Multi-component Nonlinear Schrödinger Equations with Nonzero Boundary Conditions: Higher-Order Vector Peregrine Solitons and Asymptotic Estimates

Abstract

The any multi-component nonlinear Schrödinger (alias n-NLS) equations with nonzero boundary conditions are studied. We first find the fundamental and higher-order vector Peregrine solitons (alias rational rogue waves (RWs)) for the n-NLS equations by using the loop group theory, an explicit $\left( n+1\right)$-multiple root of a characteristic polynomial of degree $(n+1)$ related to the Benjamin–Feir instability, and inverse functions. Particularly, the fundamental vector rational RWs are proved to be parity-time-reversal symmetric for some parameter constraints and classified into n cases in terms of the degree of the introduced polynomial. Moreover, a systematic approach is proposed to study the asymptotic behaviors of these vector RWs such that the decompositions of RWs are related to the so-called governing polynomials ${\mathcal {F}}_\ell (z)$, which pave a powerful way in the study of vector RW structures of the multi-component integrable systems. The vector RWs with maximal amplitudes can also be determined via the parameter vectors, which are interesting and useful in the study of RWs for multi-component nonlinear physical systems.

Appendices

Appendix A: The Proof of Lemma 1

Proof

Since the coefficient matrices ${\mathbf {U}}$ and ${\mathbf {V}}$ have the symmetric relationships:

$$\begin{aligned} {\mathbf {U}}(\lambda ;x,t)=-{\mathbf {U}}^{\dag }(\lambda ^*;x,t),\quad {\mathbf {V}}(\lambda ;x,t)=-{\mathbf {V}}^{\dag }(\lambda ^*;x,t), \end{aligned}$$

(134)

thus it follows that the matrix function $\Phi ^{\dag }(\lambda ^*;x,t)$ satisfies the adjoint Lax pair:

$$\begin{aligned} -\Phi _x^{\dag }(\lambda ^*;x,t)=\Phi ^{\dag }(\lambda ^*;x,t) {\mathbf {U}}(\lambda ;x,t),\quad -\Phi _t^{\dag }(\lambda ^*;x,t)=\Phi ^{\dag }(\lambda ^*;x,t) {\mathbf {V}}(\lambda ;x,t)\nonumber \\ \end{aligned}$$

(135)

if $\Phi (\lambda ;x,t)$ satisfies the Lax pair (4) and (5). On the other hand, the inverse matrix function $\Phi ^{-1}(\lambda ;x,t)$ also satisfies the adjoint Lax pair (135). By the uniqueness and existence of differential equations and $\Phi (\lambda ;0,0)={\mathbb {I}}$, we complete the proof. $\square$

Appendix B: The Proof of Lemma 2

Proof

The inverse matrix function ${\mathbf {N}}^{-1}(\lambda ;x,t)$ satisfies the adjoint linear spectral problem

$$\begin{aligned} \frac{\partial }{\partial x}\left( {\mathbf {N}}^{-1}(\lambda ;x,t)\right) =-{\mathbf {N}}^{-1}(\lambda ;x,t) {\mathbf {U}}(\lambda ;x,t), \end{aligned}$$

(136)

and then we have the following stationary zero-curvature equations:

$$\begin{aligned} \frac{\partial }{\partial x}\Theta (\lambda ;x,t)=\left[ {\mathbf {U}}(\lambda ;x,t),\Theta (\lambda ;x,t)\right] , \end{aligned}$$

(137)

by $\Theta (\lambda ;x,t)\equiv {\mathbf {N}}(\lambda ;x,t)\sigma _3{\mathbf {N}}^{-1}(\lambda ;x,t)={\mathbf {M}}(\lambda ;x,t)\sigma _3{\mathbf {M}}^{-1}(\lambda ;x,t).$ The above coefficients $\Theta _j(x,t)$ can be determined recursively:

$$\begin{aligned} \begin{aligned} \Theta _{j+1}^{\mathrm{off}}&=-\frac{1}{2}\sigma _3\left( \mathrm {i}\,\Theta _{j,x}^{\mathrm{off}}+\left[ {\mathbf {Q}}, \Theta _j^{\mathrm{diag}}\right] \right) , \\ \Theta _{j,x}^{\mathrm{diag}}&=\mathrm {i}\left[ {\mathbf {Q}}, \Theta _j^{\mathrm{off}}\right] . \end{aligned} \end{aligned}$$

(138)

On the other hand, it is readily to see that $\Theta ^2(\lambda ;x,t)={\mathbb {I}}$, which implies that

$$\begin{aligned} \Theta _j^{\mathrm{diag}}=-\frac{\sigma _3}{2}\sum _{s=1}^{j-1}\left( \Theta _j^{\mathrm{diag}}\Theta _{s-j}^{\mathrm{diag}}+\Theta _j^\mathrm{off}\Theta _{s-j}^{\mathrm{off}}\right) ,\qquad j\ge 2; \qquad \Theta _1^{\mathrm{diag}}=0. \end{aligned}$$

(139)

Through the first equation of (138) and equation (139), we complete the proof. $\square$

Appendix C.  MI Analysis of the Plane-Waves

Here, we utilize the squared eigenfunction method (Kaup 1976a, b to construct the solutions of linearized vector NLS equations. To this purpose, we depart from the stationary zero curvature equation:

$$\begin{aligned} \Psi _x=[{\mathbf {U}},\Psi ],\qquad \Psi _t=[{\mathbf {V}},\Psi ],\qquad \Psi =\begin{pmatrix} \Psi _{11} &{}\Psi _{12} \\ \Psi _{21} &{} \Psi _{22} \\ \end{pmatrix}, \end{aligned}$$

(140)

where the matrices $\Psi _{11}$, $\Psi _{12}$, $\Psi _{21}$ and $\Psi _{22}$ have the shape $1\times 1$, $1\times n$, $n\times 1$ and $n\times n$, respectively. Then, we write the corresponding component form:

$$\begin{aligned} \begin{aligned} \Psi _{11,x}=&\mathrm {i}\left( {\mathbf {q}}^{\dag } \Psi _{21}-\Psi _{12}{\mathbf {q}}\right) , \\ \Psi _{12,x}=&\mathrm {i}\left( 2\lambda \Psi _{12}+{\mathbf {q}}^{\dag }\Psi _{22}-\Psi _{11}{\mathbf {q}}^{\dag }\right) , \\ \Psi _{21,x}=&\mathrm {i}\left( {\mathbf {q}}\Psi _{11}-\Psi _{22}{\mathbf {q}}-2\lambda \Psi _{21}\right) , \\ \Psi _{22,x}=&\mathrm {i}\left( {\mathbf {q}}\Psi _{12}-\Psi _{21}{\mathbf {q}}^{\dag }\right) . \end{aligned} \end{aligned}$$

(141)

Further, taking the second-order derivative of $\Psi _{12}$ and $\Psi _{21}$ with respect to x, together with equation (141) we obtain that

$$\begin{aligned} \begin{aligned} \Psi _{12,xx}=&-2\left[ \left( \lambda {\mathbf {q}}^{\dag }-\frac{\mathrm {i}}{2}{\mathbf {q}}_x^{\dag }\right) \Psi _{22}-\Psi _{11}\left( \lambda {\mathbf {q}}^{\dag }-\frac{\mathrm {i}}{2}{\mathbf {q}}_x^{\dag }\right) +2\lambda ^2\Psi _{12}\right] \\&-{\mathbf {q}}^{\dag }\left( {\mathbf {q}}\Psi _{12}-\Psi _{21}{\mathbf {q}}^{\dag }\right) +\left( {\mathbf {q}}^{\dag }\Psi _{21}- \Psi _{12}{\mathbf {q}}\right) {\mathbf {q}}^{\dag }, \\ \Psi _{21,xx}=&2\left[ (\lambda {\mathbf {q}}+\frac{\mathrm {i}}{2}{\mathbf {q}}_x)\Psi _{11}-\Psi _{22}(\lambda {\mathbf {q}}+\frac{\mathrm {i}}{2}{\mathbf {q}}_x)-2\lambda ^2\Psi _{21}\right] \\&-\left( {\mathbf {q}}^{\dag } \Psi _{21}-\Psi _{12}{\mathbf {q}}\right) +\left( {\mathbf {q}}\Psi _{12}-\Psi _{21}{\mathbf {q}}^{\dag }\right) {\mathbf {q}}. \end{aligned} \end{aligned}$$

(142)

Similarly, for the t-part of stationary zero-curvature equations, we write the corresponding component form:

$$\begin{aligned} \begin{aligned} \Psi _{11,t}&=V_{12}\Psi _{21}-\Psi _{12}V_{21}, \\ \Psi _{12,t}&=V_{12}\Psi _{22}-\Psi _{11}V_{12}+V_{11}\Psi _{12}-\Psi _{12}V_{22}, \\ \Psi _{21,t}&=V_{21}\Psi _{11}-\Psi _{22}V_{21}+V_{22}\Psi _{21}-\Psi _{21}V_{11}, \\ \Psi _{22,t}&=V_{21}\Psi _{12}+V_{22}\Psi _{22}-\Psi _{21}V_{12}-\Psi _{22}V_{22}, \end{aligned} \end{aligned}$$

(143)

where

$$\begin{aligned} \begin{aligned} V_{11}=&\mathrm {i}\left( \lambda ^2-\frac{1}{2}{\mathbf {q}}^{\dag }\Lambda {\mathbf {q}}\right) , \quad V_{12}=\mathrm {i}\left( \lambda {\mathbf {q}}^{\dag }\Lambda -\frac{\mathrm {i}}{2}{\mathbf {q}}^{\dag }_x\Lambda \right) , \\ V_{21}=&\mathrm {i}\left( \lambda {\mathbf {q}}+\frac{\mathrm {i}}{2}{\mathbf {q}}_x\right) , \quad V_{22}=\mathrm {i}\left( \lambda ^2{\mathbb {I}}_n+\frac{1}{2}{\mathbf {q}}{\mathbf {q}}^{\dag }\Lambda \right) . \\ \end{aligned} \end{aligned}$$

(144)

Combining with equations (142) and (143), we will obtain the linearized equations:

$$\begin{aligned} \begin{aligned} \mathrm {i}\Psi _{21,t}&=-\left[ \frac{1}{2}\Psi _{21,xx}+\left( \Psi _{21}{\mathbf {q}}^{\dag }{\mathbf {q}}-{\mathbf {q}}\Psi _{12}{\mathbf {q}} +{\mathbf {q}}{\mathbf {q}}^{\dag }\Psi _{21}\right) \right] ,\\ \mathrm {i}\Psi _{12,t}&=\left[ \frac{1}{2}\Psi _{12,xx}+\left( \Psi _{12}{\mathbf {q}}{\mathbf {q}}^{\dag } -{\mathbf {q}}^{\dag }\Psi _{21}{\mathbf {q}}^{\dag }+{\mathbf {q}}^{\dag }{\mathbf {q}}\Psi _{12}\right) \right] . \end{aligned} \end{aligned}$$

(145)

Moreover, the symmetry relationship $\Psi _{12}=-\Psi _{21}^{\dag }$ guarantees the above linearized equation to satisfy the linearized multi-component NLS equations.

We construct the solutions of $\Psi$ by the wave-function of Lax pair. Suppose we have a vector solution $\phi _i(\lambda )$ for Lax pair, by the symmetric proposition we know that $\phi _j^{\dag }(\lambda ^*)$ satisfies the adjoint Lax pair, which implies that the matrix function $\Psi =\phi _i(\lambda )\phi _j^{\dag }(\lambda ^*)$ solves the stationary zero-curvature equations (140), where $\phi _i(\lambda )=[\phi _{i,1}(\lambda ), \phi _{i,2}^{\mathrm {T}}(\lambda )]^{\mathrm {T}}$. Similarly, we know that $\Psi =\phi _j(\lambda ^*)\phi _i^{\dag }(\lambda )$ also solves equation (145). Since equation (145) is unrelated with the spectral parameters $\lambda$. Thus, the solutions

$$\begin{aligned} \phi _{i,2}(\lambda )\phi _{j,1}^*(\lambda ^*)-\phi _{j,2}(\lambda ^*)\phi _{i,1}^*(\lambda ) \end{aligned}$$

(146)

solve the linearized multi-component NLS equations automatically.

In what follows, we consider how to use the above procedures to solve linear stability analysis for the multi-component NLS equation with the plane-wave solution (42). Suppose we consider the perturbation form:

$$\begin{aligned} \Psi _{21}=\begin{pmatrix} \left( g_1\mathrm {e}^{\mathrm {i}\xi ( x+\zeta t)}+f_1^*\mathrm {e}^{-\mathrm {i}\xi (x+\zeta ^* t)}\right) \mathrm {e}^{\mathrm {i}(b_1x+c_1t)}\\ \left( g_2\mathrm {e}^{\mathrm {i}\xi ( x+\zeta t)}+f_2^*\mathrm {e}^{-\mathrm {i}\xi (x+\zeta ^* t)}\right) \mathrm {e}^{\mathrm {i}(b_2x+c_2t)}\\ \vdots \\ \left( g_n\mathrm {e}^{\mathrm {i}\xi ( x+\zeta t)}+f_n^*\mathrm {e}^{-\mathrm {i}\xi (x+\zeta ^* t)}\right) \mathrm {e}^{\mathrm {i}(b_nx+c_nt)} \end{pmatrix}, \end{aligned}$$

(147)

and $\phi _i(\lambda )={\mathbf {G}}L_i(\lambda )\mathrm {e}^{\mathrm {i}[\xi _i(\lambda )-\lambda ]x+\eta _i(\lambda ) t}$ and $\phi _j(\lambda )={\mathbf {G}}L_j(\lambda )\mathrm {e}^{\mathrm {i}[\xi _j(\lambda )-\lambda ]x+\eta _j(\lambda ) t}$, where the vector $L_{i/j}(\lambda )$ is the eigenvector of ${\mathbf {H}}$ in equation (44), provides a solution

$$\begin{aligned} g_k=\frac{a_i}{\xi _i(\lambda )+b_k},\qquad f_k=-\frac{a_i}{\xi _j(\lambda )+b_k} \end{aligned}$$

(148)

where $k=1,2,\ldots , n$ and

$$\begin{aligned} \xi =\xi _i(\lambda )-\xi _j(\lambda )\in {\mathbb {R}},\qquad \zeta =\frac{\gamma }{2}(\xi _i(\lambda )+\xi _j(\lambda )) \end{aligned}$$

(149)

the parameter $\xi$ determines the perturbation frequency and the parameter $\zeta$ determines the gain index. As for the fixed $\xi$, the parameter $\xi _i(\lambda )$ can be determined by the following equations

$$\begin{aligned} 1+\sum _{k=1}^n\frac{s_ka_k^2}{(\xi _i(\lambda )+b_k)((\xi _i(\lambda )-\xi +b_k))}=0. \end{aligned}$$

(150)

For the multi-component NLS equations, the gain index $\zeta =\xi _i(\lambda )-\frac{\xi }{2}$ will solve the equations

$$\begin{aligned} 1+\sum _{k=1}^n\frac{s_ka_k^2}{(\zeta +b_k)^2-\frac{\xi ^2}{4}}=0. \end{aligned}$$

(151)

Appendix D.  Asymptotic Behaviors of ${\mathbf {q}}^{[1]_4}$

Case 1. We consider four simple roots of the governing polynomial ${\mathcal {F}}_4(z)$. For instance, as $\kappa _0=9, \kappa _2=-10, \kappa _1=\kappa _3=0$ are taken for the line-typed structure along x-axis, four roots of ${\mathcal {F}}_4$ are $z=\pm 1, \pm 3$. As $\kappa _0=9, \kappa _2=10, \kappa _1=\kappa _3=0$ are taken for the line-typed structure along t-axis, four roots of ${\mathcal {F}}_4$ are $z=\pm \mathrm {i}, \pm 3\mathrm {i}$. As $\kappa _0=1, \kappa _1=\kappa _2=\kappa _3=0$ are taken for the square-typed structure, four roots of ${\mathcal {F}}_4$ are $z=\mathrm {e}^{\left( 2s-1\right) \pi \mathrm {i}/4}, s=1, 2, 3, 4$. Then, we yield the formula of asymptotic behavior

$$\begin{aligned} q_j^{[1]_4}\!=\!q_j^\mathrm {bg}\bigg [1\!+\!\sum _{\delta _1=0}^1\sum _{\delta _2=0}^1R_j\left( x-x_{\delta _1, \delta _2}, t-t_{\delta _1, \delta _2}\right) \!\bigg ]\mathrm {e}^{-2\mathrm {i}\omega _j}\!+\!{\mathcal {O}}\left( \frac{1}{h^{2}}\!\right) \!, \,\, h^{2}\rightarrow +\infty .\nonumber \\ \end{aligned}$$

(152)

with the linear superposition of four single rogue waves in the leading term, where

• $x_{\delta _1, 0}=(-1)^{\delta _1}(h+(\kappa _{0, 0}+\kappa _{2, 0})/16\zeta )+(\kappa _{1, 0}+\kappa _{3, 0}-8)/16\zeta , \, x_{\delta _1, 1}=(-1)^{\delta _1}(3h-(9\kappa _{2, 0}+\kappa _{0, 0})/48\zeta )-(\kappa _{1, 0}+9\kappa _{3, 0}+8)/16\zeta , \, t_{\delta _1, \delta _2}=0$ for the case $\kappa _0=9, \kappa _2=-10, \kappa _1=\kappa _3=0$ (see Fig. 4a, b);

• $x_{\delta _1, 0}=-(\kappa _{1, 0}-\kappa _{3, 0}+4)/16\zeta , \, t_{\delta _1, 0}=(-1)^\delta (h+(\kappa _{0, 0}-\kappa _{2, 0})/16\zeta )/\zeta , \, x_{\delta _1, 1}=(\kappa _{1, 0}-9\kappa _{3, 0}+36)/16\zeta ,\, t_{\delta _1, 1}=(-1)^\delta (3h-(\kappa _{0, 0}-9\kappa _{2, 0})/48\zeta )/\zeta$ (see Fig. 4c, d);

• $x_{\delta _1, \delta _2}=(-1)^{\delta _1}(h/\sqrt{2}+\sqrt{2}(\kappa _{0, 0}-\kappa _{2, 0})/8\zeta )+(1-\kappa _{3, 0})/4\zeta , \, t_{\delta _1, \delta _2}=(-1)^{\delta _2}(h/\sqrt{2}+\sqrt{2}(\kappa _{0, 0}-\kappa _{2, 0})/8\zeta )/\zeta +(-1)^{\delta _2}(\kappa _{1, 0}+3)/4\zeta$ (see Fig. 4e, f).

Case 2. For the second case, we consider a double root and two simple roots of the governing polynomial ${\mathcal {F}}_4$, which arrange a line-typed structure. For instance, as $\kappa _2=-1, \kappa _0=\kappa _1=\kappa _3=0$ are taken, then the governing polynomial ${\mathcal {F}}_4$ has a double root $z=0$ and two simple root $z=\pm 1$. Under the constraint $\kappa _{0, 0}=0$, we deduce the formula of asymptotic behavior

$$\begin{aligned} q_j^{[1]_4}=q_j^\mathrm {bg}\left[ 1+p_{j, 2}\left( x, t\right) +\sum _{\delta =0}^1R_j\left( x-x_\delta , t\right) \right] \mathrm {e}^{-2\mathrm {i}\omega _j} +{\mathcal {O}}\left( \frac{1}{h}\right) , \,\, h\rightarrow +\infty ,\nonumber \\ \end{aligned}$$

(153)

where $x_\delta =(-1)^\delta (h-\kappa _{2, 0}/2\zeta )-(\kappa _{3, 0}+\kappa _{1, 0}+1)/2\zeta$ and the parameters of $p_{j, 2}$ are $\alpha _0=\kappa _{0, 1}, \alpha _1=\kappa _{1, 0}\mathrm {i}, \alpha _2=2\zeta$. The leading term is linear superposition of two single rogue waves located at $\left( x_{\delta }, 0\right)$ and a double rogue wave (see Fig. 5a, b).

Case 3. For the third case, we consider a two real double roots of the governing polynomial ${\mathcal {F}}_4$. For instance, we can take $\kappa _2=-2, \kappa _1=1, \kappa _0=\kappa _3=0$, then the governing polynomial ${\mathcal {F}}_4$ has two double real roots $z=\pm 1$. Under the constraint $\kappa _{3, 0}+\kappa _{1, 0}=\kappa _{2, 0}+\kappa _{0, 0}=0$, we derive the formula of asymptotic behavior

$$\begin{aligned} q_j^{[1]_4}=q_j^\mathrm {bg}\bigg [1+\sum _{\delta =0}^1p_{j, 2}\left( x+\left( -1\right) ^\delta h^{2}, t\right) +\bigg ]\mathrm {e}^{-2\mathrm {i}\omega _j} +{\mathcal {O}}\left( \frac{1}{h^{2}}\right) , \,\, h^{2}\rightarrow +\infty .\nonumber \\ \end{aligned}$$

(154)

where the parameters of $p_{j, 2}$ are $\alpha _0=\kappa _{0, 0}+\kappa _{1, 0}, \alpha _1=-2(\kappa _{0, 1}+\kappa _{1, 1}+\kappa _{2, 1}+\kappa _{0, 0}^2)\mathrm {i}, \alpha _2=4\zeta$. The leading term is linear superposition of two double rogue waves (see Fig. 5c, d).

Case 4. For the fourth case, we consider a simple root and a triple root of ${\mathcal {F}}_4$ and the corresponding rogue waves arrange in a line. As we take $\kappa _3=1, \kappa _0=\kappa _1=\kappa _2=0$, the governing polynomial ${\mathcal {F}}_4$ has a simple roots $z=-1$ and a triple root $z=0$. Under the constraint $\kappa _{0, 0}=\kappa _{1, 0}=0$, we deduce the formula of asymptotic behavior

$$\begin{aligned} q_j^{[1]_4}=q_j^\mathrm {bg}\left[ 1+p_{j, 3}\left( x, t\right) +R_j\left( x-x_0, t\right) \right] \mathrm {e}^{-2\mathrm {i}\omega _j} +{\mathcal {O}}\left( \frac{1}{h}\right) , \,\, h\rightarrow +\infty ,\nonumber \\ \end{aligned}$$

(155)

where $x_0=(2\kappa _{2, 0}-2\kappa _{3, 0}-1)/2\zeta -h$ and the parameters of $p_{j, 3}$ are $\alpha _0=\kappa _{0, 2}, \alpha _1=\mathrm {i}\kappa _{1, 1}, \alpha _2=-2\kappa _{2, 0}, \alpha _3=-6\mathrm {i}\zeta$. The leading term is linear superposition of a single rogue wave located at $\left( x_0, 0\right)$ and a triple rogue wave (see Fig. 5e, f).

Case 5. For the fifth case, we consider a quadruple root of ${\mathcal {F}}_4$. For the convenience of the representation of the leading term of asymptotic formula, we define

$$\begin{aligned} p_{j, 4}=-\frac{2\mathrm {i}\zeta }{b_j-\mathrm {i}\zeta }\frac{\left( n+1\right) L_jL_0^*-{\mathbf {L}}^\dag {\mathbf {L}}}{{\mathbf {L}}^\dag {\mathbf {L}}},\quad j=1, 2, \ldots , n, \end{aligned}$$

(156)

where ${\mathbf {L}}, L_j's, L_0$ are defined in Eqs. (65) and (67) with $\ell =4$. As we take $\kappa _0=\kappa _1=\kappa _2=\kappa _3=0$, the governing polynomial ${\mathcal {F}}_4$ has a quadruple root $z=0$. To obtain the rogue wave solution in the leading term of the asymptotic formula, the constraint $\kappa _{0, 0}=\kappa _{0, 1}=\kappa _{0, 2}=\kappa _{1, 0}=\kappa _{1, 1}=\kappa _{2, 0}=0$ is posed. Then, $q_j^{[1]_4}=q_j^\mathrm {bg}\left( 1+p_{j, 4}\right) \mathrm {e}^{-2\mathrm {i}\omega _j}$ with $\alpha _0=\kappa _{0, 3}, \alpha _1=\mathrm {i}\kappa _{1, 2}, \alpha _2=-2\kappa _{2, 1}, \alpha _3=-6\kappa _{3, 0}\mathrm {i}, \alpha _4=24\zeta$. For other quadruple root of ${\mathcal {F}}_4$, we also have the formula of asymptotic behavior as $h\rightarrow +\infty$. For instance, as $\kappa _0=1, \kappa _1=-4, \kappa _2=6, \kappa _3=-4$, the governing polynomial ${\mathcal {F}}_4=\left( z-1\right) ^4$, which has a quadruple root $z=1$. Under the constraint $3\kappa _{0, 0}=\kappa _{2,0}=-\kappa _{1,0}=-3\kappa _{3,0},2\kappa _{0,1}=-\kappa _{1, 1}=2\kappa _{2, 1},\kappa _{0,2}=-\kappa _{1,2}$, we derive the formula of asymptotic behavior

$$\begin{aligned} q_j^{[1]_4}=q_j^\mathrm {bg}\left[ 1+p_{j, 4}\left( x-h^{2}, t\right) \right] \mathrm {e}^{-2\mathrm {i}\omega _j}+{\mathcal {O}}\left( \frac{1}{h^{2}}\right) . \end{aligned}$$

(157)

where the parameters of $p_{j, 4}$ are $\alpha _0=\kappa _{0, 3}, \alpha _1=\mathrm {i}\kappa _{1, 2}, \alpha _2=-2\kappa _{2, 1}, \alpha _3=-6\kappa _{3, 0}\mathrm {i}, \alpha _4=24\zeta$.