Rogue Waves and Their Patterns in the Vector Nonlinear Schrödinger Equation

Abstract

In this paper, we study the general rogue wave solutions and their patterns in the vector (or M-component) nonlinear Schrödinger (NLS) equation. By applying the Kadomtsev–Petviashvili reduction method, we derive an explicit solution for the rogue wave expressed by $\tau$ functions that are determinants of $K\times K$ block matrices ($1\le K \le M$) with an index jump of $M+1$. Patterns of the rogue waves for $M=3,4$ and $K=1$ are thoroughly investigated. It is found that when one of the internal parameters is large enough, the wave pattern is linked to the root structure of a generalized Wronskian–Hermite polynomial hierarchy in contrast with rogue wave patterns of the scalar NLS equation, the Manakov system, and many others. Moreover, the generalized Wronskian–Hermite polynomial hierarchy includes the Yablonskii–Vorob’ev polynomial and Okamoto polynomial hierarchies as special cases, which have been used to describe the rogue wave patterns of the scalar NLS equation and the Manakov system, respectively. As a result, we extend the most recent results by Yang et al. for the scalar NLS equation and the Manakov system. It is noted that the case $M=3$ displays a new feature different from the previous results. The predicted rogue wave patterns are compared with the ones of the true solutions for both cases of $M=3,4$. An excellent agreement is achieved.

Appendices

Appendix A

In this appendix, we provide the proof of Lemma 2.1. Assume $\xi$ is a root of ${\mathcal {R}}_M(z)=0$ of multiplicity M with $\Im (\xi )\not = 0$, then we have

$$\begin{aligned} {\mathcal {R}}^{(n)}_M(\xi ) =0, \quad n=0,1,2,\dots , M-1, \end{aligned}$$

(121)

where

$$\begin{aligned} {\mathcal {R}}^{(m)}_M(\xi ) = (-1)^m (m+1)! \sum _{j=1}^M \frac{r_j }{(\xi + k_j)^{m+2}}, \quad m \ge 1. \end{aligned}$$

The system of equations (121) is linear in $r_j$, $j=1,2,\dots , M$, so we can solve for them and obtain

$$\begin{aligned} (\xi + k_j )^{M+1} = -\dfrac{1}{2} \prod _{\begin{array}{c} i=1 \\ i \not =j \end{array}}^M (k_j-k_i) r_j. \end{aligned}$$

(122)

Denote by

$$\begin{aligned} \xi = x+ \textrm{i} y, \quad -\dfrac{1}{2} \prod _{\begin{array}{c} i=1 \\ i \not =j \end{array}}^M (k_j-k_i) r_j = \lambda _j^{M+1} \exp (\textrm{i} \theta _j \pi ), \quad j=1,2,\dots , M, \end{aligned}$$

(123)

where $\lambda _j >0, x, y$ are real, $y \not = 0$ and

$$\begin{aligned} \theta _j = {\left\{ \begin{array}{ll} 0, \quad \text { if } -\dfrac{1}{2} \prod _{\begin{array}{c} i=1 \\ i \not =j \end{array}}^M (k_j-k_i) r_j >0, \\ 1, \quad \text { if } -\dfrac{1}{2} \prod _{\begin{array}{c} i=1 \\ i \not =j \end{array}}^M (k_j-k_i) r_j <0, \end{array}\right. } \end{aligned}$$

(124)

then we deduce from (122) that, for each $k_j$, there exits $l_j \in \{0,1,\dots , M\}$ such that

$$\begin{aligned} x+k_j+ \textrm{i} y = {\left\{ \begin{array}{ll} \lambda _j \exp [2l_j \pi \textrm{i}/(M+1)], \quad \text { if } \theta _j=0, \\ \lambda _j\exp [(2 l_j +1) \pi \textrm{i}/(M+1)], \quad \text { if } \theta _j=1, \end{array}\right. } \end{aligned}$$

(125)

Comparing both sides of (125) gives

$$\begin{aligned} y = {\left\{ \begin{array}{ll} \lambda _j \sin [2l_j \pi /(M+1)], \quad \text { if } \theta _j=0, \\ \lambda _j \sin [(2 l_j +1) \pi /(M+1)], \quad \text { if } \theta _j=1. \end{array}\right. } \end{aligned}$$

(126)

This implies that all the corresponding $\sin [2l_j \pi /(M+1)]$ or $\sin [(2 l_j +1) \pi /(M+1)], j=1,2,\dots , M,$ should have the same sign. Without loss of generality, we may assume $y>0$. Note that the set

$$\begin{aligned} \{1, \exp [ \pi \textrm{i}/(M+1)], \exp [ 2\pi \textrm{i}/(M+1)], \dots , \exp [2M \pi \textrm{i}/(M+1)], \exp [(2 M +1) \pi \textrm{i}/(M+1)]\} \end{aligned}$$

(127)

contains exactly M elements with positive imaginary parts, which are

$$\begin{aligned} \exp [ \pi \textrm{i}/(M+1)], \quad \exp [ 2\pi \textrm{i}/(M+1)], \dots , \quad \exp [M \pi \textrm{i}/(M+1)]. \end{aligned}$$

(128)

Since the $k_j$’s are distinct, it then follows that

$$\begin{aligned} x+k_j+ \textrm{i} y = \lambda _j \exp [\sigma _j \pi \textrm{i}/(M+1)], \end{aligned}$$

(129)

where $(\sigma _1, \sigma _2, \dots ,\sigma _M)$ can be any permutation of the set $\{1,2,\dots ,M\}$. Without loss of generality, we may take

$$\begin{aligned} \sigma _j = j, \end{aligned}$$

(130)

where $j=1,2,\dots , M$. In this circumstance, we have $\theta _j=[1+(-1)^{j+1}]/2$ and

$$\begin{aligned} x= & {} \lambda _j \cos [j \pi /(M+1)] - k_j, \end{aligned}$$

(131)

$$\begin{aligned} y= & {} \lambda _j \sin [j \pi /(M+1)], \end{aligned}$$

(132)

and hence

$$\begin{aligned} \lambda _j= & {} \lambda _1 \frac{\sin [ \pi /(M+1)] }{\sin [j \pi /(M+1)]}, \end{aligned}$$

(133)

$$\begin{aligned} k_j= & {} k_1 + \lambda _j \cos [j \pi /(M+1)] -\lambda _1 \cos [ \pi /(M+1)], \end{aligned}$$

(134)

$$\begin{aligned}= & {} k_1 + \lambda _1 \left( \sin [ \pi /(M+1)] \cot [j \pi /(M+1)] - \cos [ \pi /(M+1)]\right) \end{aligned}$$

(135)

where $j=1,2,\dots , M$. Further, we find from (123) and (133134135) that

$$\begin{aligned} r_j = 2 (-1)^{j+1} \prod _{\begin{array}{c} i=1 \\ i \not =j \end{array}}^M (k_j-k_i)^{-1} \left( \lambda _1 \frac{\sin [ \pi /(M+1)] }{\sin [j \pi /(M+1)]} \right) ^{M+1}. \end{aligned}$$

(136)

As ${\mathcal {R}}_M(z)$ is a rational function with real coefficients, it is clear that $\xi ^*$ is a root of ${\mathcal {R}}_M(z)=0$ of multiplicity M as well. This completes the proof.

Appendix B

In this appendix, we apply Hirota’s bilinear method to derive rogue wave solutions of the vector NLS equation (1) presented in Theorem 2.2 based on the KP reduction technique. For convenience, we only consider the case when $\tau _{\textbf{n}}$ given in (18) consists of $M \times M$ block matrices, i.e., $K=M$, as other cases can be treated in a similar manner. In such case, we have $I_j = j \,(j=1,2,\dots , M)$ in (18).

We first transform the vector NLS equation (1) into a set of bilinear equations

$$\begin{aligned}&\left( D_x^2+\sum _{j=1}^M \sigma _j \rho _j^2\right) f \cdot f=\sum _{j=1}^M \sigma _j \rho _j^2 g_j g_j^*,\nonumber \\&\left( \textrm{i} D_t+D_x^2+2 \textrm{i} k_j D_x\right) g_j \cdot f=0, \quad j =1,2,\cdots ,M, \end{aligned}$$

(137)

under the nonzero boundary condition at $\pm \infty$ by the variable transformation

$$\begin{aligned} u_j=\rho _j \frac{g_j}{f} \textrm{e}^{\textrm{i}\left( k_j x + w_j t\right) }, \quad j =1,2,\cdots ,M, \end{aligned}$$

(138)

where $w_j=\sum _{j=1}^{M}\sigma _j \rho _j^2 - k_j^2,$ f is a real-valued function, $g_j$ is a complex-valued function, and D is the Hirota’s bilinear operator (Hirota 2004) defined by

$$\begin{aligned} D_x^m D_t^n f \cdot g=\left. \left( \frac{\partial }{\partial x}-\frac{\partial }{\partial x^{\prime }}\right) ^m\left( \frac{\partial }{\partial t}-\frac{\partial }{\partial t^{\prime }}\right) ^n\left[ f(x, t) g\left( x^{\prime }, t^{\prime }\right) \right] \right| _{x^{\prime }=x, t^{\prime }=t}. \end{aligned}$$

Next we define

$$\begin{aligned} m^{\textbf{n}}&=\frac{1}{p+q}{\left[ \prod _{j=1}^{M} \left( -\frac{p-\textrm{i} k_j}{q+\textrm{i} k_j}\right) ^{n_j} \right] } e^{\xi +\eta }, \\ \xi&=p x+p^2 y+\sum _{j=1}^M\frac{1}{p-\textrm{i} k_j} v_j +\xi _0(p), \\ \eta&=q x-q^2 y+\sum _{j=1}^M\frac{1}{q+\textrm{i} k_j} v_j +\eta _0(q), \end{aligned}$$

where $\textbf{n}=\left( n_1, n_2, \ldots , n_M\right)$ with $n_j$ being integers, $p, q, v_j$ are arbitrary complex constants, $j=1,2,\dots , M$, and $\xi _0(p), \eta _0(q)$ are arbitrary functions of p and q, respectively. Let ${\mathcal {A}}_i$ and ${\mathcal {B}}_j$ be differential operators of order i and j, respectively, defined by

$$\begin{aligned} {\mathcal {A}}_i(p)=\frac{1}{i !}\left[ f_1(p) \partial _p\right] ^i, \quad {\mathcal {B}}_j(q)=\frac{1}{i !}\left[ f_2(q) \partial _q\right] ^j, \end{aligned}$$

where $f_1(p), f_2(q)$ are arbitrary functions of p and q, respectively. Then, it can be calculated that (Ohta and Yang 2012) the determinant

$$\begin{aligned} \tau _{\textbf{n}} = \det _{1 \le \nu , \mu \le N}\left( m_{i_\nu , j_\mu }^{\textbf{n}} \right) \end{aligned}$$

where $\left( i_1, i_2, \cdots , i_N\right)$ and $\left( j_1, j_2, \cdots , j_N\right)$ are arbitrary sequences of indices, and the matrix element $m_{i j}^{\textbf{n}}$ is defined as

$$\begin{aligned} m_{i j}^{\textbf{n}}={\mathcal {A}}_i {\mathcal {B}}_j m^{\textbf{n}}, \end{aligned}$$

(139)

would satisfy the bilinear equations

$$\begin{aligned}&\left( \frac{1}{2} D_x D_{v_j}-1\right) \tau _{\textbf{n}} \cdot \tau _{\textbf{n}}=-\tau _{\textbf{n}_{j,1}} \tau _{\textbf{n}_{j,-1}}, \quad j= 1,2,\cdots , M,\nonumber \\&\left( D_x^2-D_y+2 \textrm{i} k_j D_x\right) \tau _{\textbf{n}_{j,1}} \cdot \tau _{\textbf{n}}=0, \quad j= 1,2,\cdots , M, \end{aligned}$$

(140)

where $\textbf{n}_{j,i}=\textbf{n}+i \times \varvec{e}_j$, and $\varvec{e}_l$ is the standard unit vector in ${\mathbb {R}}^M$.

In what follows, we will establish the reductions from the bilinear equations (140) in the KP hierarchy to the bilinear equations (137), thereby obtaining rogue wave solutions of the vector NLS equation (1). This procedure consists of several steps.

• i) Dimension reduction

  Note that

  $$\begin{aligned} \left( 2 \partial _x+\sum _{k=1}^M\sigma _k \rho _k^2 \partial _{v_k}\right) m_{i j}^{\textbf{n}}={\mathcal {A}}_i {\mathcal {B}}_j\left[ {\mathcal {G}}_M(p)+{\mathcal {H}}_M(q)\right] m^{\textbf{n}}, \end{aligned}$$

  (141)

  where

  $$\begin{aligned} {\mathcal {G}}_M(p)= \sum _{j=1}^M\frac{\sigma _j \rho _j^2}{p-\textrm{i} k_j} +2 p, \quad {\mathcal {H}}_M(q)= \sum _{j=1}^M\frac{\sigma _j \rho _j^2}{q+\textrm{i} k_j} +2 q. \end{aligned}$$

  (142)

  This implies that

  $$\begin{aligned} \left( 2 \partial _x+\sum _{k=1}^M\sigma _k \rho _k^2 \partial _{v_k}\right) m_{i j}^{\textbf{n}}&=\sum _{\mu =0}^i \frac{1}{\mu !}\left[ \left( f_1 \partial _p\right) ^\mu {\mathcal {G}}_M(p)\right] m_{i-\mu , j}^{\textbf{n}}\nonumber \\&\quad +\sum _{l=0}^j \frac{1}{l !}\left[ \left( f_2 \partial _q\right) ^l {\mathcal {H}}_M(q)\right] m_{i, j-l}^{\textbf{n}}. \end{aligned}$$

  (143)

  Then, we can use the method introduced in Yang and Yang (2021b) to find $f_1(p)$ and $f_2(q)$ such that

  $$\begin{aligned} \left( f_1 \partial _p\right) ^{M+1} {\mathcal {G}}_M(p)={\mathcal {G}}_M(p), \quad \left( f_2 \partial _q\right) ^{M+1} {\mathcal {H}}_M(q)={\mathcal {H}}_M(q). \end{aligned}$$

  (144)

  Specifically, by expressing $f_1$ in the form

  $$\begin{aligned} f_1(p) = \dfrac{{\mathcal {U}}(p)}{ {\mathcal {U}}'(p)}, \end{aligned}$$

  (145)

  we can convert (144) into

  $$\begin{aligned} \partial _{\ln {\mathcal {U}}}^{M+1} {\mathcal {G}}_M(p)={\mathcal {G}}_M(p). \end{aligned}$$

  Normalize ${\mathcal {U}}(p_0) = 1$, then, under the condition that $p_0$ is a root of ${\mathcal {G}}_M$ of multiplicity M, this equation admits the unique solution

  $$\begin{aligned} {\mathcal {G}}_M(p) = \dfrac{{\mathcal {G}}_M(p(0))}{M+1} \sum _{n=1}^{M+1} \exp \left( \exp \left( \dfrac{2 n \pi \textrm{i}}{M+1}\right) \ln {\mathcal {U}}(p)\right) . \end{aligned}$$

  (146)

  Then, $f_1(p)$ can be obtained by using the relation (145) after solving this equation for ${\mathcal {U}}$. In a similar way, $f_2(q)$ can be found.

  Choosing $q_0 = p_0^*$ and using (144) and the assumption that $p_0$ is a root of ${\mathcal {G}}'_M(p)=0$ of multiplicity M, the equation (143) reduces to

  $$\begin{aligned}{} & {} \left( 2 \partial _x+\sum _{k=1}^M\sigma _k \rho _k^2 \partial _{v_k}\right) m_{i j}^{\textbf{n}} \Big |_{p=p_0,q=q_0}\nonumber \\{} & {} \quad = {\mathcal {G}}_M(p_0) \sum _{\begin{array}{c} \mu =0 \\ \mu \equiv 0 (\text {mod } (M+1)) \end{array}}^i \frac{1}{\mu !} m_{i-\mu , j}^{\textbf{n}}+{\mathcal {H}}_M(q_0) \sum _{\begin{array}{c} l=0 \\ l \equiv 0 (\text {mod } (M+1)) \end{array}}^i \frac{1}{l !} m_{i, j-l}^{\textbf{n}} \Big |_{p=p_0,q=q_0}.\nonumber \\ \end{aligned}$$

  (147)

  Let $N=N_1+N_2+\cdots +N_M$, where $N_j, j=1,2,\cdots ,M,$ are positive integers, and define the determinant $\tau _{\textbf{n}}$ by

  $$\begin{aligned} \tau _{\textbf{n}}=\det \left( \begin{array}{llll} \tau _{\textbf{n}}^{[1,1]} &{} \tau _{\textbf{n}}^{[1,2]}&{}\cdots &{}\tau _{\textbf{n}}^{[1,M]} \\ \tau _{\textbf{n}}^{[2,1]} &{} \tau _{\textbf{n}}^{[2,2]}&{}\cdots &{}\tau _{\textbf{n}}^{[2,M]} \\ \vdots &{} \vdots &{}\ddots &{}\vdots \\ \tau _{\textbf{n}}^{[M,1]} &{} \tau _{\textbf{n}}^{[M,2]}&{}\cdots &{}\tau _{\textbf{n}}^{[M,M]} \end{array}\right) , \end{aligned}$$

  (148)

  where

  $$\begin{aligned} \tau _{\textbf{n}}^{[I, J]}&=\textrm{mat}_{1 \le i \le N_I, 1 \le j \le N_J}\nonumber \\&\quad \left( \left. m_{(M+1) i-I, (M+1) j-J}^{\textbf{n}}\right| _{p=p_0, q=q_0, \xi _0=\xi _{0, I}, \eta _0=\eta _{0, J}}\right) , \quad 1 \le I, J \le M, \end{aligned}$$

  (149)

  and $m_{i,j}^{\textbf{n}}$ is given by (139).

  With (147), we can use similar argument as in Ohta and Yang (2012) to show that the determinant $\tau _{\textbf{n}}$ satisfies the dimensional reduction condition

  $$\begin{aligned} \left( 2 \partial _x+\sum _{k=1}^M\sigma _k \rho _k^2 \partial _{v_k}\right) \tau _{\textbf{n}} =N \left[ {\mathcal {G}}_M(p_0)+{\mathcal {H}}_M(q_0)\right] \tau _{\textbf{n}}. \end{aligned}$$

  (150)

  Therefore, we can use (150) to eliminate the variables $v_j, j =1,2,\cdots ,M$, from the higher-dimensional bilinear system (140). As a result of this, we have

  $$\begin{aligned}&\left( D_x^2+\sum _{j=1}^M \sigma _j \rho _j^2\right) \tau _{\textbf{n}} \cdot \tau _{\textbf{n}} = \sum _{j=1}^M \sigma _j \rho _j^2 \tau _{\textbf{n}_{j,1}} \tau _{\textbf{n}_{j,-1}}\nonumber \\&\left( \textrm{i} D_t+D_x^2+2 \textrm{i} k_j D_x\right) \tau _{\textbf{n}_{j,1}} \cdot \tau _{\textbf{n}}=0,\quad j=1,2,\cdots ,M, \end{aligned}$$

  (151)

  where $t = - \textrm{i} y$.

• ii) Complex conjugate reduction

  Impose the parameter constraint

  $$\begin{aligned} \xi _{0,I} = \eta _{0,I}^*, \end{aligned}$$

  and in view of $p_0 = q_0^*$, we have $\left[ f_1\left( p_0\right) \right] ^*=$ $f_2\left( q_0\right)$. It then follows that

  $$\begin{aligned} \tau _{\textbf{n}} = \tau _{-\textbf{n}}^*. \end{aligned}$$

  (152)

  Define

  $$\begin{aligned} f=\tau _{\textbf{n}_0}, \quad g_j=\tau _{\textbf{n}_j}, \quad j=1,2,\cdots ,M, \end{aligned}$$

  (153)

  then the complex conjugacy condition (152) implies that f is real. Therefore, from (151) and (152), we conclude that the functions f and $g_j$ satisfy the bilinear system (137), thereby yielding rational solutions to the vector NLS equation (1) via the transformation (138).

• iii) Introduction of free parameters

  We apply the method proposed in Yang and Yang (2021b) to introduce free parameters in the following form

  $$\begin{aligned} \xi _{0,I}=\sum _{n=1}^{\infty } a_{n,I} \ln ^{n} {\mathcal {U}}(p), \end{aligned}$$

  (154)

  where ${\mathcal {U}}(p)$ is defined by (145) and the $a_{n,I}$’s are free complex constants.

• iv) Simplification of solutions

  With the aid of the generator ${\mathcal {D}}$ of the differential operators $\left( p \partial _p\right) ^k\left( q \partial _q\right) ^l$ given as

  $$\begin{aligned} {\mathcal {D}}=\sum _{k=0}^{\infty } \sum _{l=0}^{\infty } \frac{\kappa ^k}{k !} \frac{\lambda ^l}{l !}\left( p \partial _p\right) ^k\left( q \partial _q\right) ^l=\exp \left( \kappa p \partial _p+\lambda q \partial _q\right) =\exp \left( \kappa \partial _{\ln p}+\lambda \partial _{\ln q}\right) , \end{aligned}$$

  (155)

  we are able to simplify the solutions expressed by (153) using differential operators into the form of Schur polynomials as presented in Theorem 2.2. Since the computations are very similar to those in the three-wave system by Yang and Yang Yang and Yang (2021b), we omit the details.

Thus, the proof of Theorem 2.2 is completed.

Appendix C

In the first part of this appendix, we provide the values of $N_1, N_2,N_3,N_4$ that appear in Theorem 2.4 in the following lemma.

Lemma 6.1

The values of $\textbf{N}=(N_1, N_2,N_3,N_4)$ involved in Theorem 2.4 are characterized as follows.

• When $m \equiv 1 \mod 5$, we have

  $$\begin{aligned}{} & {} l=4: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( N_0, 0,0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] , N_0-\left[ \frac{m}{5}\right] ,0,0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] , N_0-2\left[ \frac{m}{5}\right] \right) , &{} 2\left[ \frac{m}{5}\right] +1 \le N_0 \le 3\left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] , N_0-3\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +1 \le N_0 \le 4\left[ \frac{m}{5}\right] \\ \left( m-1-N_0, m-1-N_0,m-1-N_0,m-1-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +1 \le N_0 \le m-1\end{array}\right. } \\{} & {} l=3: \textbf{N}\\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0,N_0, 0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( 0,\left[ \frac{m}{5}\right] , N_0-\left[ \frac{m}{5}\right] ,0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] \\ \left( 0,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] , N_0-2\left[ \frac{m}{5}\right] \right) , &{} 2\left[ \frac{m}{5}\right] +1 \le N_0 \le 3\left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] -1,N_0 -3\left[ \frac{m}{5}\right] -1\right) , &{} 3\left[ \frac{m}{5}\right] +1 \le N_0 \le 4\left[ \frac{m}{5}\right] +1 \\ \left( m-1-N_0,m-1-N_0,m-1-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +2 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=2: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,N_0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( 0,0,\left[ \frac{m}{5}\right] ,N_0-\left[ \frac{m}{5}\right] \right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] -1, N_0-2\left[ \frac{m}{5}\right] -1,0\right) , &{} 2\left[ \frac{m}{5}\right] +1 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -1\right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +1 \\ \left( m-1-N_0,m-1-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +2 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=1: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,0,N_0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] -1, N_0-\left[ \frac{m}{5}\right] -1,0,0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] , N_0-2\left[ \frac{m}{5}\right] -1,0\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -1\right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +1 \\ \left( m-1-N_0,m-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +2 \le N_0 \le m-1 \end{array}\right. } \end{aligned}$$

• When $m \equiv 2 \mod 5$, we have

  $$\begin{aligned}{} & {} l=4: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( N_0, 0,0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] ,0, N_0-\left[ \frac{m}{5}\right] ,0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] \\ \left( N_0-2\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,0,\left[ \frac{m}{5}\right] \right) &{} 2\left[ \frac{m}{5}\right] +1 \le N_0 \le 3\left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,N_0-3\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +1 \le N_0 \le 4\left[ \frac{m}{5}\right] \\ \left( m-1-N_0, m-1-N_0,m-1-N_0,m-1-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +1 \le N_0 \le m-1 \end{array}\right. } \\{} & {} \quad l=3: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0,N_0, 0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( 0,\left[ \frac{m}{5}\right] , 0,N_0-\left[ \frac{m}{5}\right] \right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( N_0-2\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,0,\left[ \frac{m}{5}\right] +1\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +1 \\ \left( m-1-N_0,m-1-N_0,m-1-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +2 \le N_0 \le m-1 \end{array}\right. } \\{} & {} \quad l=2: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,N_0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( N_0-\left[ \frac{m}{5}\right] -1,0,0,\left[ \frac{m}{5}\right] \right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,0,N_0-2\left[ \frac{m}{5}\right] -1, \left[ \frac{m}{5}\right] \right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] -1,N_0 -3\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0,m-1-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \\{} & {} \quad l=1: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,0,N_0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( N_0-\left[ \frac{m}{5}\right] -1,0,0,\left[ \frac{m}{5}\right] +1\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,0, N_0-2\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] +1\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0,m-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \end{aligned}$$

• When $m \equiv 3 \mod 5$, we have

  $$\begin{aligned}{} & {} l=4: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( N_0, 0,0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( \left[ \frac{m}{5}\right] ,0,0, N_0-\left[ \frac{m}{5}\right] \right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,N_0-2\left[ \frac{m}{5}\right] -1,0,\left[ \frac{m}{5}\right] \right) &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,N_0-3\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0, m-1-N_0,m-1-N_0,m-1-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=3: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0,N_0, 0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( N_0-\left[ \frac{m}{5}\right] -1,0,\left[ \frac{m}{5}\right] , 0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,0,\left[ \frac{m}{5}\right] ,N_0-2\left[ \frac{m}{5}\right] -1\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +2 \\ \left( \left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +3 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0,m-1-N_0,m-1-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=2: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,N_0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( N_0-\left[ \frac{m}{5}\right] -1,0,\left[ \frac{m}{5}\right] +1,0\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,0, \left[ \frac{m}{5}\right] +1,N_0-2\left[ \frac{m}{5}\right] -1\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,N_0 -3\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] +1,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +2 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0,m-1-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=1: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,0,N_0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( 0,N_0-\left[ \frac{m}{5}\right] -1,0,\left[ \frac{m}{5}\right] +1\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( \left[ \frac{m}{5}\right] ,N_0-2\left[ \frac{m}{5}\right] -2,0, \left[ \frac{m}{5}\right] \right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +2 \\ \left( \left[ \frac{m}{5}\right] -1,N_0 -3\left[ \frac{m}{5}\right] -3,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +3 \le N_0 \le 4\left[ \frac{m}{5}\right] +3 \\ \left( m-1-N_0,m-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +4 \le N_0 \le m-1 \end{array}\right. } \end{aligned}$$

• When $m \equiv 4 \mod 5$, we have

  $$\begin{aligned}{} & {} l=4: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( N_0, 0,0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] \\ \left( N_0-\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] ,0,0\right) , &{} \left[ \frac{m}{5}\right] +1 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( N_0-2\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,0\right) &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +2 \\ \left( N_0-3\left[ \frac{m}{5}\right] -3,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] \right) , &{} 3\left[ \frac{m}{5}\right] +3 \le N_0 \le 4\left[ \frac{m}{5}\right] +3 \\ \left( m-1-N_0, m-1-N_0,m-1-N_0,m-1-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. } \\{} & {} l=3: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0,N_0, 0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( N_0-\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] +1,0, 0,\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +1 \\ \left( N_0-2\left[ \frac{m}{5}\right] -2, \left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] +1,0\right) , &{} 2\left[ \frac{m}{5}\right] +2 \le N_0 \le 3\left[ \frac{m}{5}\right] +2 \\ \left( N_0 -3\left[ \frac{m}{5}\right] -3,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +3 \le N_0 \le 4\left[ \frac{m}{5}\right] +2 \\ \left( m-1-N_0,m-1-N_0,m-1-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +3 \le N_0 \le m-1 \end{array}\right. }\\{} & {} l=2: \textbf{N} \\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,N_0,0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( 0,N_0-\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] +1,0\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +2 \\ \left( N_0-2\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] +1, \left[ \frac{m}{5}\right] +1,0\right) , &{} 2\left[ \frac{m}{5}\right] +3 \le N_0 \le 3\left[ \frac{m}{5}\right] +2 \\ \left( N_0 -3\left[ \frac{m}{5}\right] -3,\left[ \frac{m}{5}\right] ,\left[ \frac{m}{5}\right] +1,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +3 \le N_0 \le 4\left[ \frac{m}{5}\right] +3 \\ \left( m-1-N_0,m-1-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +4 \le N_0 \le m-1 \end{array}\right. }\\{} & {} l=1: \textbf{N}\\{} & {} \quad = {\left\{ \begin{array}{ll} \left( 0, 0,0,N_0\right) , &{} 0 \le N_0 \le \left[ \frac{m}{5}\right] +1 \\ \left( 0,0,N_0-\left[ \frac{m}{5}\right] -1,\left[ \frac{m}{5}\right] +1\right) , &{} \left[ \frac{m}{5}\right] +2 \le N_0 \le 2\left[ \frac{m}{5}\right] +2 \\ \left( 0,N_0-2\left[ \frac{m}{5}\right] -2,\left[ \frac{m}{5}\right] +1, \left[ \frac{m}{5}\right] +1\right) , &{} 2\left[ \frac{m}{5}\right] +3 \le N_0 \le 3\left[ \frac{m}{5}\right] +3 \\ \left( N_0 -3\left[ \frac{m}{5}\right] -3,\left[ \frac{m}{5}\right] +1,\left[ \frac{m}{5}\right] +1,\left[ \frac{m}{5}\right] +1\right) , &{} 3\left[ \frac{m}{5}\right] +4 \le N_0 \le 4\left[ \frac{m}{5}\right] +3 \\ \left( m-1-N_0,m-N_0,m-N_0,m-N_0\right) , &{} 4\left[ \frac{m}{5}\right] +4 \le N_0 \le m-1. \end{array}\right. } \end{aligned}$$

In the second part of this appendix, we will prove Theorems 2.3 and 2.4 for root structures of the generalized Wronskian–Hermite polynomials $W_N^{[m,k,l]}$. We only provide the proof for $k=5$, $l=4$ and $m \equiv 1,2,3,4\mod 5$, as other cases can be proved in a similar manner.

Firstly, we define a new class of special Schur polynomials $S_j^{[m]}(z; a)$ and the polynomials $\widehat{W}_N^{[m,5,4]}(z; a)$ as

$$\begin{aligned} \sum _{j=0}^{\infty } S_j^{[m]}(z; a) \epsilon ^j= & {} \exp \left[ z \epsilon +a \epsilon ^m\right] , \end{aligned}$$

(156)

$$\begin{aligned} \widehat{W}_N^{[m,5,4]}(z; a)= & {} c_N^{[m,5,4]} \left| \begin{array}{cccc} S_4^{[m]}(z; a) &{} S_3^{[m]}(z; a) &{} \cdots &{} S_{5-N}^{[m]}(z; a) \\ S_9^{[m]}(z; a) &{} S_8^{[m]}(z; a) &{} \cdots &{} S_{10-N}^{[m]}(z; a) \\ \vdots &{} \vdots &{} \vdots &{} \vdots \\ S_{5 N-1}^{[m]}(z; a) &{} S_{5 N-2}^{[m]}(z; a) &{} \cdots &{} S_{4 N}^{[m]}(z; a) \end{array}\right| ,\nonumber \\ \end{aligned}$$

(157)

where a is a parameter, $c_N^{[m,5,4]}$ is a constant defined in (3637), and $S_j^{[m]}(z; a) \equiv 0$ when $j<0$. Compared with the generalized Wronskian–Hermite polynomials $W_N^{[m,k,l]}$, the polynomials $\widehat{W}_N^{[m,5,4]}(z; a)$ have a new parameter a. Note that the polynomials $S_j^{[m]}(z; a)$ are related to $p_j^{[m]}(z)$ by

$$\begin{aligned} S_j^{[m]}(z ; a)=a^{j / m} p_j^{[m]}(\hat{z}), \quad \hat{z}=a^{-1 / m} z. \end{aligned}$$

(158)

In addition, we have

$$\begin{aligned} \widehat{W}_N^{[m,5,4]}(z ; a)=a^{2N(N+1)/m} W_N^{[m,5,4]}(\hat{z}). \end{aligned}$$

(159)

From (158) and (159), we find that each term in the polynomial $\widehat{W}_N^{[m]}(z; a)$ is a constant multiple of $a^s z^k$ and $k+ms=2N(N+1)$. This indicates that when the power of a is larger, the power of z is lower. Thus, to find the lowest order of z, it suffices to find the highest order of a. To this end, we rewrite the polynomials $S_j^{[m]}(z; a)$ as

$$\begin{aligned} S_j^{[m]}(z ; a)=\sum _{n=0}^{[j / m]} \frac{a^n}{n !(j-n m) !} z^{j-n m} \end{aligned}$$

(160)

and substitute (160) into the determinant (156157). Note that coefficients of each $a^sz^k$ in each row are proportional to each other, thus we can ignore them. In particular, for $m=5r+1$, where r is positive integer, we get

$$\begin{aligned} \widehat{W}_N^{[m,5,4]}(z ; a) \sim c_N\left| \begin{array}{cccccccc} z^4 &{} z^3 &{} \cdots \\ \vdots &{} \vdots &{} \ddots \\ z^{5r-1} &{} z^{5r-2} &{} \cdots \\ a z^3 + \cdots &{} a z^2 + \cdots &{} \cdots \\ \vdots &{} \vdots &{} \ddots \\ a z^{5r-2}+z^{5r-2+m} &{} a z^{5r-3}+z^{5r-3+m} &{} \cdots \\ a^2 z^2 +a z^{2+m} + \cdots &{} a^2 z^1 +a z^{1+m} + \cdots &{} \cdots \\ \vdots &{} \vdots &{} \ddots \\ a^2 z^{5r-3} +a z^{5r-3+m} + \cdots &{} a^2 z^{5r-4} +a z^{5r-4+m} + \cdots &{} \cdots \\ a^3 z^1 +a^2 z^{1+m} + \cdots &{} a^3 z^0 +a^2 z^m + \cdots &{} \cdots \\ \vdots &{} \vdots &{} \ddots \\ a^3 z^{5r-4} +a^2 z^{5r-4+m} + \cdots &{} a^3 z^{5r-5} +a^2 z^{5r-5+m} + \cdots &{} \cdots \\ a^4 z^0 + a^3 z^m + \cdots &{} a^3 z^{5r} + \cdots &{} \cdots \\ \vdots &{} \vdots &{} \ddots \\ a^4 z^{5r}+a^3 z^{5r+m} + \cdots &{} a^4 z^{5r-1}+a^3 z^{5r-1+m} + \cdots &{} \cdots \\ \vdots &{} \vdots &{} \ddots \end{array}\right| .\nonumber \\ \end{aligned}$$

(161)

Next, we perform row operations to (161), which consist of several steps.

• 1. Note that the coefficients of the highest-order terms in a in the first column of (161) are periodic and one period is given by

  $$\begin{aligned} z^4 \cdots z^{5r-1};\quad z^3 \cdots z^{5r-2};\quad z^2 \cdots z^{5r-3};\quad z^1 \cdots z^{5r-4};\quad z^0 \cdots z^{5r}. \end{aligned}$$

  (162)

  According to this periodicity, we divide the determinants (161) into $\left[ N/m\right]$ block matrices of size $m \times N$ and one $N_0 \times N$ block matrix, where $N_0 \equiv N \mod m$. In addition, we divide the first column in each block into five parts, which have distinct initial powers in z, and the difference of the powers in z of consecutive terms in each part is 5. We denote the number of parts starting with power j by $N_{5-j}, j=1, \dots , 4$.

• 2. We are only concerned with the first column of (161), since other columns have similar structures. According to the above discussions,

  we may use each part of the first block to cancel the highest-order terms in a of the corresponding parts for the subsequent blocks. After the first round row operations, the coefficients of the highest-order terms in a in the first column of the second block become

  $$\begin{aligned} z^{4+m} \cdots z^{5r-1+m}; z^{3+m} \cdots z^{5r-2+m}; z^{2+m} \cdots z^{5r-3+m}; z^{1+m} \cdots z^{5r-4+m}; z^m \cdots z^{5r+m}, \end{aligned}$$

  (163)

  and from the third to the last blocks, the corresponding coefficients change to $z^{i+m}$ from $z^i$. In the second round, we can use the second block to cancel the highest-order terms in a of the blocks below. Then, we continue this process until the last block. At the end of these operations, we can exchange the rows in each block such that the highest-order terms in a of the first column are

  $$\begin{aligned} z^0, z^1, z^2, \cdots , z^{5r}; z^m, z^{m+1}, z^{m+2},\cdots , z^{m+5r}; \cdots ; z^{km}, z^{km+1},z^{km+2},\cdots ,z^{km+5r};\cdots . \end{aligned}$$

  and the determinant (161) becomes

  $$\begin{aligned} \widehat{W}_{N \times N}^{[m,5,4]} \sim \left( \begin{array}{ccc} \textbf{L}_{\left( N-N_0\right) \times \left( N-N_0\right) } &{} \textbf{0}_{\left( N-N_0\right) \times N_0} \\ \textbf{M}_{N_0 \times \left( N-N_0\right) } &{} \widehat{W}_{N_0 \times N_0} \end{array}\right) \end{aligned}$$

  where $\textbf{L}_{\left( N-N_0\right) \times \left( N-N_0\right) }$ is a lower triangular matrix whose diagonal entries are all 1.

• 3. Therefore, to calculate the lowest power of z of $\widehat{W}_N^{[m,5,4]}(z; a)$, it suffices to compute the power of the reduced $N_0 \times N_0$ determinant $\widehat{W}_{N_0 \times N_0}$ and the final result is $\Gamma$ as given in (42).

Next, we derive the factorization of $W_N^{[m,5,4]}(z)$ provided in Theorem 2.4. Since the multiplicity of the zero root of $W_N^{[m,5,4]}(z)$ is $\Gamma$, we can write

$$\begin{aligned} W_N^{[m,5,4]}(z)=z^{\Gamma } q_N^{[m]}(z). \end{aligned}$$

(164)

Note that

$$\begin{aligned} p_j^{[m]}(\omega z)=\omega ^j p_j^{[m]}(z), \end{aligned}$$

(165)

where $\omega$ is any of the m-th root of 1, i.e., $\omega ^m=1$. From (164) and (165), we immediately have

$$\begin{aligned} W_N^{[m,5,4]}(\omega z)=\omega ^{2N(N+1)} W_N^{[m,5,4]}(z) \end{aligned}$$

and

$$\begin{aligned} q_N^{[m]}(\omega z)=\omega ^{2N(N+1)-\Gamma } q_N^{[m]}(z). \end{aligned}$$

Since $2N(N+1)-\Gamma$ is a multiple of m, we have $\omega ^{2N(N+1)-\Gamma }=1$, and hence,

$$\begin{aligned} q_N^{[m]}(\omega z)=q_N^{[m]}(z). \end{aligned}$$

This completes the proof.

Remark 9

We note that the row operations performed above are similar to those in the proof of rogue patterns in the inner region. For some special cases, such as $m=4r+2$ in the three-component NLS equation, the proof needs some modifications similar to the proof of Theorem 2.3.