Diverse Forms of Breathers and Rogue Wave Solutions for the Complex Cubic Quintic Ginzburg Landau Equation with Intrapulse Raman Scattering

Abstract

This manuscript consist of diverse forms of lump: lump one stripe, lump two stripe, generalized breathers, Akhmediev breather, multiwave, M-shaped rational and rogue wave solutions for the complex cubic quintic Ginzburg Landau (CQGL) equation with intrapulse Raman scattering (IRS) via appropriate transformations approach. Furthermore, it includes homoclinic, Ma and Kuznetsov-Ma breather and their relating rogue waves and some interactional solutions, including an interactional approach with the help of the double exponential function. We have elaborated the kink cross-rational (KCR) solutions and periodic cross-rational (KCR) solutions with their graphical slots. We have also constituted some of our solutions in distinct dimensions by means of 3D and contours profiles to anticipate the wave propagation. Parameter domains are delineated in which these exact localized soliton solutions exit in the proposed model.

1. Introduction

The analysis of the solitary wave solutions (SWs) for various nonlinear partial differential Equations (NLPDEs) play a significant role in different aspects of mathematical and physical phenomena [1,2,3,4,5]. Mainly natural phenomena arising in applied science, such as nuclear physics, chemical reactions, optical fibres [6,7,8,9,10], fluid mechanics, plasma, physics and ecology, can sometimes be modeled and described by NLPDEs [11,12,13,14,15,16,17,18,19,20,21]. Constructing the SWs of these equations has become a global interest in recent years. Hence, an enormous number of mathematical experts have attempted to invent various approaches by which one can obtain the exact solutions of such equations. Nowadays, some new effective techniques have been residential and well known [22]. To learn the mechanism of phenomena for the NLPDEs in physics and engineering, their SWs are calculated. There are many integration architectonics, such as Lie symmetry analysis [23], Backlund transformations [24], conservation laws, symmetry bifurcation [25], extended tanh-function, spontaneous symmetry [26], Painleve and Lie symmetries [27], CESTAC Method [28], polynomial law [29], computational architectonic, Semi inverse technique [30], HBM [31], mapping algorithm [32], $(G^{\prime }/G)$ expansion algorithm [33], Kudryashove architectonic [34], auxiliary equation scheme [35] and $exp\left((-{\phi }^{\prime }/\phi )\eta \right)$-expansion scheme [36]. The Riccati-Bernoulli sub-ODE method, optimal homotopy asymptotic approach, Exp-function algorithm, sine-cosine process, tanh-sech mechanism, extended tanh-scheme, F-expansion method, homogeneous balance technique, Jacobi elliptic function mechanism and several others have been developed to obtain SWs. A massive number of NLPDEs can be purely solved by the abovementioned methods. However, there is no a specific approach by which we can deal with all NLPDEs. In addition, some NLPDEs cannot be effortlessly solved by most traditional methods. The proposed method, which allows us to execute tedious and sophisticated algebraic calculations, is utilized to establish solitary wave solutions, peaked wave solutions and exact wave solutions for NLPDEs.

The NLPDEs are mainly valuable zones for nonlinear optics to reveal the proliferation distinctively short pulse in ultra-fast signal routing and telecommunication and light pulse propagation in condensed matter [37,38,39,40,41]. There are lots of recognized model, such as the modified KP-Equation [24], Fokas Equation [23], rth Dym Equation [21], Pochhammer-Chree model [14], modified Equation [11], fractional NLSE [20], fiber Bragg gratings model [29], Einstein’s vacuum field Equation [27], double-chain model [13], Wazwaz Benjamin model [12], modified Veronese Web Equation [42], (KMN)-Equation [30], Sawada Kotera Equation [31] and Fokas–Lenells model.

Recently, lump and interactional solutions (LISs) have shown significance to depict the wave features for various NLPDEs. For instance, LISs were studied by Zhou et al. with the Hirota Satsuma model [43], LIsS were found by Wang et al. with the Burgers model [44], Wu et al. worked on lump, periodic lump solutions in the KP model [45] and, similarly, Li et al. studied various lumps for BLMP model [46]. Breather soliton is a nonlinear wave in which energy is localized in space but oscillates in time, or vice versa, and has been newly reported in an optical fiber cavity. Cavity solitons (CSs) are localized pulses of light that can be wound up in nonlinear optical resonators and have sparked imperative study curiosity in the perspective of micro resonator-based frequency comb generation, and are found in a range of subfields of natural science, for instance fluid dynamics, solid-state physics, plasma physics, molecular biology, chemistry and nonlinear optics [47]. Recently, Rizvi et al. investigated breathers for NLEE [15], Seadawy et al. interpreted breather solutions for NLEE [42], Ahmed et al. studied breathers for the general $(2+1)$-rth dispersionless Equation [21], and Ahmed et al. found kinky breathers for the nonlinear model [48], among many other studies. Multiwave solutions (MS) for nonlinear models have its own worth. Seadawy et al. worked on MS for the HS-Equation [15], Ahmed et al. studied MS for the $(2+1)$-rth dispersionless Equation [21], Rizvi et al. reported MS for NLEE [42], Seadawy et al. worked on MS for the nonlinear model [48], Wazwaz analyzed rogue wave and breathers [49], etc.

In this template, we begin our analysis by taking the CQGL-equation with IRS term [22];

$$\begin{array}{c}\hfill i{\mathsf{\Delta }}_z+\frac{1}{2}{\mathsf{\Delta }}_{tt}+\gamma |\mathsf{\Delta }{|}^2\mathsf{\Delta }=i\delta \mathsf{\Delta }+i\beta {\mathsf{\Delta }}_{tt}+i\epsilon |\mathsf{\Delta }{|}^2\mathsf{\Delta }-\nu |\mathsf{\Delta }{|}^4\mathsf{\Delta }+i\mu |\mathsf{\Delta }{|}^4\mathsf{\Delta }+T_r{\left(|\mathsf{\Delta }{|}^2\right)}_t\mathsf{\Delta },\end{array}$$

(1)

where z is the normalized propagation distance, t is the retarded time and $\mathsf{\Delta }$ is the normalized envelope of the pulse. For a laser system, the interpretation of distinct coefficients is as follows: $\beta$ shows spectral filtering or gain dispersion, $\mu$ expresses higher-order correction to the nonlinear absorption or amplification, $ϵ$ shows nonlinear gain, $\nu$ shows a higher-order correction term to the nonlinear refractive index, $T_r$ shows the IRS coefficient, $\gamma$ displays the positive Kerr effect (or negative Kerr effect if negative) and $\delta$ is a constant gain (or loss if negative). The stated equation is a canonical model for weakly nonlinear, dissipative systems and one of the most studied nonlinear equations in the physics community. It can be used to describe a vast variety of nonlinear phenomena, such as Bose–Einstein condensation, superconductivity, strings in field theory, superfluidity, lasers and liquid crystals.

In order to solve Equation (1), we insert $\mathsf{\Delta }=p+iq$, where $\left|\mathsf{\Delta }\right|=\sqrt{p^2+q^2}$. Thus, Equation (1) may be converted into real and imaginary parts:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{p}^3\gamma +q^2\gamma p+p^5\nu +2p^3{q}^2\nu +q^4\nu p+\delta q+p^2\epsilon q+q^3\epsilon +p^4\mu q+2p^2{q}^3\mu +q^5\mu +\frac{1}{2}{p}_{tt}+\beta q_{tt}-q_z=0,\hfill \\ -\delta p-p^3\epsilon -q^2\epsilon p-\mu p^5-2\mu p^3{q}^2-\mu q^{4}p+\gamma p^{2}q+\gamma q^3+\nu p^{4}q+2\nu p^2{q}^3+\nu q^5-\beta p_{tt}+\frac{1}{2}{q}_{tt}+p_z=0.\hfill \end{array}\right.\end{array}$$

(2)

The document for the upcoming sections will be detailed in a sequence: in Section 2, we will evaluate the lump solutions for the proposed model form with few graphical slots. In Section 3, there will be a concise discussion of lump one stripe solutions with some 3D and contour graphical slots. In Section 4, we will construct lump two stripe results with some suitable profiles. Section 5 consist of a Ma-breather (MB) and its relating rogue wave. Similarly, in Section 6, we will evaluate the Kuznetsov-Ma breather (KMB) with some suitable 3D and contour shapes. In Section 7, we will find the generalized breathers (GB) for proposed equation with their relating figures. Section 8 includes Akhmediev breathers (AB) along with some profiles for the concerned model. Similarly, Section 9 will detail the procedure to construct standard rogue waves. In Section 10, we will explain the methodology for finding multiwave solutions. In the same way, we will compute homoclinic breathers for the proposed equation in Section 11. There will be M-shaped solitons in Section 12. In Section 13, there will be an interaction approach for the proposed model. We will find kink cross-rational (KCR) solutions in Section 14. Section 15 includes periodic cross-rational (PCR) solutions along with some 3D and contour profiles for the concerned model. Section 16 contains the results and a discussion about our newly achieved solutions and we will make an suitable comparison with earlier work. Finally, in Section 17, we will provide some concluding annotations.

2. Lump Solution

For the lump solutions of Equation (2), we apply the subsequent ansatz [43,44]:

$$p=\frac{6}{\rho }{\left(lng\right)}_z,q=\frac{6}{\omega }{\left(lnh\right)}_z,$$

(3)

and get the proceeding form:

$$\begin{array}{c}\hfill 2p^2\gamma \omega g^2{h}^3{g}_z+2q^2\gamma \omega g^2{h}^3{g}_z+2p^4\nu \omega g^2{h}^3{g}_z+4p^2{q}^2\gamma \omega g^2{h}^3{g}_z+2q^4\nu \omega g^2{h}^3{g}_z\\ \hfill +2\omega h^3{g}_t^2{g}_z-\omega g{h}^3{g}_{tt}{g}_z+2\delta \rho g^3{h}^2{h}_z+2p^2\epsilon \rho g^3{h}^2{h}_z+2q^2\epsilon \rho g^3{h}^2{h}_z+2p^4\mu \rho g^3{h}^2{h}_z\\ \hfill +4p^2{q}^2\mu \rho g^3{h}^2{h}_z+2q^4\mu \rho g^3{h}^2{h}_z+4\beta \rho g^3{h}_t^2{h}_z+\dots +2\beta \rho g^3{h}^2{h}_{ztt}-2\rho g^3{h}^2{h}_{zz}=0.\end{array}$$

(4)

Now, the function g and h in Equation (4) can be considered as [43,44]:

$$g={\xi }_1^2+{\xi }_2^2+a_2,h={\xi }_1^2+{\xi }_2^2+a_3,$$

(5)

where ${\xi }_1=a_{0}z+t$, ${\xi }_2=a_{1}z+t.$, while $a_i(1\le i\le 3)$ are specific real parameters. Now, using g and h into Equation (4) and solving the coefficients of the z and t implies:

Set I. When

$$a_0=\left(-4+\sqrt{15}\right)a_1,a_1=a_1,a_2=0,\rho =\rho ,\omega =\omega .$$

(6)

These generated parameters make the lump solution:

$${\mathsf{\Delta }}_1=\frac{-6\left(-4+\sqrt{15}\right)R_1+{\left(-4+\sqrt{15}\right)}^2{a}_1^2\left(2a_1\left(t+a_{1}z\right)+2\left(-4+\sqrt{15}\right)a_1{R}_2\right)}{a_1^2\left({\left(t+a_{1}z\right)}^2+{\left(t+\left(-4+\sqrt{15}\right)a_{1}z\right)}^2\right)}+{\mathsf{\Omega }}_1,$$

(7)

where $R_1=a_1^2+\left(-4+\sqrt{15}\right)a_1^2$, ${\mathsf{\Omega }}_1=\frac{6i\left(2a_1\left(t+a_{1}z\right)+2\left(-4+\sqrt{15}\right)a_1\left(t+\left(-4+\sqrt{15}\right)a_{1}z\right)\right)}{\left(\frac{{\left(-4+\sqrt{15}\right)}^2{a}_1^3}{a_1^3-\left(-4+\sqrt{15}\right)a_1^3+\left(-4+\sqrt{15}\right)a_1^3}+{\left(t+a_{1}x\right)}^2+{\left(t+\left(-4+\sqrt{15}\right)a_{1}z\right)}^2\right)\omega }$ and $R_2=t+\left(-4+\sqrt{15}\right)a_{1}z$.

3. Lump One Stripe Solution

To get the lump one stripe solution, we apply the transformation shown in Equation (4) [50]:

$$g={\xi }_1^2+{\xi }_2^2+a_2+b_0{e}^{k_{1}z+k_{2}t},h={\xi }_1^2+{\xi }_2^2+a_3+b_0{e}^{k_{1}z+k_{2}t},$$

(8)

where ${\xi }_1=a_{0}z+t$, ${\xi }_2=a_{1}z+t.$, while $a_i(1\le i\le 3)$, $k_1,k_2$ and $b_0$ are any specific real parameters. Now, using g and h in Equation (4) and solving the coefficients of the z and t:

Set I. When $a_2=0$:

$$a_0=-a_1,a_3=\frac{-6b_0\omega }{b_0\omega \left(\nu p^6{q}^2+\nu q^4+\gamma p^2+\gamma q^2\right)},a_5=a_5,\omega =\omega ,\rho =\rho ,k_2=0.$$

(9)

These parameters exhibit the required solution to Equation (2):

$${\mathsf{\Delta }}_2=\frac{6\left(b_0{e}^{k_{1}z}{k}_1-2a_1\left(t-a_{1}z\right)+2a_1\left(t+a_{1}z\right)\right)}{\rho \left(b_0{e}^{k_{1}z}+{\left(t-a_{1}z\right)}^2+{\left(t+a_{1}z\right)}^2\right)}+{\mathsf{\Omega }}_2.$$

(10)

where ${\mathsf{\Omega }}_2=\frac{6i\left(b_0{e}^{k_{1}z}{k}_1-2a_1\left(t-a_{1}z\right)+2a_1\left(t+a_{1}z\right)\right)}{\left(b_0{e}^{k_{1}z}+{\left(t-a_{1}z\right)}^2+{\left(t+a_{1}z\right)}^2-\frac{6}{p^2\gamma +q^2\gamma +p^4\nu +2p^2{q}^2\nu +q^4\nu }\right)\omega }.$

4. Lump Two Stripe Solution

To obtain the lump two stripe solution, we assume the subsequent transformation in Equation (4) [50]:

$$g={\xi }_1^2+{\xi }_2^2+a_2+b_0{e}^{k_{1}z+k_{2}t}+b_1{e}^{k_{3}z+k_{4}t},h={\xi }_1^2+{\xi }_2^2+a_3+b_0{e}^{k_{1}z+k_{2}t}+b_1{e}^{k_{3}z+k_{4}t},$$

(11)

where ${\xi }_1=a_{0}z+t$, ${\xi }_2=a_{1}z+t.$, while $a_i(1\le i\le 3)$, $k_1,k_2,k_3,k_4,b_0$ and $b_1$ are any specific real parameters. Now, using g and h in Equation (4) and solving the coefficients of the z and t:

Set I. When $a_1=a_2=a_3=0$:

$$a_0=a_0,a_5=a_5,b_1=\frac{-5k_3\delta }{7k_3^2-5k_3\delta },\beta =\frac{b_1\omega \left(7{k_3}^2-5k_3\delta \right)}{10b_1{k}_3\delta \rho },\rho =\frac{-4k_3^2+5k_3}{5k_3}.$$

(12)

These parameters exhibits the required solution to Equation (2):

$${\mathsf{\Delta }}_3=\frac{6\left(b_0{e}^{k_{2}t+k_{1}z}{k}_1-\frac{e^{k_{4}t+k_{3}z}{k}_3\left(5k_3-4k_3^2\right)}{-5k_3+11k_3^2}+R_3\right)}{\rho \left(b_0{e}^{k_{2}t+k_{1}z}{k}_1-\frac{e^{k_{4}t+k_{3}z}{k}_3\left(5k_3-4k_3^2\right)}{-5k_3+11k_3^2}+R_4\right)}+\frac{6i\left(b_0{e}^{k_{2}t+k_{1}z}{k}_1-\frac{e^{k_{4}t+k_{3}z}{k}_3\left(5k_3-4k_3^2\right)}{-5k_3+11k_3^2}+R_3\right)}{\omega \left(b_0{e}^{k_{2}t+k_{1}z}{k}_1-\frac{e^{k_{4}t+k_{3}z}{k}_3\left(5k_3-4k_3^2\right)}{-5k_3+11k_3^2}+R_4\right)},$$

(13)

where $R_3=2a_0\left(t+a_{0}z\right)$ and $R_4=t^2+{\left(t+a_{0}z\right)}^2$.

5. Ma-Breather (MB) and Its Relating Rogue Wave

We assume g and h in Equation (4) as [44]:

$$g=1+{\alpha }_1+e^{i\left(p_{1}x\right)}+e^{-i\left(p_{1}x\right)}{e}^{{\lambda }_{1}t+{\gamma }_1}+{\beta }_1{e}^{2({\lambda }_{1}t+{\gamma }_1)},h=1+{\alpha }_2+e^{i\left(p_{2}x\right)}+e^{-i\left(p_{2}x\right)}{e}^{{\lambda }_{2}t+{\gamma }_2}+{\beta }_2{e}^{2({\lambda }_{2}t+{\gamma }_2)},$$

(14)

where ${\alpha }_1$, ${\alpha }_2$, $p_1$, $p_2$, ${\lambda }_1$, ${\lambda }_2$, ${\gamma }_1$ and ${\gamma }_2$ are any parameters. Now, using g and h in Equation (4) and letting the coefficients of exp and cos functions be zero:

Set I. When ${\gamma }_1={\beta }_2=0$:

$$\begin{array}{c}\hfill {\alpha }_1={\alpha }_1,{\alpha }_2={\alpha }_2,\mu =\frac{i{p}_2-2\delta -2p^2\epsilon -2q^2\epsilon -5\beta {\lambda }_2^2}{2{\left(p^2+q^2\right)}^2},p_1=p_1,a_4=a_4.\end{array}$$

(15)

These parameters form the Ma-breather solution to Equation (1):

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_4=\frac{6e^{t{\lambda }_1}\left(-i{e}^{-i{p}_{1}z}{p}_1+i{e}^{i{p}_{1}z}{p}_1\right){\alpha }_1}{\left(1+e^{t{\lambda }_1}\left(e^{-i{p}_{1}z}+e^{i{p}_{1}z}\right){\alpha }_1\right)\rho }+\frac{6i{e}^{{\gamma }_2+t{\lambda }_2}\left(-i{e}^{-i{p}_{2}z}{p}_2+i{e}^{i{p}_{2}z}{p}_2\right){\alpha }_2}{\left(1+e^{{\gamma }_2+t{\lambda }_2}\left(e^{-i{p}_{2}z}+e^{i{p}_{2}z}\right){\alpha }_2+e^{2\left({\gamma }_2+t{\lambda }_2\right)}{\beta }_2\right)\omega }.\end{array}$$

(16)

6. Kuznetsov-Ma Breather (KMB) and Its Relating Rogue Wave

We assume g and h in Equation (4) as [44]:

$$\begin{array}{c}\hfill g=e^{-p_1(b_{2}z-b_{1}t)}+a_{1}cos\left(p(b_{2}z+b_{1}t)\right)+a_{2}cos\left(p(b_{2}z-b_{1}t)\right),\\ \hfill h=e^{-p_2(b_{3}z-b_{4}t)}+a_{3}cos\left(p(b_{3}z+b_{4}t)\right)+a_{4}cos\left(p(b_{3}z-b_{4}t)\right),\end{array}$$

(17)

where $p_1$, $p_2$, $b_1$, $b_2$, $b_3$, $b_4$, $a_1$, $a_2$, $a_3$ and $a_4$ are any parameters to be found. Now, using g and h in Equation (4) and letting the coefficients of $exp$ and $\cos$ functions be zero follows:

Set I. When:

$$\begin{array}{c}\hfill p_1=p_1,\nu =\frac{-p_1^2\left({\gamma }^2+\mu \right)}{p^2},a_1=a_1,\gamma =\gamma ,a_3=a_3,\rho =\rho .\end{array}$$

(18)

These parameters form the proposed solution to Equation (1):

$$\begin{array}{c}\hfill {\mathsf{\Delta }}_5=\frac{6\left(-b_2{e}^{-p_1\left(-b_{1}t+b_{2}z\right)}{p}_1+a_2{b}_2{e}^{p_1\left(-b_{1}t+b_{2}z\right)}{p}_1-a_1{b}_{2}psin\left(p\left(b_{1}t+b_{2}z\right)\right)\right)}{\rho \left(e^{-p_1\left(-b_{1}t+b_{2}z\right)}+a_2{e}^{p_1\left(-b_{1}t+b_{2}z\right)}+a_{1}cos\left(p\left(b_{1}t+b_{2}z\right)\right)\right)}+{\mathsf{\Omega }}_3,\end{array}$$

(19)

where ${\mathsf{\Omega }}_3=\frac{6i\left(-b_3{e}^{-p_2\left(-b_{3}t+b_{4}z\right)}{p}_2+a_4{b}_3{e}^{-p_2\left(-b_{3}t+b_{4}z\right)}{p}_2-a_3{b}_{3}psin\left(p\left(b_{4}t+b_{3}z\right)\right)\right)}{\omega \left(e^{-p_2\left(-b_{3}t+b_{4}z\right)}+a_4{e}^{p_2\left(-b_{3}t+b_{4}z\right)}+a_{3}cos\left(p\left(b_{4}t+b_{3}z\right)\right)\right)}.$

7. Generalized Breathers (GB)

In order to obtain generalized breathers we use ansatz [51]:

$$\mathsf{\Delta }(z,t)=2bc{\left(\frac{6}{\kappa }ln\mathsf{\Psi }\left(z,t\right)\right)}_z+m,$$

(20)

where b, c and m are any particular constants. Inserting Equation (20) into Equation (1), we have:

$$\begin{array}{c}\hfill m^3\gamma {\kappa }^5{\psi }^5-im\delta {\kappa }^5{\psi }^5-i{m}^3\epsilon {\kappa }^5{\psi }^5+m^5{\kappa }^5\nu {\psi }^5+m^5{\kappa }^5\nu {\psi }^5+36bc{m}^2\gamma {\kappa }^4{\psi }^4{\psi }_z-12ibc\delta {\kappa }^4{\psi }^4{\psi }_z\\ \hfill -36ibc{m}^2\epsilon {\kappa }^4{\psi }^4{\psi }_z-60ibc{m}^4\mu {\kappa }^4{\psi }^4{\psi }_z+60bc{m}^4\nu {\kappa }^4{\psi }^4{\psi }_z+12bcm{T}_r{\kappa }^4{\psi }^3{\psi }_t{\psi }_z+12bc{\kappa }^4{\psi }^2{\psi }_z{\psi }_t^2\\ \hfill -24ibc\beta {\kappa }^4{\psi }^2{\psi }_t^2{\psi }_z-6bc{\kappa }^4{\psi }^3{\psi }_{tt}{\psi }_z+12ibc\beta {\kappa }^4{\psi }^3{\psi }_{tt}{\psi }_z+\dots +12ibc{\kappa }^4{\psi }^4{\psi }_{zz}-12ibc\beta {\kappa }^4{\psi }^4{\psi }_{zt}=0.\end{array}$$

(21)

For finding the required solutions, we use the following assumption in Equation (21):

$$\psi =\frac{\left(1-4c\right)cosh\left(\sigma t\right)+\sqrt{2c}cos\left(\rho z\right)+i\sigma sinh\left(\sigma t\right)}{\sqrt{2c}cos\left(\rho z\right)-cosh\left(\sigma t\right)}{e}^{it},$$

(22)

where $\sigma$, $\rho$ and c are constants to be found. The coefficients of cosh, sinh and exp functions are defined as follows:

$$a=a,b=b,m=0,c=\frac{1}{2},\rho =\rho ,\sigma =\sigma .$$

(23)

These values implies the following GB profiles of Equation (1):

$${\mathsf{\Delta }}_6=\frac{6bi{e}^{-it}\left(cos\left(\rho z\right)-cosh\left(\sigma t\right)\right)\left(\frac{-e^{it}\rho sin\left(\left(\rho z\right)\right)}{cos\left(\rho z\right)-cosh\left(\sigma t\right)}+\frac{e^{it}\rho sin\left(\left(\rho z\right)\right)\left(cos\left(\left(\rho z\right)\right)-cosh\left(\sigma t\right)+i\sigma sinh\left(\sigma t\right)\right)}{{\left(cos\left(\rho z\right)-cosh\left(\sigma t\right)\right)}^2}\right)}{\kappa cos\left(\rho z\right)-cosh\left(\sigma t\right)+i\sigma sinh\left(\sigma t\right)}.$$

(24)

8. Akhmediev Breathers (AB)

We use the following transformation in Equation (21) [52]:

$$\psi =\sqrt{p_0}\frac{\left(1-4a\right)cosh\left(bz\right)+ibsinh\left(bz\right)+\sqrt{2a}cos\left({\omega }_{mod}T\right)}{\sqrt{2a}cos\left({\omega }_{mod}T\right)-cosh\left(b\xi \right)},$$

(25)

where ${\omega }_{mod}$ interprets the perturbation frequency (PF) with $p_0$ as the power. The coefficients a and b depends on ${\omega }_{mod}$ and are defined by $2a=1-{\left(\frac{{\omega }_{mod}}{{\omega }_c}\right)}^2$ and $b={\left[8a(1-2a)\right]}^2$ with ${\omega }_c^2=\frac{4p_0\gamma }{|{\beta }_2|}.$ Setting the coefficients of trigonometric and hyperbolic functions be zero:

$$a=a,b=b,c=c,\omega =\sqrt{\frac{4-82}{2i\beta }},\rho =\rho ,p_0=p_0.$$

(26)

These values imply the AB of Equation (1) to be as follows:

$${\mathsf{\Delta }}_7=m+\frac{12bc\left(\sqrt{2a}cos\left(\frac{t\sqrt{-\frac{i\left(4-8a\right)}{\beta }}}{\sqrt{2}}\right)-cosh\left(bz\right)\right){\mathsf{\Omega }}_4}{\sqrt{p_0}\kappa \left(\sqrt{2a}cos\left(\frac{t\sqrt{-\frac{i\left(4-8a\right)}{\beta }}}{\sqrt{2}}\right)+\left(1-4a\right)cosh\left(bz\right)+ibsinh\left(bz\right)\right)},$$

(27)

where:

$${\mathsf{\Omega }}_4=\left(\frac{b\sqrt{p_0}\sinh \left(bz\right)\left(\sqrt{2a}cos\left(\frac{t\sqrt{-\frac{i\left(4-8a\right)}{\beta }}}{\sqrt{2}}\right)+\left(1-4a\right)cosh\left(bz\right)+ibsinh\left(bz\right)\right)}{{\left(\sqrt{2a}cos\left(\frac{t\sqrt{-\frac{i\left(4-8a\right)}{\beta }}}{\sqrt{2}}\right)-cosh\left(bz\right)\right)}^2}+\frac{\sqrt{p_0}\left(i{b}^{2}cosh\left(bz\right)+\left(1-4a\right)bsinh\left(bz\right)\right)}{\sqrt{2a}cos\left(\frac{t\sqrt{-\frac{i\left(4-8a\right)}{\beta }}}{\sqrt{2}}\right)-cosh\left(bz\right)}\right)$$

.

9. Standard Rogue Wave (SRW) Solutions

For evaluating the SRW, we apply the subsequent assumption in Equation (21) [44]:

$$\psi =-\left(1-\frac{4\left(1+2it\right)}{1+4z^2+4t^2}\right)e^{it},$$

(28)

Setting the coefficients of exponential function, z and t be zero will follow:

$$b=b,\beta =\frac{-i}{2},m=\sqrt{\frac{-3i\epsilon +3\gamma +\sqrt{20\delta \mu -9{\epsilon }^2-18i\gamma \epsilon +20i\delta \nu +9{\gamma }^2}}{10\left(i\mu -\nu \right)}},c=c,\kappa =\kappa .$$

(29)

These values implies the SRW to Equation (1):

$${\mathsf{\Delta }}_8=-\frac{384bc\left(1+2it\right)z}{\left(1+4t^2+4z^2\right)\left(-1+\frac{4\left(1+2it\right)}{1+4t^2+4z^2}\right)\kappa }+\sqrt{\frac{-3i\epsilon +3\gamma +\sqrt{20\delta \mu -9{\epsilon }^2-18i\gamma \epsilon +20i\delta \nu +9{\gamma }^2}}{10\left(i\mu -\nu \right)}}.$$

(30)

10. Multiwaves Solutions (MS)

For these type of results, we use the preceding transformation in Equation (2) [48]:

$$\begin{array}{c}\hfill \mathsf{\Delta }(z,t)=\psi \left(\xi \right)e^{\iota \theta },\xi =k_{1}z-c_{1}t,\theta =k_{2}z-c_{2}t.\end{array}$$

(31)

Using the above transformation, we obtain the real and imaginary parts of equal Equation (2), by considering the real part only:

$$\begin{array}{c}\hfill \gamma {\psi }^3+\nu {\psi }^5+c_1{T}_r\psi {\psi }^{{}^{\prime }}-\frac{1}{2}{c}_1^2\psi +\frac{1}{2}{c}_1^2{\psi }^{{}^{″}}+2\beta c_1{c}_2{\psi }^{{}^{\prime }}=0.\end{array}$$

(32)

Now, by way of following the assumption in Equation (32):

$$\psi =2{\left(lnf\right)}_{\xi },$$

(33)

we obtain:

$$\begin{array}{c}\hfill -c_2^2{f}^4{f}^{{}^{\prime }}-4\beta c_1{c}_2{f}^3{f^{{}^{\prime }}}^2+2c_1^2{f}^2{f^{{}^{\prime }}}^3+8\gamma f^2{f^{{}^{\prime }}}^3-4c_1{T}_r{f}^2{f^{{}^{\prime }}}^3+32\nu {f^{{}^{\prime }}}^5+4c_1{c}_2\beta f^4{f}^{{}^{″}}-3c_1^2{f}^3{f}^{{}^{\prime }}{f}^{{}^{″}}\\ \hfill +4c_1{f}^3{T}_r{f}^{{}^{\prime }}{f}^{{}^{″}}+c_1^2{f}^4{f}^{{}^{″\prime }}=0\end{array}$$

(34)

To get the MS of Equation (34), we use anstaz [48]:

$$f=b_{0}cosh(a_1\xi +a_2)+b_{1}cos(a_3\xi +a_4)+b_{2}cosh(a_5\xi +a_6),$$

(35)

where $a_1$, $a_2$, $a_3$, $a_4$, $a_5$ and $a_6$ are any specific constants. Substituting Equation (35) into Equation (34) with Mathematica and letting the coefficients of hyperbolic and trigonometric functions to zero:

Set I. When:

$$\begin{array}{c}\hfill a_1=a_1,a_2=a_2,a_3=\frac{-1}{2}{a}_5,a_4=a_4,a_5=a_5,b_0=0,b_1=b_1,c_1=c_1.\end{array}$$

(36)

Using the above values, we have:

$${\mathsf{\Delta }}_9=\frac{2e^{i\left(-c_{2}t+k_{2}z\right)}\left(\frac{1}{2}{a}_5{b}_{1}cos\left(a_4-\frac{1}{2}{a}_5\left(-c_{1}t+k_{1}z\right)\right)sin\left(a_4-\frac{1}{2}{a}_5\left(-c_{1}t+k_{1}z\right)\right)+{\mathsf{\Omega }}_5\right)}{b_{1}cos\left(a_4-\frac{1}{2}{a}_5\left(-c_{1}t+k_{1}z\right)\right)+b_{2}cosh\left(a_6+a_5\left(-c_{1}t+k_{1}z\right)\right)},$$

(37)

where ${\mathsf{\Omega }}_5=a_5{b}_{2}cosh\left(a_6+a_5\left(-c_{1}t+k_{1}z\right)\right)sinh\left(a_6+a_5\left(-c_{1}t+k_{1}z\right)\right).$

11. Homoclinic Breather (HB)

In this approach we assume f the form [48]:

$$f=e^{-p\left(a_2+a_1\xi \right)}+b_1{e}^{p\left(a_4+a_3\xi \right)}+b_{0}cos\left(p_1\left(a_6+a_5\xi \right)\right),$$

(38)

where $a_i^{\prime }s$ denotes any particular constants. Inserting Equation (38) into Equation (34) and collecting coefficients of exponential and trigonometric functions to be zero yields:

Set I. When:

$$a_1=\frac{1}{2}{a}_5,a_2=a_2,a_3=a_3,c_1=\frac{-2a_5^2\nu p^2}{T_r},b_1=b_1,a_5=a_5.$$

(39)

Via the above values we obtain:

$${\mathsf{\Delta }}_{10}=\frac{2\left(\frac{-1}{2}{a}_5{b}_{1}p{e}^{p\left(a_4-\frac{1}{2}{a}_4\left(k_{1}z+\frac{2a_5^2{p}^{2}t\nu }{T_r}\right)\right)}-\frac{1}{2}{a}_{5}p{e}^{-p\left(a_2-\frac{1}{2}{a}_5\left(k_{1}z+\frac{2a_5^2{p}^{2}t\nu }{T_r}\right)\right)}\right)e^{i\left(-c_{2}t+k_{2}z\right)}}{b_1{e}^{p\left(a_4-\frac{1}{2}{a}_4\left(k_{1}z+\frac{2a_5^2{p}^{2}t\nu }{T_r}\right)\right)}+e^{-p\left(a_2-\frac{1}{2}{a}_5\left(k_{1}z+\frac{2a_5^2{p}^{2}t\nu }{T_r}\right)\right)}}.$$

(40)

12. M-Shaped Rational Solitons

For these solutions, we consider the form [48,53]:

$$f={(d_1\xi +d_2)}^2+{(d_3\xi +d_4)}^2+d_5,$$

(41)

where $d_i(1\le i\le 5),$ are any parameters. Put f into Equation (34) and solving coefficients of $\xi$ to get subsequent result on parameters:

Set I. Whenever $d_5=d_2=0$:

$$d_1=i{d}_3,d_3=d_3,d_4=\frac{c_1^2{d}_3\sqrt{c_1^4{d}_3^2}+24\beta c_1{c}_2\gamma }{6\beta c_1{c}_2},c_1=c_1,c_2=c_2.$$

(42)

Using the above values, we obtain:

$${\mathsf{\Delta }}_{11}=\frac{2e^{i\left(-c_{2}t+k_{2}z\right)}\left(-2d_3^2\left(-c_{1}t+k_{1}z\right)+2d_3\left(d_3\left(-c_{1}t+k_{1}z\right)+\frac{c_1^2{d}_3\sqrt{c_1^4{d}_3^2+24\beta c_1{c}_2\gamma }}{6\beta c_1{c}_2}\right)\right)}{-{d_3}^2{\left(-c_{1}t+k_{1}z\right)}^2+{\left(d_3\left(-c_{1}t+k_{1}z\right)+\frac{c_1^2{d}_3\sqrt{c_1^4{d}_3^2+24\beta c_1{c}_2\gamma }}{6\beta c_1{c}_2}\right)}^2}.$$

(43)

13. Interactional Solutions with Double Exponential Form

We use the following hypothesis [48]:

$$f=b_1{e}^{-a_1\xi +a_2}+b_2{e}^{a_3\xi +a_4}.$$

(44)

where $a_1$, $a_2$, $a_3$ and $a_4$ are some constants. Inserting Equation (44) into Equation (34) and solving coefficients of exponential functions, a system of equations is obtained. By solving it:

Set I.

$$a_1=\left(2-\sqrt{3}\right)a_3,a_2=a_2,c_1=\frac{8a_3^2\nu \left(7-4\sqrt{3}\right)}{T_r},a_3=a_3,a_4=a_4.$$

(45)

Using the above values we have:

$${\mathsf{\Delta }}_{12}=\frac{2\left(a_3{b}_2{e}^{a_4+a_3\left(k_{1}z-\frac{8\left(7-4\sqrt{3}\right)a_3^{2}t\nu }{T_r}\right)}+\left(2-\sqrt{3}\right)a_3{b}_1{e}^{a_2+\left(2-\sqrt{3}\right)a_3\left(k_{1}z-\frac{8\left(7-4\sqrt{3}\right)a_3^{2}t\nu }{T_r}\right)}\right)e^{i\left(-c_{2}t+k_{2}z\right)}}{b_2{e}^{a_4+a_3\left(k_{1}z-\frac{8\left(7-4\sqrt{3}\right)a_3^{2}t\nu }{T_r}\right)}+b_1{e}^{a_2+\left(2-\sqrt{3}\right)a_3\left(k_{1}z-\frac{8\left(7-4\sqrt{3}\right)a_3^{2}t\nu }{T_r}\right)}}.$$

(46)

14. Kink Cross-Rational (KCR) Solutions

For KCR solutions, we consider f as [54,55]:

$$f=g_0+e^{-\left(a_1\xi +a_2\right)}+k_1{e}^{a_1\xi +a_2}+{\left(b_1\xi +b_2\right)}^2+{\left(b_3\xi +b_4\right)}^2,$$

(47)

where $a_i$ and $b_i$ are some constants. Inserting Equation (47) into Equation (34) and solving coefficients of exponential functions:

Set I.

$$a_1=\sqrt{\frac{-3}{32\mu }}{c}_1,b_1=b_1,b_2=b_2,k_1=0,a_2=a_2,\nu =\frac{2}{5}\mu ,T_r=\frac{3\sqrt{\frac{-3}{32\mu }}{c}_1^2+8\beta c_2}{4\sqrt{\frac{-3}{32\mu }}{c}_1},b_3=b_3,b_4=b_4.$$

(48)

Using the above values we have:

$${\mathsf{\Delta }}_{13}=\frac{2e^{i\left(-c_{2}t+k_{2}z\right)}\left(-2b_1\left(b_2-b_1{c}_{2}t\right)+2b_3\left(b_4-b_3{c}_{2}t\right)-\frac{1}{4}\sqrt{-\frac{3}{2\mu }}{c}_2{e}^{-a_2+\frac{1}{4}\sqrt{-\frac{3}{2\mu }}{c}_2^{2}t}\right)}{e^{-a_2+\frac{1}{4}\sqrt{-\frac{3}{2\mu }}{c}_2^{2}t}+g_0+{\left(b_2-b_1{c}_{2}t\right)}^2+{\left(b_4-b_3{c}_{2}t\right)}^2}.$$

(49)

15. Periodic Cross-Rational (PCR) Solutions

We use the following hypothesis [54,55]:

$$f=g_0+{\left(a_1\xi +a_2\right)}^2+{\left(a_3\xi +a_4\right)}^2+k_{1}cos\left(b_1\xi +b_2\right)+k_{2}cosh\left(b_3\xi +b_4\right),$$

(50)

where $a_i$ and $b_i$ are some constants. Inserting Equation (50) into Equation (34) and solving coefficients of exponential, trigonometric and hyperbolic functions:

Set I.

$$a_1=a_1,b_3=\frac{-4\gamma }{3k_2{c}_1^2},c_2=0,T_r=\frac{3}{4}{c}_1,b_1=0,b_4=b_4.$$

(51)

Using the above values we have:

$${\mathsf{\Delta }}_{14}=\frac{2e^{i\left(k_{2}z\right)}\left(2a_1{\left(a_2+a_1\left(-c_{1}t+k_{1}z\right)\right)}^2+2a_3{\left(a_4+a_3\left(-c_{1}t+k_{1}z\right)\right)}^2-\frac{4\gamma sinh\left(b_4-\frac{4\left(-c_{1}t+k_{1}z\right)\gamma }{3c_1^2{k}_2}\right)}{3c_1^2}\right)}{g_0+{\left(a_2+a_1\left(-c_{1}t+k_{1}z\right)\right)}^2+{\left(a_4+a_3\left(-c_{1}t+k_{1}z\right)\right)}^2+k_{2}cosh\left(b_4-\frac{4\left(-c_{1}t+k_{1}z\right)\gamma }{3c_1^2{k}_2}\right)}.$$

(52)

16. Result and Discussions

A lot of work has been done on the proposed model: Akhmediev et al. found singularities via a simple approach [56], Soto Crespo et al. studied pulse solutions for the case of normal group-velocity dispersion [57], Yan et al. found stable transmission of solitons for the concerned model via the asymmetric method [58], Biswas et al. worked on Dromion-like structures for the variable-coefficients CQGL-equation by using the asymmetric method [59], Gurevich et al. investigated soliton explosions for the CQGL-equation via explosion modes [60], Uzunov et al. studied pulsating solutions for the CQGL-equation by using the variation method and the method of moments [61], Nikolov et al. interpreted the influence of the higher-order effects on the solutions for the concerned model [62], Mihalache et al. analyzed the coaxial vortex solitons forthe CQGL-Equation [63], Fang et al. worked on soliton dynamics [64], Djoko et al. investigated the effects of the septic nonlinearity [65], Mou et al. studied discrete localized excitations [66] and Liu et al. analyzed harmonic and damped motions of dissipative solitons for the proposed model [67]. However, in this work, we have applied the appropriate transformations method to obtain the stated solutions for the governing model.

This article contains five classes of breather solutions (i.e., MB, KBM, GB, AB and homoclinic breather solutions), as well as lump, lump one stripe, lump two stripe and rogue wave solutions. Furthermore, a detailed analysis of SRW solution is made. Multiwave, M-shaped and interactional solutions are computed for ensuing model. These type of solutions, utilized in diverse fields of sciences, i.e., optics, engineering, physics and biology etc. [11,12,13,14,15,16,17,18,19,20,21]. A breather is a nonlinear localized wave and is a periodic solution of discrete lattice equations. Our newly attained results show a discrepancy of their shapes by appropriate choices of parameters. Now, we can definitely understand the geometric structure from Figure 1, which shows the lump profiles with one bright and dark soliton of the solution ${\mathsf{\Delta }}_1$ in Equation (7) via distinct parameters $\omega =1$. The bright and dark soliton behavior of three-dimensional profiles steadily increases the value of (i) $a_1=5,$ (ii) $a_1=10$ and (iii) $a_1=-3$. Figure 2 shows the contour shapes for Figure 1 successively. The lump one stripe profiles of the solution ${\mathsf{\Delta }}_2$ in Equation (10) are interpreted via distinct values of $\omega =5$, $k_1=2$, $\rho =4$, $\gamma =-1$, $p=2$, $q=-1$, $\nu =1$ and $b_0=3$. Three-dimensional profiles are shown in Figure 3 at (i) $a_1=5,$ (ii) $a_1=10$ and (iii) $a_1=-2$. Figure 4 shows the contour shapes for Figure 3 successively. Similarly, Figure 5 shows the lump two stripe graphs of the solution ${\mathsf{\Delta }}_3$ in Equation (13) for the distinct values of $\omega =5$, $k_1=2$, $\rho =4$, $\gamma =-1$, $k_2=1$, $k_3=1$, $k_4=2$ and $b_0=3$, with three-dimensional profiles at (i) $a_0=5,$ (ii) $a_0=10$ and (iii) $a_0=-2$. Figure 6 builds contour profiles for Figure 5 successively. The MB graphs of the solution ${\mathsf{\Delta }}_4$ in Equation (16) are interpreted via distinct values of $\omega =5$, $p_1=2$, $p_2=3$, ${\alpha }_1=1$, ${\alpha }_2=2.5$, ${\lambda }_1=1$, ${\lambda }_2=2$, ${\beta }_2=3$, $\rho =4$ and ${\gamma }_2=1$. Three-dimensional profiles at (i) $a_0=5,$ (ii) $a_0=10$ and (iii) $a_0=-1$ are shown in Figure 7. Figure 8 shows the contour profiles for Figure 7. In the same way, Figure 9 presents the KMB graphs of solution ${\mathsf{\Delta }}_5$ in Equation (19) through values of $a_1=2$, $a_3=1$, $a_4=3$, $\omega =5$, $p_1=2$, $p_2=3$, $b_1=1$, $b_2=2.5$, $b_3=1$, $b_4=2$ and $\rho =4$, with three-dimensional profiles at (i) $a_2=5,$ (ii) $a_2=20$ and (iii) $a_2=-1$. Figure 10 shows the contour profiles for Figure 9. The GB profiles of the solution ${\mathsf{\Delta }}_6$ in Equation (24) are formed through values of $\kappa =5,b=1$ and $\rho =4$. Three-dimensional graphs at (i) $\sigma =0.2,$ (ii) $\sigma =0.8$ and (iii) $\sigma =-0.1$ are shown in Figure 11. Similarly, Figure 12 shows the contour profiles for Figure 11. The AB profiles of the solution ${\mathsf{\Delta }}_7$ in Equation (27) are formed through values of $a=5$, $b=1$, $c=3$, $p_0=4$ and $m=5$. Three-dimensional graphs at (i) $\beta =5,$ (ii) $\beta =10$ and (iii) $\beta =-3$ are shown in Figure 13, while Figure 14 shows the contour graphs for Figure 13. The SRW profiles of the solution ${\mathsf{\Delta }}_8$ in Equation (30) are constructed for values of $ϵ=0.5$, $b=1$, $c=3$, $\gamma =0.1$, $\nu =10$, $\mu =2$ and $\kappa =2$. Three-dimensional graphs at (i) $\delta =-5,$ (ii) $\delta =10$ and (iii) $\delta =20$ are shown in Figure 14. Figure 15 shows the contour graphs for Figure 14. Similarly, MS graphs of solution ${\mathsf{\Delta }}_9$ in Equation (37) are formed with $a_6=0.2$, $c_1=1$, $k_1=3$, $a_4=0.1$, $b_1=10$, $b_2=2$, $c_2=2$ and $k_2=3$. Three-dimensional graphs at (i) $a_5=-5,$ (ii) $a_5=3$ and (iii) $a_5=7$ are shown in Figure 15. Figure 16 shows the contour shapes for Figure 15. The HB profiles of solution ${\mathsf{\Delta }}_{11}$ in Equation (43) are constructed for values of $c_1=2$, $c_2=1$, $k_1=3$, $k_2=0.1$, $\gamma =1$ and $\beta =5$. Three-dimensional graphs at (i) $d_3=-5,$ (ii) $d_3=3$ and (iii) $d_3=15$ are shown in Figure 17. Figure 18 shows the contour shapes for Figure 17, while the MS profiles of the solution ${\mathsf{\Delta }}_{11}$ in Equation (40) are constructed for values of $b_1=0.5$, $p=1$, $a_4=3$, $a_2=2$, $k_1=0.1$, $\nu =1$ and $T_r=2$. Three-dimensional graphs at (i) $a_5=-5,$ (ii) $a_5=3$ and (iii) $a_5=10$ are shown in Figure 19. Figure 20 shows the contour shapes for Figure 19. When the value of $a_5$ steadily increases, we can see that the waves come closer to interact with each other. In the same manner, Figure 21 presents the soliton profiles of the solution ${\mathsf{\Delta }}_{12}$ in Equation (46) for values of $a_4=2$, $b_2=1$, $b_1=3$, $a_2=0.1$, $T_r=1$, $c_2=5$, $k_2=2$ and $\nu =3$, with three-dimensional graphs at (i) $a_3=-5,$ (ii) $a_3=0.1$ and (iii) $a_5=4$. When the value of $k_2$ steadily increases, we can see from the behavior of the M-shaped wave that the waves come closer to interact with each other. Similarly, Figure 22 shows the contour shapes for Figure 21 successively. The KCR profiles of the solution ${\mathsf{\Delta }}_{13}$ in Equation (49) are formed for values of $b_2=1$, $b_1=3$, $a_2=0.1$, $\mu =2$, $g_0=3$, $b_3=2$, $b_4=5$, $c_2=4$ and $k_2=2$. Three-dimensional graphs at (i) $a_2=-5,$ (ii) $a_2=1$ and (iii) $b_2=40$ are shown in Figure 23. Figure 24 shows the contour shapes for Figure 23 successively. The PCR graphs of the solution ${\mathsf{\Delta }}_{14}$ in Equation (52) are formed for particular values of $b_2=1$, $a_4=2$, $a_2=-3$, $\mu =2$, $g_0=7$, $b_2=1$, $c_1=3$, $k_1=0.1$, $c_2=6$, $\gamma =2$ and $k_2=1$. Three-dimensional graphs at (i) $a_1=-4,$ (ii) $a_1=0$ and (iii) $a_1=30$ are shown in Figure 25. Finally, Figure 26, Figure 27 and Figure 28 shows the contour shapes for Figure 25 successively.

17. Concluding Remarks

We have considered the CQGL-equation under the influence of intrapulse Raman scattering (IRS) and constructed distinct localized wave solutions by employing test functions with the aid of an appropriate transformations method. Five classes of breather solutions (i.e., Ma, Kuznetsov-Ma, GB, AB, homoclinic breather solutions), as well as lump, lump one stripe, lump two stripe and rogue wave solutions were successfully evaluated. Furthermore, a detailed analysis of SRW solution was performed. Multiwave, M-shaped, interactional solutions, KCR solutions and PCR solutions are computed for the ensuing model. Interaction behaviors between multiple-lump waves and soliton were also discussed. Multiple-lump wave evolution with time was also observed. We graphically presented many valuable results obtained here. The conditions imposed on the parameters have been explicitly demonstrated to guarantee that they were well defined and that the solutions were localized. To our knowledge, the results are new for the governing equation. These results may be useful for the experimental realization of undistorted transmission of optical waves in optical fibers and further understanding of their optical transmission properties. Finally, we hope that the exact nature of these solitary waves interpreted here may be profitably exploited in designing the optimal Raman fiber laser experiments.