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Original Research ARTICLE

Front. Phys., 04 February 2020 | https://doi.org/10.3389/fphy.2019.00255

Rogue Wave Solutions and Modulation Instability With Variable Coefficient and Harmonic Potential

Safdar Ali1 and Muhammad Younis2*
  • 1Department of Mathematics and Statistics, The University of Lahore, Lahore, Pakistan
  • 2Punjab University College of Information Technology, University of the Punjab, Lahore, Pakistan

This article studies the propagation of rogue waves with a nonautonomous NLSE in the presence of external potential. This model is considered to be an important model for many physical phenomena in quantum mechanics and optical fiber. The obtained waves are of first and second order and are investigated using similarity transformation. The nonlinear dynamic behavior of these waves is also demonstrated with different parameter values for the magnetic and gravity fields. The results show the influence of these fields over density, width, and peak heights. Moreover, the modulation instability is also discussed.

1. Introduction

One of the interesting known models with a time-dependent coefficient is the nonautonomous NLSE with a harmonic potential. This is expressed as:

iqt+α(t)2qxx+(-iγ(t)+ω(t)r22+β(t)|q|2)q=0.    (1)

The function q is a wave profile in a homogeneous nonlinear medium, α(t) is the dispersion coefficient, β(t) is the measure of the Kerr nonlinearity, γ(t) is considered as the distributed gain/loss coefficient, and the harmonic potential is given by ω(t)r2/2. This model describes many physical phenomena in nonlinear sciences.

This article studies the first- and second-order rogue wave solutions. It is a single giant wave whose amplitude is two to three times higher than those of the surrounding waves. The interesting fact regarding this wave is that it appears from nowhere and disappears without a trace. The similarity transformation (ST) is utilized to construct the solutions. These waves are also found in deep and shallow water and, beyond oceanic expanses, in optical fibers [18], super fluids, and so on [918]. In recent times, the theoretical study of these kinds of waves has become an interesting part of the field of nonlinear sciences [1934]. The following section deals with the extraction of wave solutions with ST.

2. Rogue Wave Solutions

The envelope field q is considered in the following form [33]:

q=(qR+iqI)eiϕ,    (2)

where qR, qI, q, and ϕ are all dependent functions of x and t, while the intensity is defined by:

|q|2=|qR|2+|qI|2.    (3)

The use of Equations (2)–(3) in (1) yields an equation with variable coefficients. After solving and simplification, we can split this equation into its real and imaginary equations. For the real functions qR, qI, and ϕ, which depend on x and t, the variables ξ(x, t) and τ(t) are introduced. Thus, the new transformations for qR, qI, and ϕ are constructed in this manner: qR = A(t) + B(t)P(ξ(x, t), τ(t)), qI = C(t) + D(t)Q(ξ(x, t), τ(t)), and ϕ = ζ(x, t) + λ τ(t), where λ is a constant. Substituting this new transformation into the real and imaginary part equations, the following equations are obtained:

-2(A+BP)(ζt+λτt)-2(Ct+DtQ+DQξξt+DQττt)-α(t)(C+DQ)ζxx-α(t)(A+BP)ζx2-2α(t)DQξξxζx+α(t)(BPξξξx2+BPξξxx)+2β(t)((A+BP)2+(C+DQ)2)(A+BP)+2γ(t)(C+DQ)+ω(t)x2(A+BP)=0,    (4)
-2(C+DQ)(ζt+λτt)+2(At+BtP+BPξξt+BPττt)+α(t)(A+BP)ζxx-α(t)(C+DQ)ζx2+2α(t)BPξξxζx+α(t)(DQξξξx2+DQξξxx)+2β(t)((A+BP)2+(C+DQ)2)(C+DQ)-2γ(t)(A+BP)+ω(t)x2(C+DQ)=0.    (5)

Simplifying the above equations, we perform the similarity reduction in the following way.

ξxx=0,    (6)
ξt+α(t)ξxζx=0,    (7)
ω(t)x2-2ζt-α(t)ζx2=0,    (8)
2σt+(α(t)ζxx-2γ(t))σ=0,for (σ=A,B,C,D),    (9)
-2(A+BP)λτt-2DQττt+α(t)BPξξξx2+2β(t)(A+BP)(|A+BP|2+|C+DQ|2)=0,    (10)
-2(C+DQ)λτt+2BPττt+α(t)DQξξξx2+2β(t)(C+DQ)(|A+BP|2+|C+DQ|2)=0.    (11)

where ξ(x, t), ζ(x, t), A(t), B(t), C(t), D(t), P(ξ, τ), and Q(ξ, τ) are different functions and are determined later. After algebraic computation, the above equations produce the following results.

ξ=δ(t)x+δ0(t),    (12)
ω=2ζt+α(t)ζx2x2,    (13)
ζ(x,t)=-1α(t)(δ(t)t2δ(t)x2+δ0(t)δ(t)x),    (14)
A(t)=a0 exp [120t(δ(k)kδ(k)+2γ(k))dk],B(t)=bA,D(t)=dA,    (15)

where a0, b, and d are constants, and C = 0. The variables τ(t) and β(t) are given by

τ(t)=120tα(k)δ2(k)dk,    (16)
β(t)=α(t)δ22A2.    (17)

To further reduce to Equations (4) and (5) to the partial differential equations, we require

-2(1+bP)λ-2dQτ+α(t)bPξξ+2β(t)(1+bP)(|1+bP|2+|1+dQ|2)=0,    (18)
-2(c+dQ)λ+2bPτ+α(t)dQξξ+2β(t)(c+dQ)(|1+bP|2+|1+dQ|2)=0.    (19)

According to the direct method, we obtain the first-order rational solution

P(ξ,τ)=-4R1(ξ,τ)b ,Q(ξ,τ)=-8τR1(ξ,τ)d,    (20)

where R1=1+2ξ2+4τ2. Moreover, the second-order solution is obtained as

P(ξ,τ)=P1(ξ,τ)R2(ξ,τ)b ,Q(ξ,τ)=Q1(ξ,τ)τR2(ξ,τ)d,    (21)
P1(ξ,τ)=38-9τ2-3ξ22-6ξ2τ2-10τ4-ξ42,    (22)
Q1(ξ,τ)=-154+2τ2-3ξ2+4ξ2τ2+4τ4+ξ4,    (23)
R2=332+338τ2+9ξ216-3ξ2τ22+9τ42+ξ48           2ξ63+ξ2τ6+ξ4τ22+ξ612.    (24)

The direct reduction solution is considered in the following form:

q=A(1+bP+idQ)ei(ζ+τ),    (25)

where ξ(x, t), ζ(x, t), A(t), τ(t), P(ξ, τ), and Q(ξ, τ) are expressed by the relations given in Equations (12), (14)–(16), and (20), respectively.

The rogue wave solution of first order to Equation (1) can be obtained using Equations (20) and (25); thus, after simplification, we may have the following form:

q=a0(-3+2ξ2+4τ2-8iτ1+2ξ2+4τ2)       × exp [120t(δ(k)kδ(k)+2γ(k))dk]ei(ζ,τ),    (26)

whose amplitude can be written as

|q|2=a02[3+2(δ(t)x+δ0(t))2+4τ2]2+64τ2(t)[1+2(δ(t)x+δ0(t))2+4τ2(t)]2              ×exp [0t(δ(k)kδ(k)+2γ(k))dk].    (27)

The rogue wave (rational-like) solution of second order to Equation (1) can be obtained using Equations (21) and (25); thus, after simplification, we may have the following form:

q=a0(1-4(-3+4ξ4+72τ2+80τ4+12ξ2(1+4τ2))3+18ξ2+4ξ4+24ξ6+4(33+4ξ2(-3+ξ2))τ2+144τ4+32ξ2τ6        +i8τ(4ξ2(-3+ξ2+4τ2)+(-5+8t2))3+18ξ2+4ξ4+24ξ6+4(33+4ξ2(-3+ξ2))τ2+144τ4+32ξ2τ6)        × exp [120t(δ(k)kδ(k)+2γ(k))dk]ei(ζ+τ),    (28)

whose intensity is written as

|q|2=a02[(1-(4(-3+4ξ4+72τ2+80τ4+12ξ2(1+4τ2))/(3+18ξ2               +4ξ4+24ξ6+4(33+4ξ2(-3+ξ2))τ2+144τ4+32ξ2τ6))2               +(8τ(4ξ2(-3+ξ2+4τ2)+(-5+8t2))/(3+18ξ2+4ξ4               +24ξ6+4(33+4ξ2(-3+ξ2))τ2+144τ4+32ξτ6)2]               × exp[0t(δ(k)kδ(k)+2γ(k))dk],    (29)

The following section discusses the dynamical behavior of waves.

3. Dynamical Behavior of Waves

The behavior of constructed waves is demonstrated using the relation δ(t) = b + l cos(ωt). The first term on the right-hand side represents the gravity field (GF) b = δmg with the real parameter δ, and the second term on the same side is the external magnetic field (EMF) and is given by l cos(ωt).

There are two possibilities for the occurrence of the waves in the presence of GF. The first is that when the GF (i.e., b ≠ 0 and l = 0) is acting, and the second is that when both the GF and EMF are present (i.e., b ≠ 0 and l ≠ 0).

Now, we discuss the first possibility for nonlinear dynamical behavior, when there is only the GF. Say δ(t) = b, and the amplitude (corresponding to l = 0) is given by the following relation:

|q|2=a02[3+2(bx+δ0(t))2+4τ2]2+64τ2(t)[1+2(bx+δ0(t))2+4τ2(t)]2               × exp [0t(δ(k)kδ(k)+2γ(k))dk].         (30)

The behavior of the second-order rogue wave is considered when there is only the GF. Then, the value of δ(t) = b, so the amplitude (corresponding to l = 0) is given by

|q|2=a02[(1(4(3+4(bx+δ0(t))4+72τ2+80τ4              +12(bx+δ0(t))2(1+4τ2))/(3+18(bx+δ0(t))2              +4(bx+δ0(t))4+24(bx+δ0(t))6+4(33+4(bx              +δ0(t))2(3+(bx+δ0(t))2))τ2+144τ4+32(bx               +δ0(t))2τ6))2+(8τ(4(bx+δ0(t))2(3+(bx              +δ0(t))2+4τ2)+(5+8t2))/(3+18(bx+δ0(t))2              +4(bx+δ0(t))4+24(bx+δ0(t))6              +4(33+4(bx+δ0(t))2(3+(bx+δ0(t))2))τ2+144τ4               +32(bx+δ0(t))τ6)2]              × exp [0t(δ(k)kδ(k)+2γ(k))dk].    (31)

4. Analysis of Modulation Instability

In this section, we study the modulation instability (MI). The linear stability analysis technique [34] has been applied, and we suppose that Equation (1) has the perturbed steady-state (PSS) solution in the following form:

q(x,t)={P+χ(x,t)}×e(iφNL), φNL=βPx,    (32)

where χ < < P, P is the incident optical power, and φNL is the phase component. The perturbation χ(x, t) is examined by using linear stability analysis. Now, we substitute Equation (32) into Equation (1) and, after linearizing it, we obtain

iχt+12α(t)2χx2+β(t)P(χ+χ*)+(-iγ(t)+ω(t)x22)χ=0,       (33)

where “*” denotes a complex conjugate. Consider that the solution of Equation (33) has of the form

χ(x,t)=η1ei(kx-νt)+η2e-i(kx-νt),    (34)

where ν and k are the frequency of perturbation and normalized wave number, respectively. After putting Equation (34) into Equation (33) and by separating the obtained equation into its real and imaginary parts, we get the dispersion relation:

-ν2+ανk2-2iγν-α24k4+iαγk2+βPωr2+γ2+ω2r44=0.    (35)

The dispersion relation given in Equation (35) has the following solutions in terms of frequency ν after taking the modulus of the above equation. We have

ν=12αk2±12      -4γ2+ω2r4+4βPr2ω±4-γ2ω2r4-4βPr2ωγ2.    (36)

The above dispersion relation determines the PSS stability, and that depends on the harmonic potential or distributed gain (loss) coefficient of the model. If the frequency ν has an imaginary part, the PSS solution is unstable since the perturbations grow exponentially. On the other hand, if ν is real, then the PSS solution is stable against small perturbations. The necessary condition for the existence of MI is

γ2ωr2(ωr2+4βP)>0,    (37)

or

(-4γ2+ω2r4+4βPr2ω±4-γ2ω2r4-4βPr2ωγ2)<0.    (38)

The MI gain spectrum is given as

g(ν)=2Im(ν)         =-4γ2+ω2r4+4βPr2ω±4-γ2ω2r4-4βPr2ωγ2.    (39)

The MI is significantly affected by P. If P is increased, the growth rate of MI will appear to disperse.

5. Graphical Results and Discussion

The graphical representation of the amplitude defined by Equation (30) considering a0 = 1, α = t, and γ(t) = sin3(0.005t) is depicted in Figures 1A,B, with the values of only GF b (0.5 and 0.79) and δ0 (0.5 and 0.61). The graph with the maximum peak can be obtained at b = 0.5 and δ0 = 0.5. For the second possibility, when the GF and the EMF are both present, we discuss the graphical behavior of the solutions. For this, let us consider δ(t)=0.7+0.9 cos(0.1t),δ0(t)=0.5t2, δ(t)=0.86+1.2 cos(0.1t),δ0(t)=0.35t2 and δ0(t)=0.35t2, and δ(t) = 0.1 + 1.2 cos(0.1t) and δ0(t)=0.35t2. The graphical representations are demonstrated in Figures 1C–E, respectively.

FIGURE 1
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Figure 1. 3D graphical representations of first-order rogue waves. (A) b = 0.5 and δ0(t) = 0.5, (B) b = 0.79 and δ0(t) = 0.61, (C) δ0(t)=0.5t2 and δ(t) = 0.7 + 0.9 cos(0.1t), (D) δ0(t)=0.35t2 and δ(t) = 0.86 + 1.2 cos(0.1t), and (E) δ0(t)=0.35t2, δ(t) = 0.1 + 1.2 cos(0.1t), and δ0(t)=0.35t2.

The results show that there are no different effects of GF on first- and second-order rogue waves. Graphical representations of the amplitude given by Equation (31) at a0 = 1 and γ(t) = sin3(0.005t) with different values of GF and δ0(t) is depicted in Figures 2A–C. Six small peaks appear around the one high peak of the second-order solution. Graphical representations of second-order rogue waves with both GF and EMF are also shown in Figures 2D–F.

FIGURE 2
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Figure 2. 3D graphical representations of second-order waves. (A) δ(t) = 0.5 and δ0(t) = 0.1, (B) δ(t) = 0.4 and δ0(t) = 0.1, (C) δ(t) = 0.7 and δ0(t) = 0.2 exp(sech(0.2t)), (D) δ(t) = 0.5sech(0.2t) and δ0(t) = 0.5 exp(sech(0.2t)), (E) δ(t) = 0.5 + 1.2 cos(0.005t) and δ0(t)=0.35t2, and (F) δ(t) = 0.1 + 1.2 cos(0.1t) and δ0(t)=0.35t2.

Graphical representations of the amplitudes given by equation (30) at a0 = 1 and γ(t) = t are depicted in Figures 3A–D with the different parameter values. The curves in Figures 3A,B are formed under the GF, and those in Figures 3C,D are formed when both the GF and EMF are present.

FIGURE 3
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Figure 3. 3D graphical representations of first-order rogue waves. The figures correspond with (A) b = 1.3, δ0(t) = exp(0.5 + 0.5 cos t) and α = tan2(0.02t), (B) b = 1.5, δ0(t) = exp(0.05 + 0.5 cos t) and α = tan2(0.02t), (C) δ(t) = 1.4 + 0.05 cos t, δ0(t) = exp(0.002 + 0.4 cos t) and α = tan2(0.02t), and (D) δ(t) = 1.3 + 0.01 cos t, δ0(t) = exp(0.05 + 0.5 cos t), and α = tan2(0.02t).

Graphical representations of the amplitude given by Equation (31) at a0 = 1 and γ(t) = t with different values of GF and δ0(t) are depicted in Figures 4A,B. Small lumps appear in the graph of the second-order solution. Graphical demonstrations of second-order rogue waves with both GF and EMF are shown in Figures 4C,D.

FIGURE 4
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Figure 4. 3D graphical representations of second-order rogue waves. These are constructed with (A) b = 0.9, δ0(t) = 0.001t and α = 5 tan2(0.05t), (B) b = 0.9, δ0(t) = 0.001t and α = 5 tan2(0.05t), (C) δ(t) = 1.5 + 0.4 cos t, δ0(t) = 0.001t and α = 25 tan2(0.05t), and (D) δ(t) = 1.5 + 0.001 cos t, δ0(t) = 0.001t and α = 35 tan2(0.05t).

6. Conclusion

This article studies the construction of rogue waves in NLSE with a variable coefficient in the presence of harmonic potential. The graphical demonstration shows that the dynamical behavior of waves under the influence of gravity and magnetic fields in linear potential. It is observed that in the presence of GF, the density remains constant, while peak height and width remain invariant. The obtained solutions are of first and second order and are constructed using the ST approach. Moreover, the MI is calculated and is significantly affected by incident optical power.

Author Contributions

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

Conflict of Interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

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Keywords: rogue wave solutions, modulation instability, similarity transformation, NLSE, harmonic potential

Citation: Ali S and Younis M (2020) Rogue Wave Solutions and Modulation Instability With Variable Coefficient and Harmonic Potential. Front. Phys. 7:255. doi: 10.3389/fphy.2019.00255

Received: 21 October 2019; Accepted: 31 December 2019;
Published: 04 February 2020.

Edited by:

Mustafa Inc, Firat University, Turkey

Reviewed by:

Aly R. Seadawy, Taibah University, Saudi Arabia
Abdullahi Yusuf, Federal University, Dutse, Nigeria

Copyright © 2020 Ali and Younis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Muhammad Younis, younis.pu@gmail.com

These authors have contributed equally to this work