Rogue wave patterns in the nonlinear Schrödinger equation

The name of rogue waves first appeared in oceanography, where it referred to large spontaneous and unexpected water wave excitations that are a threat even to big ships Ocean_rogue_review ; Pelinovsky_book . Later, their counterparts in optics were also reported Solli_Nature ; Wabnitz_book . Due to their physical importance, rogue waves have received intensive theoretical and experimental studies in the past decade. On the theoretical front, analytical expressions of rogue waves have been derived in a wide variety of integrable physical models, such as the nonlinear Schrödinger (NLS) equation for wave-packet propagation in the ocean and optical systems Benney ; Peregrine ; AAS2009 ; DGKM2010 ; KAAN2011 ; GLML2012 ; OhtaJY2012 , the derivative NLS equations for circularly polarized nonlinear Alfvén waves in plasmas and short-pulse propagation in a frequency-doubling crystal Kaup_Newell ; KN_Alfven1 ; Wise2007 ; KN_rogue_2011 ; KN_rogue_2013 ; YangDNLS2019 , the Manakov equations for light transmission in randomly birefringent fibers Menyuk ; BDCW2012 ; ManakovDark ; LingGuoZhaoCNLS2014 ; Chen_Shihua2015 ; ZhaoGuoLingCNLS2016 , and the three-wave resonant interaction equations Ablowitz_book ; BaroDegas2013 ; DegasLomba2013 ; ChenSCrespo2015 ; WangXChenY2015 ; ZhangYanWen2018 . On the experimental front, various rogue wave solutions in the NLS equation and defocusing Manakov equations have been observed in water tanks and optical fibers Tank1 ; Tank2 ; Fiber1 ; Fiber2 ; Fiber3 . In these experiments, intimate knowledge of theoretical rogue wave solutions in the underlying nonlinear wave equations was utilized, which highlights the importance of theoretical developments on rogue waves for practical rogue wave verifications and predictions.

The study of rogue wave patterns is important as this information allows for the prediction of later rogue wave events from earlier wave forms. Although graphs of low-order rogue waves have been plotted for many integrable equations, and simple patterns such as triangles and rings have been reported, richer patterns arising from high-order rogue wave solutions have received little attention. For the NLS equation, preliminary investigations on rogue wave patterns were made in KAAN2011 ; HeFokas ; KAAN2013 through Darboux transformation and numerical simulations. It was observed in KAAN2011 that if a $N$-th order rogue wave exhibits a single-shell ring structure, then the center of the ring is a $(N-2)$-th order rogue wave. This observation was explained analytically in HeFokas . In KAAN2013 , it was observed that NLS rogue patterns could be classified according to the order of the rogue waves and the parameter shifts applied to the Akhmediev breathers in the rogue-wave limit. This latter observation allowed the authors to extrapolate the shapes of rogue waves beyond order six, where numerical plotting of rogue waves became difficult. However, an analytical and quantitative prediction of NLS rogue patterns at arbitrary orders is still nonexistent.

In this article, we analytically investigate rogue wave patterns in the NLS equation at arbitrary rogue-wave orders. We show that if any internal parameter in the rogue waves (which controls the shape of initial small perturbations to the uniform background) is large, then these waves would exhibit clear geometric structures, which comprise Peregrine (fundamental) rogue waves forming shapes such as triangle, pentagon, heptagon and nonagon, with a possible lower-order rogue wave at the center. These rogue patterns are analytically predicted by the root structures of the Yablonskii-Vorob’ev polynomial hierarchy, and their orientations are controlled by the phase of the large internal parameter. We also show that if multiple internal parameters in the rogue waves are large but satisfy certain constraints, then the same rogue patterns would still hold. These results reveal a deep connection between rogue patterns and the Yablonskii-Vorob’ev polynomials, and drastically improve our analytical understanding and quantitative description of rogue events. As a small application of our analytical results, the numerical observation in KAAN2011 on single-shell ring structures is explained. Comparison between true rogue patterns and our analytical predictions is also presented, and excellent agreement is observed.

This paper is structured as follows. In Sec. 2, we present a simplified bilinear expression of general rogue waves in the NLS equation, as well as the Yablonskii-Vorob’ev polynomial hierarchy and its root structure, which we will utilize in later texts. In Sec. 3, we present our main theorems on rogue wave patterns when a single internal parameter in the rogue waves is large. In Sec. 4, we graphically illustrate rogue patterns under a large parameter and compare these patterns to our theoretical predictions. In Sec. 5, we prove the theorems stated in Sec. 3. In Sec. 6, we generalize our analytical results to cases where multiple internal parameters in the rogue waves are large but meet certain constraints. Sec. 7 concludes the paper.

The nonlinear Schrödinger (NLS) equation

$$\textrm{i}u_{t}+\frac{1}{2}u_{xx}+|u|^{2}u=0$$

(1)

arises in numerous physical situations such as water waves and optics Benney ; Ablowitzbook ; Kivsharbook . In this article, we consider its rogue wave solutions, which are rational solutions which approach a constant-amplitude continuous-wave background as $x,t\to\pm\infty$. Since this equation admits Galilean and scaling invariances, we can set the boundary conditions of these rogue waves as

$$u(x,t)\to e^{\textrm{i}t},\quad x,t\to\infty,$$

(2)

without any loss of generality.

Rogue wave solutions in Theorem 1 contain $N-1$ free internal complex parameters $a_{3},a_{5},\cdots,a_{2N-1}$. In this section, we consider asymptotics of these rogue solutions when one of these internal parameters is large, while the other parameters remain $O(1)$. Generalizations to cases where multiple internal parameters are large but satisfy certain constraints will be made in Sec. 6.

Suppose $a_{2m+1}$ is large, where $1\leq m\leq N-1$, and the other parameters are $O(1)$. Then our results on the large-$a_{2m+1}$ asymptotics of rogue waves in Theorem 1 are summarized by the following two theorems.

Theroem 3. Far away from the origin, with $\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, the $N$-th order rogue wave $u_{N}(x,t)$ in Eq. (3) separates into $N_{p}$ fundamental (Peregrine) rogue waves, where $N_{p}$ is given in Eq. (30). These Peregrine waves are $\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip 1.42271pte^{\textrm{i}t}$, where

$$\hat{u}_{1}(x,t)=1-\frac{4(1+2\textrm{i}t)}{1+4x^{2}+4t^{2}},$$

(31)

and their positions $(\hat{x}_{0},\hat{t}_{0})$ are given by

$\displaystyle\hat{x}_{0}+\textrm{i}\hskip 1.42271pt\hat{t}_{0}=z_{0}\left(-\frac{2m+1}{2^{2m}}a_{2m+1}\right)^{\frac{1}{2m+1}},$

(32)

with $z_{0}$ being any one of the $N_{p}$ simple nonzero roots of $Q_{N}^{[m]}(z)$. The error of this Peregrine wave approximation is $O(|a_{2m+1}|^{-1/(2m+1)})$. Expressed mathematically, when $\left[(x-\hat{x}_{0})^{2}+(t-\hat{t}_{0})^{2}\right]^{1/2}=O(1)$, we have the following solution asymptotics

$$u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})=\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip 1.42271pte^{\textrm{i}t}+O\left(|a_{2m+1}|^{-1/(2m+1)}\right),\quad|a_{2m+1}|\gg 1.$$

(33)

When $(x,t)$ is not in the neighborhood of any of these $N_{p}$ Peregrine waves, or $\sqrt{x^{2}+t^{2}}$ is larger than $O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, $u_{N}(x,t)$ asymptotically approaches the constant background $e^{\textrm{i}t}$ as $|a_{2m+1}|\to\infty$.

Theroem 4. In the neighborhood of the origin, where $\sqrt{x^{2}+t^{2}}=O(1)$, $u_{N}(x,t)$ is approximately a lower $N_{0}$-th order rogue wave $u_{N_{0}}(x,t)$, where $N_{0}$ is given in Theorem 2 with $0\leq N_{0}\leq N-2$, and $u_{N_{0}}(x,t)$ is given by Eq. (3) with its internal parameters $a_{3},a_{5},\cdots,a_{2N_{0}-1}$ being the first $N_{0}-1$ values in the parameter set $(a_{3},a_{5},\cdots,a_{2N-1})$ of the original rogue wave $u_{N}(x,t)$. The error of this lower-order rogue wave approximation $u_{N_{0}}(x,t)$ is $O(|a_{2m+1}|^{-1})$. Expressed mathematically, when $\sqrt{x^{2}+t^{2}}=O(1)$,

$$u_{N}(x,t;a_{3},a_{5},\cdots,a_{2N-1})=u_{N_{0}}(x,t;a_{3},a_{5},\cdots,a_{2N_{0}-1})+O\left(|a_{2m+1}|^{-1}\right),\quad|a_{2m+1}|\gg 1.$$

(34)

If $N_{0}=0$, then there will not be such a lower-order rogue wave in the neighborhood of the origin, and $u_{N}(x,t)$ asymptotically approaches the constant background $e^{\textrm{i}t}$ there as $|a_{2m+1}|\to\infty$.

These two theorems will be proved in Sec. 5.

Remark 1. Theorem 3 predicts that when $a_{2m+1}$ is large, the $N$-th order rogue wave (3) far away from the origin comprises $N_{p}$ Peregrine waves. The rogue pattern formed by these Peregrine waves has the same geometric shape as the root structure of the polynomial $Q_{N}^{[m]}(z)$, and thus this rogue pattern has $2\pi/(2m+1)$-angle rotational symmetry. The only difference between the predicted rogue pattern and the root structure of $Q_{N}^{[m]}(z)$ is a dilation and rotation between them due to the multiplication factor on the right side of Eq. (32). The angle of rotation is equal to the angle of the complex number $-a_{2m+1}$ divided by $2m+1$, and the dilation factor is equal to $[(2m+1)2^{-2m}|a_{2m+1}|]^{1/(2m+1)}$.

Remark 2. On the right side of Eq. (32), we can pick any one of the $(2m+1)$-th root of $-(2m+1)2^{-2m}a_{2m+1}$, because roots $z_{0}$ of the polynomial $Q_{N}^{[m]}(z)$ have $2\pi/(2m+1)$-angle rotational symmetry, see the comment in the paragraph below Theorem 2.

As a small application of the above two theorems, we explain the numerical observations in Ref. KAAN2011 . Under our bilinear rogue solution (3), a $N$-th order rogue wave exhibits a ring structure when $a_{2N-1}$ is large (see Fig. 2 in the next section). In this case, $m=N-1$, and $N_{0}=N-2$ from Eq. (29). Then, our theory predicts that the center of this $N$-th order rogue wave is a $(N-2)$-th order rogue wave, surrounded by $N_{p}=2N-1$ Peregrine waves that are evenly spaced on a ring due to the $2\pi/(2m+1)$-, i.e., $2\pi/(2N-1)$-angle rotational symmetry (see Remark 1 above). This is precisely what was observed in KAAN2011 .

In this section, we compare true rogue patterns with our analytical predictions. For this purpose, we first show in Fig. 2 true rogue wave solutions (3) from the 2rd to 7th order, with large $a_{3}$, $a_{5}$, $a_{7}$, $a_{9}$, $a_{11}$ and $a_{13}$ in the first to sixth columns respectively. The specific value of the large parameter in each panel of this figure is listed in Table 1, and the other parameters in each solution are chosen as zero.

It is seen that these rogue waves comprise a number of Peregrine waves forming triangular patterns for large $a_{3}$, pentagon patterns for large $a_{5}$, heptagon patterns for large $a_{7}$, nonagon patterns for large $a_{9}$, hendecagon (eleven-sided polygon) patterns for large $a_{11}$, and tridecagon (thirteen-sided polygon) patterns for large $a_{13}$. In the literature, patterns on the diagonal of Fig. 2 are sometimes called single-shell ring structures KAAN2011 . In addition to these Peregrine waves away from the origin, some of the rogue waves also contain a lower-order rogue wave at their centers. For instance, for the 7-th order rogue waves in the bottom row of Fig. 2, the first and fourth panels (with large $a_{3}$ and $a_{9}$ respectively) exhibit a Peregrine wave in their centers; the second panel (with large $a_{5}$) exhibits a second-order rogue wave in the center; the fifth panel (with large $a_{11}$) exhibits a third-order rogue wave in the center; and the last panel (with large $a_{13}$) exhibits a fifth-order rogue wave in the center. For our choices of parameters in rogue waves of Fig. 2, these lower-order rogue waves in the center are all super-rogue waves, i.e., rogue waves with the highest peak amplitude of their orders. We note by passing that the first five rows of rogue patterns in Fig. 2 resemble those plotted in Ref. KAAN2013 from Akhmediev breathers in the rogue-wave limit, although orientations between the two sets of patterns are very different.

Now, we compare these true rogue patterns in Fig. 2 with our analytical predictions. Our prediction $|u_{N}^{(p)}(x,t)|$ from Theorems 3 and 4 can be assembled into a simple formula,

$$\left|u_{N}^{(p)}(x,t)\right|=\left|u_{N_{0}}(x,t)\right|+\sum_{j=1}^{N_{p}}\left(\left|\hat{u}_{1}(x-\hat{x}_{0}^{(j)},t-\hat{t}_{0}^{(j)})\right|-1\right),$$

(35)

where $\hat{u}_{1}(x,t)$ is the Peregrine wave given in (31), their positions $(\hat{x}_{0}^{(j)},\hat{t}_{0}^{(j)})$ given by (32) with $z_{0}$ being every one of the $N_{p}$ simple nonzero roots of $Q_{N}^{[m]}(z)$, and $u_{N_{0}}(x,t)$ is the lower-order rogue wave in Eq. (34) whose internal parameters $(a_{3},a_{5},\cdots,a_{2N_{0}-1})$ are the first $N_{0}-1$ values in the parameter set $(a_{3},a_{5},\cdots,a_{2N-1})$ of the original rogue wave $u_{N}(x,t)$. For true rogue waves in Fig. 2, all internal parameters except for $a_{2m+1}$ were chosen as zero, and $N_{0}\leq m$ (see Theorem 2). Then, all internal parameters in the predicted lower-order rogue wave $u_{N_{0}}(x,t)$ at the origin are also zero.

Our predicted $(N_{p},N_{0})$ values for rogue waves of Fig. 2 are displayed in Table 2, where $m=1,2,\cdots,6$ correspond to large $a_{3},a_{5},\cdots,a_{13}$ respectively. These $(N_{p},N_{0})$ values provide our predictions for the number of Peregrine waves away from the origin $(x,t)=(0,0)$, as well as the order of the reduced rogue wave in the neighborhood of the origin. Visual comparison between Table 2 and Fig. 2 shows complete agreement.

We further compare our predicted whole solutions (35) with the true solutions of Fig. 2 for the same sets of $(a_{3},a_{5},\cdots)$ parameter values. These predicted whole solutions (35) are displayed in Fig. 3, with identical $(x,t)$ intervals as in Fig. 2’s true solutions. It is seen that the predicted patterns are strikingly similar to the true ones. In particular, since our predicted Peregrine locations (32) in the $(x,t)$ plane are given by all the non-zero roots of the Yablonskii-Vorob’ev polynomials $Q_{N}^{[m]}(z)$, multiplied by a fixed complex constant, predicted patterns formed by these Peregrine waves then are simply the root structures of these Yablonskii-Vorob’ev polynomials under certain rotation and dilation, as is evident by comparing predicted rogue waves in Fig. 3 to the Yablonskii-Vorob’ev root structures in Fig. 1 for $N=6$. These predicted Peregrine patterns clearly match the true ones in Fig. 2 very well. This visual agreement shows the deep connection between NLS rogue patterns and root structures of the Yablonskii-Vorob’ev hierarchy, as our theorem 3 predicts.

Regarding our predictions $u_{N_{0}}(x,t)$ for centers of the rogue waves $u_{N}(x,t)$ in Fig. 2, we can show that our bilinear rogue wave solution (3) in Theorem 1 with all internal parameters set as zero gives the super-rogue wave. This means that our predictions $u_{N_{0}}(x,t)$ for the centers of true rogue waves are all lower-order super-rogue waves, which agree with centers of true solutions shown in Fig. 2.

Theorem 3 reveals that the orientation of the rogue pattern formed by Peregrine waves is controlled by the phase of the large parameter $a_{2m+1}$. Specifically, the rogue-pattern orientation is the one of the root pattern of $Q_{N}^{[m]}(z)$ rotated by an angle of $\mbox{arg}(-a_{2m+1})/(2m+1)$, where “arg” represents the argument (angle) of a complex number. To check this prediction, we choose the 4-th order pentagon-shaped rogue waves, where $a_{5}$ is large and the other parameters are set as zero. For three choices of the $a_{5}$ value with the same modulus but different arguments, namely, $500e^{-\textrm{i}\pi/3}$, $500e^{\textrm{i}\pi/3}$ and $500e^{4\textrm{i}\pi/3}$, true rogue patterns from solutions (3) are displayed in the upper row of Fig. 4. As expected, orientations of these pentagon patterns indeed change as the argument of $a_{5}$ varies. Using our formula (32), predicted locations of Peregrine waves in the rogue pattern are shown in the lower row of Fig. 4. Comparison of the upper and lower rows of Fig. 4 shows that the predicted orientations are in perfect agreement with the true ones.

Next, we make quantitative comparisons between true rogue waves and our predictions for large $a_{2m+1}$, and verify the error decay rate of $O(|a_{2m+1}|^{-1/(2m+1)})$ for the prediction of Peregrine-wave locations far away from the origin in Theorem 3, and the error decay rate of $O(|a_{2m+1}|^{-1})$ for the prediction of the lower-order rogue wave at the center in Theorem 4.

For the quantitative comparison on Peregrine-wave locations away from the origin, we choose two patterns of 3rd-order rogue waves. One is a triangle pattern from large $a_{3}$, and we set arg$(a_{3})=-\pi/4$; and the other is a pentagon pattern from large $a_{5}$, and we set $a_{5}$ to be real positive. In each pattern, we choose all other parameters of the rogue wave solutions to be zero. These triangul and pentagon patterns are shown schematically in Fig. 5(a, c) respectively. In each of these two patterns, we pick one of its Peregrine waves, which is marked by an arrow, and quantitatively compare its true $(x_{0},t_{0})$ location with our analytical prediction (32) as $|a_{3}|$ or $|a_{5}|$ increases. Here, the true location of the Peregrine wave is defined as the $(x_{0},t_{0})$ location where this Peregrine wave attains its maximum amplitude, and the error of our asymptotic prediction $(\hat{x}_{0},\ \hat{t}_{0})$ in Eq. (32) is defined as

$$\mbox{error of Peregrine location}=\sqrt{\left(\hat{x}_{0}-x_{0}\right)^{2}+\left(\hat{t}_{0}-t_{0}\right)^{2}}.$$

These errors of Peregrine locations versus $|a_{3}|$ or $|a_{5}|$ are plotted as solid lines in panels (b) and (d) of Fig. 5 for the triangular and pentagon patterns respectively. For comparison, the decay rates of $|a_{3}|^{-1/3}$ and $|a_{5}|^{-1/5}$ are also displayed in these panels as dashed lines. We see that these errors of Peregrine locations indeed decay at the rate of $|a_{2m+1}|^{-1/(2m+1)}$, thus confirming the analytical error estimates (33) in Theorem 3.

To quantitatively compare our prediction in Theorem 4 on the lower-order rogue wave at the center with the true solution, we choose a fifth-order rogue wave $u_{5}(x,t)$ with large $a_{9}$ and the other internal parameters set as zero. This $|u_{5}(x,t)|$ solution with $a_{9}=-5000\textrm{i}$ is displayed in Fig. 6(a). The center region of this wave marked by a dashed-line box in panel (a) is amplified and replotted in panel (b). In the present case, $N=5$ and $m=4$. Since $5\equiv-4\hskip 2.84544pt\mbox{mod}\hskip 2.84544pt9$, we get $N_{0}=3$ from Eq. (29). Thus, according to Theorem 4, this $u_{5}(x,t)$ solution contains a 3rd-order rogue wave $u_{3}(x,t)$ in its center region, where all internal parameters $(a_{3},a_{5})$ in this $u_{3}(x,t)$ solution are zero. Such a $u_{3}(x,t)$ solution is a third-order super rogue wave. This predicted $|u_{3}(x,t)|$ solution is displayed in Fig. 6(c), with the same $(x,t)$ internals as in the true center-region solution displayed in panel (b). Visually, this predicted center solution in (c) is identical to the true center solution in (b). Quantitatively, we have also obtained the errors in our predicted solution $u_{3}(x,t)$ at $x=t=0.5$ of the center region as $a_{9}$ increases in magnitude with arg$(a_{9})=-\pi/2$. Our error is defined as

$$\mbox{error of center region prediction}=\left|u_{5}(x,t)-u_{3}(x,t)\right|_{x=t=0.5}.$$

The dependence of this error on $|a_{9}|$ is plotted in Fig. 6(d). Comparison of this error decay with the $|a_{9}|^{-1}$ decay [shown as a dashed line in panel (d)] indicates that this error is indeed of $O(|a_{9}|^{-1})$, confirming the error prediction (34) in Theorem 4.

In this section, we prove the analytical predictions on NLS rogue patterns in Theorems 3 and 4 of Sec. 3. Our proof is based on an asymptotic analysis of the rogue wave solution (3), or equivalently, the determinant $\sigma_{n}$ in Eq. (4), in the large $a_{2m+1}$ limit.

Proof of Theorem 3.   When $a_{2m+1}$ is large and the other parameters $O(1)$ in the rogue wave solution (3), at $(x,t)$ where $\sqrt{x^{2}+t^{2}}=O\left(|a_{2m+1}|^{1/(2m+1)}\right)$, by denoting

$$\lambda=a_{2m+1}^{-1/(2m+1)}$$

(36)

and recalling the expression of Schur polynomials in Eq. (9), we have

	$\displaystyle S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=S_{k}\left(x_{1}^{+},\nu s_{2},x_{3}^{+},\nu s_{4},\cdots\right)=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,\nu s_{2}\lambda^{2},x_{3}^{+}\lambda^{3},\nu s_{4}\lambda^{4},\cdots\right)$
	$\displaystyle\sim\lambda^{-k}S_{k}\left[(x+\textrm{i}t)\lambda,0,\cdots,0,1,0,\cdots\right]=S_{k}\left(x+\textrm{i}t,0,\cdots,0,a_{2m+1},0,\cdots\right).$		(37)

Thus,

$$S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})\sim S_{k}(\textbf{v}),\quad\quad|a_{2m+1}|\gg 1,$$

(38)

where

$$\textbf{v}=(x+\textrm{i}t,0,\cdots,0,a_{2m+1},0,\cdots).$$

(39)

From the definition of Schur polynomials (8), $S_{k}(\textbf{v})$ is given by

$$\sum_{k=0}^{\infty}S_{k}(\textbf{v})\epsilon^{k}=\exp\left[(x+\textrm{i}t)\epsilon+a_{2m+1}\epsilon^{2m+1}\right].$$

(40)

Thus, it is related to the polynomial $p_{k}^{[m]}(z)$ in (20) as

$$S_{k}(\textbf{v})=A^{k/(2m+1)}p_{k}^{[m]}(z),$$

(41)

where

$$A=-\frac{2m+1}{2^{2m}}a_{2m+1},\hskip 8.5359ptz=A^{-1/(2m+1)}(x+\textrm{i}t).$$

(42)

Using these formulae, we find that

$$\det_{1\leq i,j\leq N}\left[S_{2i-j}(\textbf{\emph{x}}^{+}(n)+j\textbf{\emph{s}})\right]\sim c_{N}^{-1}A^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(z),\quad\quad|a_{2m+1}|\gg 1.$$

(43)

Similarly,

$$\det_{1\leq i,j\leq N}\left[S_{2i-j}(\textbf{\emph{x}}^{-}(n)+j\textbf{\emph{s}})\right]\sim c_{N}^{-1}\left(A^{*}\right)^{\frac{N(N+1)}{2(2m+1)}}Q_{N}^{[m]}(z^{*}),\quad\quad|a_{2m+1}|\gg 1.$$

(44)

Hereafter, $S_{k}=0$ when $k<0$.

To proceed further, we use determinant identities and the Laplace expansion to rewrite $\sigma_{n}$ in Eq. (4) as OhtaJY2012

$\displaystyle\sigma_{n}=\sum_{0\leq\nu_{1}<\nu_{2}<\cdots<\nu_{N}\leq 2N-1}\det_{1\leq i,j\leq N}\left[\frac{1}{2^{\nu_{j}}}S_{2i-1-\nu_{j}}(\textbf{\emph{x}}^{+}(n)+\nu_{j}\textbf{\emph{s}})\right]\times\det_{1\leq i,j\leq N}\left[\frac{1}{2^{\nu_{j}}}S_{2i-1-\nu_{j}}(\textbf{\emph{x}}^{-}(n)+\nu_{j}\textbf{\emph{s}})\right].$

(45)

Since the highest order term of $a_{2m+1}$ in this $\sigma_{n}$ comes from the index choices of $\nu_{j}=j-1$, then

$$\sigma_{n}\sim|\alpha|^{2}\hskip 1.42271pt|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}\left|Q_{N}^{[m]}(z)\right|^{2},\quad\quad|a_{2m+1}|\gg 1,$$

(46)

where

$$\alpha=2^{-N(N-1)/2}c_{N}^{-1}\left(-\frac{2m+1}{2^{2m}}\right)^{\frac{N(N+1)}{2(2m+1)}}.$$

(47)

Since $\alpha$ is independent of $n$, the above equation shows that for large $a_{2m+1}$, $\sigma_{1}/\sigma_{0}\sim 1$, i.e., the solution $u(x,t)$ is on the unit background, except at or near $(x,t)$ locations $\left(\hat{x}_{0},\hat{t}_{0}\right)$ where

$$z_{0}=A^{-1/(2m+1)}(\hat{x}_{0}+\textrm{i}\hat{t}_{0})$$

(48)

is a root of the polynomial $Q_{N}^{[m]}(z)$, and such $\left(\hat{x}_{0},\hat{t}_{0}\right)$ locations are given by Eq. (32) in view of Eq. (42).

Next, we show that when $(x,t)$ is in the neighborhood of each of the $\left(\hat{x}_{0},\hat{t}_{0}\right)$ locations given by Eq. (32), i.e., when $\left[(x-\hat{x}_{0})^{2}+(t-\hat{t}_{0})^{2}\right]^{1/2}=O(1)$, the rogue wave $u_{N}(x,t)$ in Eq. (3) approaches a Peregrine wave $\hat{u}_{1}(x-\hat{x}_{0},t-\hat{t}_{0})\hskip 1.42271pte^{\textrm{i}t}$ for large $a_{2m+1}$. The asymptotic analysis above indicates that when $(x,t)$ is in the neighborhood of $\left(\hat{x}_{0},\hat{t}_{0}\right)$, the highest power term $|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}$ in $\sigma(x,t)$ vanishes. Thus, in order to determine the asymptotics of $u_{N}(x,t)$ in that $(x,t)$ region, we need to derive the leading order term of $a_{2m+1}$ in Eq. (45) whose order is lower than $|a_{2m+1}|^{\frac{N(N+1)}{2m+1}}$. For this purpose, we notice from Eq. (5) that when $(x,t)$ is in the neighborhood of $\left(\hat{x}_{0},\hat{t}_{0}\right)$, we have a more refined asymptotics for $S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})$ as

	$\displaystyle S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=\lambda^{-k}S_{k}\left(x_{1}^{+}\lambda,0,\cdots,0,1,0,\cdots\right)\left[1+O(\lambda^{2})\right]$
	$\displaystyle\hskip 59.75095pt=S_{k}\left(x_{1}^{+},0,\cdots,0,a_{2m+1},0,\cdots\right)\left[1+O(\lambda^{2})\right],$		(49)

i.e.,

$$S_{k}(\textbf{\emph{x}}^{+}(n)+\nu\textbf{\emph{s}})=S_{k}(\hat{\textbf{v}})\left[1+O\left(a_{2m+1}^{-2/(2m+1)}\right)\right],$$

(50)

where

$$\hat{\textbf{v}}=(x+\textrm{i}t+n,0,\cdots,0,a_{2m+1},0,\cdots).$$

(51)

The polynomials $S_{k}(\hat{\textbf{v}})$ are related to $p_{k}^{[m]}(z)$ in (20) as

$$S_{k}(\hat{\textbf{v}})=A^{k/(2m+1)}p_{k}^{[m]}(\hat{z}),$$

(52)

where $A$ is as given in Eq. (42), and $\hat{z}=A^{-1/(2m+1)}(x+\textrm{i}t+n)$.