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:

Equation (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

If 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,

Following Eq. (3), we first write the Lagrangian,

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

Clearly, 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

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

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

For convenience, we split $L_d$ into three parts:

We 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 ${{}_{y_1\to \infty }{}^{\lim }L}_{d1}$ shows that it equals to

Using 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 $r(x)=1+4B^2{x}^2$ 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

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

Again, setting $r(x)=1+4B^2{x}^2$ 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

This 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, $t_0^{\prime \prime }(x)=\frac{8{\tau }_R}{15w_0^4}$, 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

 

Fig. 1. Plot of the relative soliton offset, $t_0/{\tau }_R$, showing the effect of Raman delay. We compare the numerically found result (blue curve) with the analytic result of Eq. (29) (red curve). Here, delay is given by ${\tau }_R=0.02$ and width $w_0=0.4$.

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Fig. 2. Plot of the soliton, Eq. (23), under the effect of Raman delay, using Eqs. (24)–(30). Here, delay is given by ${\tau }_R=0.02$ and width $w_0=0.4$. At the ends ($x=\pm 3$), the $t$ offset is about 1.8.

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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 $z=\sqrt{X}=\sqrt{1+4x^2}$ and $r(z)=s^2(z)$. This gives

We find that the solution is

Note that this remains real even when $c$ is small. In fact, ${\lim }_{c\to 0}s(z)=z$, so ${\lim }_{c\to 0}r(z)=z^2$. 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.

 

Fig. 3. Plot of function $r(x)$ for various $c$ values. The blue curves are for $c<0$. This solution corresponds to the expansion in Eq. (37) with $c=-0.04$ (top blue curve) and $c=-0.02$ (next blue curve). The green curve, namely $r(x)=z^2=1+4x^2$, is for $c=0$. The two red curves are plots of Eqs. (35) and (36). The upper red curve is for $c=0.02$. The lower red curve is for $c=0.04$. In each case, the expansion, Eq. (37), is superimposed on the exact result. The curves are indistinguishable.

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Fig. 4. Plot of function $t_0(x)/{\tau }_R$, from Eq. (46) for various $c$: $c=0.04$ (blue curve), $c=0.4$ (magenta), $c=0.2$ (red), and $c=0.1$ (green).

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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, $r(x)=(1+4x^2)$. 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

The $r(x)$ equation is

Now, if we take $r(x)=4x^2+1$ 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 ${\tau }_R<0.002$.

 

Fig. 5. Plot of functions $t_0(x)$, Eq. (46), $r(x)$, using the results of Section 6, including Eq. (38), and $\theta (x)$, using Eq. (53), together with the same functions found numerically by solving Eqs. (33)–(41). Here, $c=0.04$, and the delay is given by ${\tau }_R=0.005$. The two upper curves give $r(x)$, while the two lower positive ones give $t_0(x)$ and the two negative ones show $\theta (x)$. In each case, the exact numeric solution and the approximation for it are very close.

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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.

 

Fig. 6. Plot of the rogue wave, Eq. (7), using Eqs. (46)–(53) for delay given by ${\tau }_R=0.005$. Here $c=0.04$.

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Fig. 7. Contour plot of the rogue wave, Eq. (7), using Eqs. (46)–(53) for delay given by ${\tau }_R=0.005$. Here $c=0.04$ and the propagation direction, $x$, is the horizontal axis, while the transverse axis, $t$, is vertical.

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Now, $\frac{t_0^{\prime \prime }(0)}{{\tau }_R}=\frac{2}{c}(6+66c+215c^2)$, 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.

 

Fig. 8. Plot of functions $t_0^{\prime \prime }(x)/{\tau }_R$ (even function of $x$ with $t_0(0)\approx c_0=300$ here), $t_0^{\prime }(x)/{\tau }_R$ (odd function of $x$) and $t_0(x)/{\tau }_R$ (even function of $x$ with $t_0(0)=0$) from original Eqs. (46), (45), and (48), respectively, together with a comparison with the Gaussian-based approximations for them, given by Eqs. (49), (50), and (51), all for $c=0.1$. Here, $c_0=\frac{2}{c}(6+66c+215c^2)$.

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So, for small $x$,

For $c_0=\frac{2}{c}(6+66c+215c^2)$, 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