## Abstract

Rogue waves can appear in optical fibers and other optical systems as well as in natural events like water waves. Their mathematical description is based on partial differential equations that have solutions that are localized both in time and space. One example is the “Peregrine” solution of the nonlinear Schrödinger equation (NLSE). When higher-order terms in the equation are involved, the solution becomes distorted, but its main features remain localized in space and time. Although exact solutions are not obtained in all cases, approximations that describe the solutions with reasonable accuracy do exist. Here, we consider approximate rogue wave solutions of the NLSE with an optically relevant Raman delay term.

© 2018 Optical Society of America

## 1. INTRODUCTION

Rogue waves were first found in optics as high-amplitude pulses in supercontinuum generation [1]. However, these are not the only type of rogue waves in optics. They can also be found in the output of laser radiation [2–6] and in other types of optical cavities [7–10]. They can be influenced by Brillouin scattering [11] or by the Raman effect [12]. Indeed, there is a multiplicity of forms of rogue waves [13–17], and this can make it difficult to classify them at present. Mathematically, the simplest type can be represented in the form of the “Peregrine” solution of the nonlinear Schrödinger equation (NLSE) [18]. This is a solution that is localized both in time and in space. This fundamental solution can be extended to cover more complicated cases [19–22]. Rogue waves involving partial nonlinearity have been studied [23], and effects in media with decreasing dispersion or diffraction have been considered in [24]. Akhmediev breathers and rogue waves can also appear in parity-time-symmetric coupled waveguides [25]. However, presenting rogue wave solutions in exact form is not always possible.

Light propagation in optical fibers is subject to various higher-order effects [26,27], such as third-order dispersion, ${\psi}_{ttt}$; fourth-order dispersion, ${\psi}_{tttt}$; a “quintic” adjustment to the Kerr nonlinearity, ${|\psi |}^{4}\psi $; and the Raman downshift effect, ${\tau}_{R}\psi ({|\psi |}^{2}{)}_{t}$. Similarly, water wave propagation in “deep” water can be influenced by assorted physical phenomena [28], such as, wind, viscosity, and bottom friction, introducing terms like ${\psi}_{ttt}$, ${|\psi |}^{2}{\psi}_{t}$, and ${|\psi |}^{1-\sigma}\psi $. These higher-order effects can potentially influence the dynamics of rogue waves [29] in ways that differ from their effect on solitons. Such impacts can significantly change the detected shapes. In this work, we consider the effects of Raman delay and other disturbances on the shape of rogue waves, and, where possible, compare the results with predictions from other possible approaches.

## 2. BASIC EQUATIONS

We start with the extended NLSE that describes propagation of ultrashort pulses in optical fibers:

where $R$ indicates the additional terms in the formEquation (1) in general is not integrable and cannot be solved exactly. Therefore, we have a choice of finding approximate solutions. Among the techniques that allows us to do that we can list the Lagrangian approach and the “method of moments.” The Lagrangian technique was introduced into soliton theory by Anderson [30]. It can be used to understand the effects of various disturbances on a known system. The method of moments [31] can also be used for reducing the dimensionality of soliton dynamical systems. The Lagrangian approach is based on minimization of “action.” It can give results which are similar to those obtained by the method of moments [32]. The two methods were compared in regard to the soliton solutions of the cubic–quintic complex Ginzburg–Landau equation in [33]. They have also been used to study self-focusing and self-defocusing two-dimensional beams in dissipative media [34].

It is less obvious that the Lagrangian approach can be applied to rogue wave solutions which are localized both in space and time, in contrast to soliton solutions that are localized only in either time or space. In this work, we show that the technique can be applied to rogue waves when the exact solutions may not be available. In particular, it can be applied to nonintegrable equations such as higher-order extensions of the NLSE.

The use of the variational integral and Euler equations has been explained in [35], where the Lagrange density is introduced and its relation to the Euler–Lagrange equations for the Schrödinger equation and other conservative equations is also given. We employ the Lagrangian

where ${L}_{d}$ is the Lagrangian density, and we haveIf we apply the Euler–Lagrange equations to Eq. (3), we obtain

For a trial solution containing several parameters ${c}^{(j)},j=1,2\dots $, the standard variational approach can be modified to allow for dissipative terms [32,33,36,37]. In this way, we obtain a separate equation

## 3. SIMPLE EXAMPLES

#### A. Peregrine Rogue Wave

In order to illustrate the application of the above technique to solutions which are localized both in time and space, we first apply it to the simplest case, namely, the lowest-order rational solution of the NLSE. Thus, we first consider the case $R=0$, that is, the basic NLSE,

and illustrate the idea using its known exact solution. Namely, we write the solution in the form of the trial function with the intention of finding the unknown function, $c(x)$.Following Eq. (3), we first write the Lagrangian,

Equation (4) allows us to write the equation relative to the unknown function $c(x)$:

which results in givingClearly, this is the well-known rogue wave solution of the NLSE [19]. In this case, the result is exact, as we deal with the integrable NLSE and we knew the form of the solution in advance. In general, the solutions are expected to be approximate.

#### B. “Moving” Rogue Wave

As a more complicated example, we consider a rogue wave with an internal “velocity” or “skewness.” From previous experience, we know that this occurs for rogue waves of the Hirota equation [20] and for a multiplicity of extended equations with odd higher-order terms [38]. In order to describe this type of rogue wave, we take the solution in the form

where and with the function ${t}_{0}(x)$ being responsible for the lateral motion.We now need to find $L$ and then the four unknown functions $a(x),{t}_{0}(x),r(x),\theta (x)$. We find:

We obtain the four ordinary differential equations (ODEs), and then solve the $a(x)$ equation by setting $a(x)={t}_{{0}^{\prime}}(x)$. The first (i.e., $\theta $) equation found is

and we use its simplest solution, namely, $r(x)=4{x}^{2}+1$.Then the only remaining nonzero equation is

Now ${t}_{0}^{\prime}(x)$ is a velocity, and the simplest solution is to take it as an arbitrary constant velocity, $v$, so ${t}_{0}^{\prime}(x)=v$. Hence ${t}_{0}(x)=vx$ and $\theta (x)=-{v}^{2}x/2$, so we have

The Galilean transformation of the NLSE is well known {e.g., see Eq. (2.6) of [39]}. The solution given by Eq. (11) agrees with that found by applying such a transform to the basic rogue wave of Eq. (6) in Section 3.A. It reduces to that lowest-order rogue wave when $v=0$. We can see that the Lagrangian approach works well for these simple illustrative cases. This encourages us to move up to more complicated cases, as given below.

## 4. MORE GENERAL FORM OF THE TRIAL FUNCTION

As a next step, we allow for nonzero functional R in the trial function. One of the consequences may be resizing of the rogue wave, say, in the $x$-direction. In order to take this into account, we extend the result of Eq. (10) to allow for a stretching factor, $B$, by taking

with ${D}_{m}(x,t)$ still given by Eq. (8) and ${F}_{m}(x,t)$ given by Eq. (9). Now, we apply the Euler–Lagrange formalism to the more general trial function (12).For convenience, we split ${L}_{d}$ into three parts:

whereWe need to integrate these terms over an infinite range in ${y}_{1}\equiv t-{t}_{0}(x)$. The terms in odd powers of ${y}_{1}$ will integrate out to zero. In the remaining part of ${L}_{d1}$, a part is subtracted so that its limit is zero for high $|{y}_{1}|$. Then, taking ${L}_{j}={\int}_{-\infty}^{\infty}{L}_{dj}\mathrm{d}t$, we have $L={L}_{1}+{L}_{2}+{L}_{3}$.

Taking the limit ${{}_{{y}_{1}\to \infty}{}^{\mathrm{lim}}L}_{d1}$ shows that it equals to

with unknown functions $a(x),{t}_{0}^{\prime}(x),{\theta}^{\prime}(x)$. Setting this to zero shows that the constants needed here are ${a}_{1}={h}_{1}={u}_{1}=1$. Now, ${L}_{d2}$ clearly approaches zero for large $t$ or ${y}_{1}$, and so needs no level adjustment. For ${L}_{d3}$, we note that ${}_{{y}_{1}\to \infty}{}^{\mathrm{lim}}|\psi |=1$, hence we subtract this background before integrating. We then findUsing Eq. (4), we define the left-hand-side terms of the equation relating to parameter ${c}^{(j)}$ as

So, using Eq. (13), the $a$ term is

The $\theta (x)$ term is

The $r(x)$ term is

Using Eq. (4), we define the influence of any functional $R$ on parameter ${c}^{j}$ as

Once $R$ is given, we thus need to solve the four ODEs, $S[{c}^{j}]=J[{c}^{j};R]$, with ${c}^{j}=\{a(x),{t}_{0}(x),r(x),\theta (x)\}$.

## 5. MORE EXAMPLES OF APPLICATION

#### A. Rogue Waves of Full Hirota Equation

The “full” Hirota equation [40] is

The right-hand sides of the reduced dynamical system (4) for this equation take the form

Setting $r(x)=1+4{B}^{2}{x}^{2}$ reduces the dynamical system and leaves only the equation for $r$, namely $S[r]=J[r;{R}_{h}]$, to solve. It is straightforward to show that it can be solved with $a(x)=k$, an arbitrary constant, $B=6{\alpha}_{3}k+1$, $\theta (x)=x{c}_{0}$, and ${t}_{0}=-{v}_{s}x$, where ${c}_{0}={k}^{2}({c}_{1}+{c}_{2}{\alpha}_{3}k)$ while

Hence,

This equation must be valid for all $k$ and all ${\alpha}_{3}$. Consequently, we obtain ${c}_{1}=-\frac{1}{2}$, ${v}_{1}=6$, and ${c}_{2}=-2$. Then, the functions ${t}_{0}(x)$ and $\theta (x)$ in the trial function (12) become

This outcome agrees with the known exact result (with ${\alpha}_{2}=\frac{1}{2}$) found in [38]. The velocity factor, ${t}_{0}^{\prime}(x)$, can be zero for some combinations of $k$ and ${\alpha}_{3}$, thus cancelling the effect of higher-order terms in the Hirota equation.

#### B. Rogue Wave of Lakshmanan—Porsezian—Daniel Equation

The even higher-order Lakshmanan—Porsezian—Daniel equation [41,42] is

whereFor this equation, the right-hand sides of the dynamical system (4) are

Again, setting $r(x)=1+4{B}^{2}{x}^{2}$ reduces the system and leaves only the $r$ equation, $S[r]=J[r;{R}_{l}]$, to solve. It is straightforward to show that it can be solved with $a(x)=k$, an arbitrary constant, $B={\alpha}_{4}({b}_{1}{k}^{2}+{b}_{2})+{b}_{0}$, $\theta (x)=x\{3{\alpha}_{4}[{({k}^{2}-2)}^{2}+{n}_{2}]+{k}^{2}{n}_{1}\}$, and ${t}_{0}=-{v}_{e}x$, where ${v}_{e}=k[{\alpha}_{4}({j}_{1}+{j}_{2}{k}^{2})-1]$.

Expanding $S[r]=J[r;{R}_{l}]$, we readily find the constants, ${b}_{0}=1$, ${b}_{1}=-12$, ${b}_{2}=12$, ${j}_{1}=-24$, ${j}_{2}=4$, ${n}_{1}=-1/2$, ${n}_{2}=-6$. Hence $B=1-12{\alpha}_{4}({k}^{2}-1)$, and

This outcome also agrees with the known exact result (with ${\alpha}_{2}=\frac{1}{2}$) found in [38]. Again, the velocity factor, ${t}_{0}^{\prime}(x)$, is zero for some combinations of $k$ and ${\alpha}_{4}$.

## 6. SOLITON UNDER INFLUENCE OF RAMAN DELAY

The previous examples are related to integrable equations in order for us to be able to compare the results of the variational approach with exact solutions. However, the technique itself is fully justified only when there is no possibility of obtaining exact solutions. Thus, below, we consider more involved cases relevant to practical situations. First, to show the principle, we consider a soliton solution under the influence of Raman delay. This has been done earlier in [43,44]. Suppose $R={\tau}_{R}\psi ({|\psi |}^{2}{)}_{t}\equiv {R}_{r}$ is the only nonzero term in Eq. (2). Various approximations have been given to estimate the soliton shape for small delay, ${\tau}_{R}$, in this case [45–47]. The delay makes the soliton solution nonsymmetric in $t$, and this requires a new trial function. Here we take

The approximating procedure described in Section 1 leads to five ODEs. The $\theta (x)$ equation is

showing that $Q(x)={Q}_{0}$ is a constant. The $a(x)$ equation is showing that $a(x)={t}_{0}^{\prime}(x)$. The $w(x)$ equation is showing that $w(x)=2/{Q}_{0}={w}_{0}$. Finally, the ${t}_{0}(x)$ equation is showing thatThis result for time shift is in agreement with Eqs. (5b) and (13) in [44]. Plainly, ${t}_{0}(x)$ is a close analogue of vertical projectile motion under gravity, where the velocity ${t}_{0}^{\prime}(x)$ changes sign at $x=0$, but the acceleration, ${t}_{0}^{\prime \prime}(x)=\frac{8{\tau}_{R}}{15{w}_{0}^{4}}$, does not. This as expected, since the acceleration is caused by the Raman delay and this does not change sign.

The remaining $Q(x)$ equation is

showing that the rate of change of phase is ${\theta}^{\prime}(x)=-\frac{1}{2{w}_{0}^{2}}-\frac{1}{2}{(\frac{8{\tau}_{R}x}{15{w}_{0}^{4}})}^{2}$, as in [43] (where $\theta =-\varphi $). If ${\tau}_{R}=0$, we obtain## 7. ROGUE WAVE UNDER INFLUENCE OF RAMAN DELAY

Now, instead of a soliton, we consider a rogue wave solution which is localized in time. For this, we use the same trial function as in Eqs. (7), (8), and (9).

The ODEs will naturally differ from the previous cases. The $\theta (x)$ equation is

To solve it, we set $z=\sqrt{X}=\sqrt{1+4{x}^{2}}$ and $r(z)={s}^{2}(z)$. This gives

We find that the solution is

where where $c$ is an arbitrary real solution parameter.Note that this remains real even when $c$ is small. In fact, ${\mathrm{lim}}_{c\to 0}s(z)=z$, so ${\mathrm{lim}}_{c\to 0}r(z)={z}^{2}$. Also, $r(x)$ is even in $c$, so we only need to use $c\ge 0$. This means that $r(0)\le 1$ for any $c$. We plot the function $r(x)$ in Fig. 3.

Since $c$ is generally small ($c<0.1$), it is convenient to expand this result. For $c\ge 0$:

That is,

So, when $c=0$, the solution reduces to the known NLSE result, $r(x)=(1+4{x}^{2})$. In solving the four ODEs, we take $c$ to be arbitrary but larger than ${\tau}_{R}$.

Using Eq. (4) with $S[{t}_{0}]$ from Eq. (17) and $J[{t}_{0};{R}_{r}]$ from Eq. (19), for ${t}_{0}(x)$, we find

so the ${t}_{0}(x)$ equation isThe $r(x)$ equation is

Now, if we take $r(x)=4{x}^{2}+1$ to solve the last equation, then the second one cannot be solved. So, we need to take $a(x)={t}_{0}^{\prime}(x)$ to solve it. This reduces Eq. (39) to

We can solve the resultant ODEs, namely, Eqs. (33), (42), and (43), to various orders in ${\tau}_{R}$. We take $c$ and ${\tau}_{R}$ to be small and have similar order (around ${10}^{-3}$). We assume that the offset, $c$, is not much smaller than ${\tau}_{R}$, so we cannot have ${\tau}_{R}/c\gg 1$. This excludes $c=0$. In fact, we expect $c>{\tau}_{R}$.

Using Eq. (38), we have

Then

Now, taking ${t}_{0}(0)=0$, we obtain (see Fig. 4)

So, for small $x$, we have

In the soliton case, we also found that ${t}_{0}(x)$ varied as ${\tau}_{R}{x}^{2}$. If $c$ is small, say $c<0.1$, then

Hence, we obtain the following simple form for the acceleration:

Thus, the acceleration, ${t}_{0}^{\prime \prime}(x)$, of the ridge takes the same sign as that for the soliton case found in Eq. (27) and is also an even function, but it has a more complicated form. As for the soliton case, it is proportional to ${\tau}_{R}$. The estimates for the function ${t}_{0}(x)$ are shown in Fig. 5 along with the results of numerical simulations for the same set of parameters. As can be seen from this figure, numeric solution of the four ODEs give results which are reasonably close to the above approximate solutions for small ${\tau}_{R}$, say ${\tau}_{R}<0.002$.

The Raman effect causes a delay, so we expect deceleration. It is analogous to the deceleration found for the Raman soliton in Eq. (28). Examples of evolution are shown in Figs. 6 and 7. Acceleration here is clearly seen.

Now, $\frac{{t}_{0}^{\prime \prime}(0)}{{\tau}_{R}}=\frac{2}{c}(6+66c+215{c}^{2})$, and we define this to be ${c}_{0}$. On plotting the acceleration, we see that it plainly has a Gaussian-type form. We find that it can be approximated by

Integrating shows that

The comparisons in Fig. 8 show that this is a convenient and accurate approximation, giving a simple view of the ridge offset during propagation.

So, for small $x$,

For ${c}_{0}=\frac{2}{c}(6+66c+215{c}^{2})$, we find that ${c}_{0}$ reaches its minimum of 275 when $c=0.17$, and $275<{c}_{0}<295$ for $0.1<c<0.28$. For ${c}_{0}=\frac{2}{c}(6+66c)$, we find that ${c}_{0}=250$ when $c=0.1$, and then ${c}_{0}$ decreases monotonically as $c$ increases. Now ${t}_{0}(x)\approx \frac{{c}_{0}}{2}{\tau}_{R}{x}^{2}$, so we need ${\tau}_{R}<c$ to get realistic (not too high). For typical values of $c$, say $c=0.2$, we have ${c}_{0}$ around 250, and hence

for small $x$.Equations (46)–(48) are valid for all $x$; thus position ${t}_{0}(x)$ and acceleration ${t}_{0}^{\prime \prime}(x)$ are even functions of $x$, while velocity ${t}_{0}^{\prime}(x)$ is an odd function in $x$. It thus corresponds to a more complicated form than the motion under gravity of Section 6, as here the analogous particle would be subject to a force proportional to $\frac{2{\tau}_{R}}{c}[6{X}^{-5/2}+66c{X}^{-2}+215{c}^{2}{X}^{-3/2}]$. From Eq. (49), this force varies roughly as ${c}_{0}{\tau}_{R}{e}^{-6{x}^{2}}$. It approaches zero for large $|x|$ but does not change sign.

Now, let us turn to the exponent term $\theta (x)$. Setting $\theta (0)=0$, we have

The expansion,

agrees with the results from the partial differential equation (PDE) simulations. Now where $X=1+4{x}^{2}$. Plainly, this is almost constant for small $x$.In the next section, we plot ${t}_{0}(x)/{\tau}_{R}$ for various $c$ on the same diagram. We also plot $\theta (x)/c$ and $(r(x)-1-4{x}^{2})/c$ to estimate the next term in $r(x)$.

## 8. COMPARISON WITH NUMERICAL SIMULATIONS OF PDE

For $x=0$, we have $a(x)=\theta (x)=0$, so $\psi $ is real. It maximum value, ${\psi}_{m}$, occurs when $t=0$ and is given by ${\psi}_{m}=\frac{4}{r(0)}-1\approx 3+8c$. As noted earlier, this is a function of $c$ only, as $r(0)=1-2c$. Using the model, at fixed $x$, the intensity, $I$, is

We note that $r(x)$ is close to $1+4{x}^{2}$ for small $c$. By setting $\frac{\partial I}{\partial t}=0$, we find that there are three stationary points. The first is at $t={t}_{0}(x)$, and it is a maximum since $\frac{{\partial}^{2}I}{\partial {t}^{2}}<0$, as $4+16{x}^{2}\gg r(x)$. On the either side, we have $t={t}_{0}(x)\pm \frac{1}{2}\sqrt{16{x}^{2}+4-r(x)}$. These are minima, since $\frac{{\partial}^{2}I}{\partial {t}^{2}}>0$ for each.

In the simulations of the PDE we can start with initial conditions with $r(0)$ close to 1 and ${t}_{0}(0)=\theta (0)=0$. At each $x$, the value of ${t}_{0}$ giving the maximum intensity is then labelled as ${t}_{0}(x)$. This shows the dependence of ${t}_{0}(x)$ on $p$. At this point, we use the intensity maximum, ${I}_{m}=\frac{{[r(x)-4]}^{2}+64{x}^{2}}{r{(x)}^{2}}$, to find $r(x)$ by solving the quadratic equation. This shows the dependence of $r(x)$ on $p$. At each such point, we have $x$ and $r(x)$ and the complex number $\psi (x,t={t}_{0}(x))=[\frac{4(1+2ix)}{r(x)}-1]{e}^{i[x-\theta (x)]}$; we directly find $\theta (x)$. So we can compare the PDE results with the predicted results found by solving the four ODEs approximately. Using the numerical PDE runs, we now find how $r(x)$, ${t}_{0}(x)$, and $\theta (x)$ depend on $c$ and ${\tau}_{R}$.

From numerics, we can plot $r(x)-1-4{x}^{2}$ for various values of $c$, and then observe if it can be expressed as ${c}^{n}f(x)$ for some integer $n$, that is, whether the dependence on $c$ and $x$ can be separated for a suitably chosen value of $n$. Results show that $n=1$ with $f(x)=-2{(1+4{x}^{2})}^{k}$ works well for $k=1.865$, thus roughly supporting the result of Eq. (38), which had $k=3/2$. Here, $c$ is arbitrarily chosen, and it determines how much $r(0)$ differs from 1.

Similarly, we can plot ${t}_{0}(x)/{\tau}_{R}$ for various values of $c$, and then observe if it can be expressed as $j[c]g(x)$, and plot $\theta (x)$ for various values of $c$, and then observe if it can be expressed as $ch(x)$. For small $x$, we find that ${t}_{0}(x)/{\tau}_{R}\equiv {x}^{2}$, agreeing with the analytic form.

For small $c$, the numerics show that

for offset $c$. So, when $c$ is not too small, ${t}_{0}(x)$ is of the same order as the analytic form, Eq. (52), found earlier. An example with $c=0.1$ is given in Fig. 9.## 9. NUMERICAL DETERMINATION OF FUNCTIONS

Suppose $\theta (x)=-{c}^{n}y(x)$ and $r(x)=4{x}^{2}+1+{c}^{n}{r}_{1}(x)$ for some unknown $n$. Then we find, in this model, that on the “ridge” formed by the maxima at each $x$,

Using numerics, we subtract one from the other. Suppose the difference $\mathrm{\Delta}$ is

Along the ridge, we measure this $\mathrm{\Delta}$ as a complex function of $x$. So, by finding a few values of $\mathrm{\Delta}$, we determine the dependence on $c$, and immediately find the value of $n$, which turns out to be 1.

Then, expanding ${e}^{ix}(\varphi [x,t={t}_{0}(x);c]-\varphi [x,t]-\mathrm{\Delta})$ for small $c$ gives the first nonzero term. Equating it to zero gives

Solving this equation, we obtain ${r}_{1}(x)$, $y(x)$ explicitly:

andWe use this procedure to produce curves for ${r}_{1}(x)$ and $y(x)$ for various values of $c$. From Fig. 10 and curves for other $c$ values, we find ${r}_{1}(x)\approx -2{(4{x}^{2}+1)}^{1.865}$. This compares with $-2{(4{x}^{2}+1)}^{3/2}$ found from the analytic result, so the approximate result is reasonably accurate.

From Fig. 11 and curves for other $c$ values, we find $y(x)\approx 35x$ and hence $\theta (x)\approx -35cx$ for $x<1$. For small $x$, we find

Plainly, for small $x$, which is the main region of interest for the localized rogue wave, we have $\theta (x)\approx -35cx$, which is in good agreement with the value from the expression found numerically.

## 10. CONCLUSIONS

In this work, we have found rogue wave solutions of higher-order extensions of the NLSE. Our approach can be useful when exact solutions of the evolution equation cannot be found. As an example, approximate solutions in the form of rogue waves have been obtained and analyzed for an optical fiber with a Raman delay term. Based on this example, the technique can be applied to other higher-order terms in the extended NLSE.

In order to confirm the applicability of the technique, we tested it on a few integrable equations and compared the results with the exact solutions. The equation with the Raman term does not have the exact solution in the form of a rogue wave. In this case, we conducted numerical simulations confirming that the approximate solutions are sufficiently accurate. The approach presented here could also be used to study other optical systems that are only described by nonintegrable equations, for example, ultrafast nonlinear quadratic systems.

Rogue waves may be difficult to detect with existing equipment in optics. However, new techniques have been developed recently that may help in such measurements [48]. Namely, time stretching analog-to-digital converters [49,50], time microscopes [51], and time lens magnifiers [52] may be able to resolve individual ultrashort pulses that could not be done with traditional pulse accumulating methods. In this sense, theory and experiment are progressing equally well, and this may allow us to witness experimental observations of Raman rogue waves soon. The first experimental observation, using a single shot technique, has already been made in [12]. Fine details of the pulse profile that we studied here could also be observed in the near future.

## Funding

Australian Research Council (ARC) (DP140100265, DP150102057, DE1312345); Volkswagen Foundation.

## Acknowledgment

The authors thank Wonkeun Chang for assistance.

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