{j−3}. Of course, when the *K*_{j} and differences are known, the set *S*_{j} can be generated. So, the pattern of these hierarchies is now more evident. The results of this section are illustrated schematically in .

Table 1: Relations between hierarchies, summarising the results of Section 4.

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