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

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:

$\begin{array}{ll}i{q}_t+\frac{\alpha (t)}{2}{q}_{xx}+(-i\gamma (t)+\frac{\omega (t)r^2}{2}+\beta (t)|q{|}^2)q=0.& (1)\end{array}$

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 [1–8], super fluids, and so on [9–18]. In recent times, the theoretical study of these kinds of waves has become an interesting part of the field of nonlinear sciences [19–34]. 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]:

$\begin{array}{ll}q=(q_R+i{q}_I)e^{i\varphi },& (2)\end{array}$

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

$\begin{array}{ll}|q{|}^2=|q_R{|}^2+|q_I{|}^2.& (3)\end{array}$

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 q_R, q_I, and ϕ, which depend on x and t, the variables ξ(x, t) and τ(t) are introduced. Thus, the new transformations for q_R, q_I, and ϕ are constructed in this manner: q_R = A(t) + B(t)P(ξ(x, t), τ(t)), q_I = 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:

$\begin{array}{l}-2(A+BP)({\zeta }_t+\lambda {\tau }_t)-2(C_t+D_{t}Q+D{Q}_{\xi }{\xi }_t+D{Q}_{\tau }{\tau }_t)\\ -\alpha (t)(C+DQ){\zeta }_{xx}-\alpha (t)(A+BP){\zeta }_x^2-2\alpha (t)D{Q}_{\xi }{\xi }_x{\zeta }_x\\ +\alpha (t)(B{P}_{\xi \xi }{\xi }_x^2+B{P}_{\xi }{\xi }_{xx})+2\beta (t)({(A+BP)}^2\\ +{(C+DQ)}^2)(A+BP)+2\gamma (t)(C+DQ)\\ +\omega (t)x^2(A+BP)=0,& (4)\end{array}$

$\begin{array}{l}-2(C+DQ)({\zeta }_t+\lambda {\tau }_t)+2(A_t+B_{t}P+B{P}_{\xi }{\xi }_t+B{P}_{\tau }{\tau }_t)\hfill \\ +\alpha (t)(A+BP){\zeta }_{xx}-\alpha (t)(C+DQ){\zeta }_x^2+2\alpha (t)B{P}_{\xi }{\xi }_x{\zeta }_x\hfill \\ +\alpha (t)(D{Q}_{\xi \xi }{\xi }_x^2+D{Q}_{\xi }{\xi }_{xx})+2\beta (t)({(A+BP)}^2\hfill \\ +{(C+DQ)}^2)(C+DQ)-2\gamma (t)(A+BP)\hfill \\ +\omega (t)x^2(C+DQ)=0.\hfill & (5)\end{array}$

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

$\begin{array}{ll}{\xi }_{xx}=0,& (6)\end{array}$

$\begin{array}{ll}{\xi }_t+\alpha (t){\xi }_x{\zeta }_x=0,& (7)\end{array}$

$\begin{array}{ll}\omega (t)x^2-2{\zeta }_t-\alpha (t){\zeta }_x^2=0,& (8)\end{array}$

$\begin{array}{ll}2{\sigma }_t+(\alpha (t){\zeta }_{xx}-2\gamma (t))\sigma =0,\text{for }(\sigma =A,B,C,D),\hfill & (9)\end{array}$

$\begin{array}{l}-2(A+BP)\lambda {\tau }_t-2D{Q}_{\tau }{\tau }_t+\alpha (t)B{P}_{\xi \xi }{\xi }_x^2\hfill \\ +2\beta (t)(A+BP)(|A+BP{|}^2+|C+DQ{|}^2)=0,\hfill & (10)\end{array}$

$\begin{array}{l}-2(C+DQ)\lambda {\tau }_t+2B{P}_{\tau }{\tau }_t+\alpha (t)D{Q}_{\xi \xi }{\xi }_x^2\hfill \\ +2\beta (t)(C+DQ)(|A+BP{|}^2+|C+DQ{|}^2)=0.& (11)\end{array}$

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.

$\begin{array}{ll}\xi =\delta (t)x+{\delta }_0(t),& (12)\end{array}$

$\begin{array}{ll}\omega =\frac{2{\zeta }_t+\alpha (t){\zeta }_x^2}{x^2},& (13)\end{array}$

$\begin{array}{ll}{\zeta }_{(x,t)}=-\frac{1}{\alpha (t)}(\frac{\delta {(t)}_t}{2\delta (t)}{x}^2+\frac{{\delta }_0(t)}{\delta (t)}x),& (14)\end{array}$

$\begin{array}{l}A(t)=a_0\text{ exp }\left[\frac{1}{2}{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right],\hfill \\ B(t)=bA,D(t)=dA,\hfill & (15)\end{array}$

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

$\begin{array}{ll}\tau (t)=\frac{1}{2}{\displaystyle {\int }_0^t}\alpha (k){\delta }^2(k)dk,& (16)\end{array}$

$\begin{array}{ll}\beta (t)=\frac{\alpha (t){\delta }^2}{2A^2}.& (17)\end{array}$

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

$\begin{array}{l}-2(1+bP)\lambda -2d{Q}_{\tau }+\alpha (t)b{P}_{\xi \xi }\hfill \\ +2\beta (t)(1+bP)(|1+bP{|}^2+|1+dQ{|}^2)=0,\hfill & (18)\end{array}$

$\begin{array}{l}-2(c+dQ)\lambda +2b{P}_{\tau }+\alpha (t)d{Q}_{\xi \xi }\hfill \\ +2\beta (t)(c+dQ)(|1+bP{|}^2+|1+dQ{|}^2)=0.\hfill & (19)\end{array}$

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

$\begin{array}{ll}P(\xi ,\tau )=-\frac{4}{R_1(\xi ,\tau )b},Q(\xi ,\tau )=-\frac{8\tau }{R_1(\xi ,\tau )d},& (20)\end{array}$

where $R_1=1+2{\xi }^2+4{\tau }^2$. Moreover, the second-order solution is obtained as

$\begin{array}{ll}P(\xi ,\tau )=\frac{P_1(\xi ,\tau )}{R_2(\xi ,\tau )b},Q(\xi ,\tau )=\frac{Q_1(\xi ,\tau )\tau }{R_2(\xi ,\tau )d},& (21)\end{array}$

$\begin{array}{ll}{P}_1(\xi ,\tau )=\frac{3}{8}-9{\tau }^2-\frac{3{\xi }^2}{2}-6{\xi }^2{\tau }^2-10{\tau }^4-\frac{{\xi }^4}{2},& (22)\end{array}$

$\begin{array}{ll}{Q}_1(\xi ,\tau )=-\frac{15}{4}+2{\tau }^2-3{\xi }^2+4{\xi }^2{\tau }^2+4{\tau }^4+{\xi }^4,& (23)\end{array}$

$\begin{array}{l}{R}_2=\frac{3}{32}+\frac{33}{8}{\tau }^2+\frac{9{\xi }^2}{16}-\frac{3{\xi }^2{\tau }^2}{2}+\frac{9{\tau }^4}{2}+\frac{{\xi }^4}{8}\\ \frac{2{\xi }^6}{3}+{\xi }^2{\tau }^6+\frac{{\xi }^4{\tau }^2}{2}+\frac{{\xi }^6}{12}.& (24)\end{array}$

The direct reduction solution is considered in the following form:

$\begin{array}{ll}q=A(1+bP+idQ)e^{i(\zeta +\tau )},& (25)\end{array}$

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:

$\begin{array}{l}q=a_0\left(\frac{-3+2{\xi }^2+4{\tau }^2-8i\tau }{1+2{\xi }^2+4{\tau }^2}\right)\\ \times \text{ exp }\left[\frac{1}{2}{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right]e^{i(\zeta ,\tau )},& (26)\end{array}$

whose amplitude can be written as

$\begin{array}{l}|q{|}^2=a_0^2\frac{{[-3+2{(\delta (t)x+{\delta }_0(t))}^2+4{\tau }^2]}^2+64{\tau }^2(t)}{{[1+2{(\delta (t)x+{\delta }_0(t))}^2+4{\tau }^2(t)]}^2}\\ \times \text{exp }\left[{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right].& (27)\end{array}$

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:

$\begin{array}{l}q=a_0(1-\frac{4(-3+4{\xi }^4+72{\tau }^2+80{\tau }^4+12{\xi }^2(1+4{\tau }^2))}{3+18{\xi }^2+4{\xi }^4+24{\xi }^6+4(33+4{\xi }^2(-3+{\xi }^2)){\tau }^2+144{\tau }^4+32{\xi }^2{\tau }^6}\\ +i\frac{8\tau (4{\xi }^2(-3+{\xi }^2+4{\tau }^2)+(-5+8t^2))}{3+18{\xi }^2+4{\xi }^4+24{\xi }^6+4(33+4{\xi }^2(-3+{\xi }^2)){\tau }^2+144{\tau }^4+32{\xi }^2{\tau }^6})\\ \times \text{ exp }\left[\frac{1}{2}{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right]e^{i(\zeta +\tau )},& (28)\end{array}$

whose intensity is written as

$\begin{array}{l}|q{|}^2=a_0^2\left[\right(1-(4(-3+4{\xi }^4+72{\tau }^2+80{\tau }^4+12{\xi }^2(1+4{\tau }^2))/(3+18{\xi }^2\\ +4{\xi }^4+24{\xi }^6+4(33+4{\xi }^2(-3+{\xi }^2)){\tau }^2+144{\tau }^4+32{\xi }^2{\tau }^6){)}^2\\ +(8\tau (4{\xi }^2(-3+{\xi }^2+4{\tau }^2)+(-5+8t^2))/(3+18{\xi }^2+4{\xi }^4\\ +24{\xi }^6+4(33+4{\xi }^2(-3+{\xi }^2)){\tau }^2+144{\tau }^4+32\xi {\tau }^6{)}^2]\\ \times \text{ exp}\left[{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right],& (29)\end{array}$

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:

$\begin{array}{l}|q{|}^2=a_0^2\frac{{[-3+2{(bx+{\delta }_0(t))}^2+4{\tau }^2]}^2+64{\tau }^2(t)}{{[1+2{(bx+{\delta }_0(t))}^2+4{\tau }^2(t)]}^2}\\ \times \text{ exp }\left[{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right].& (30)\end{array}$

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

$\begin{array}{l}|q{|}^2=a_0^2\left[\right(1-(4(-3+4{(bx+{\delta }_0(t))}^4+72{\tau }^2+80{\tau }^4\\ +12{(bx+{\delta }_0(t))}^2(1+4{\tau }^2))/(3+18{(bx+{\delta }_0(t))}^2\\ +4{(bx+{\delta }_0(t))}^4+24{(bx+{\delta }_0(t))}^6+4(33+4(bx\\ +{\delta }_0(t){)}^2(-3+{(bx+{\delta }_0(t))}^2)){\tau }^2+144{\tau }^4+32(bx\\ +{\delta }_0(t){)}^2{\tau }^6){)}^2+(8\tau (4{(bx+{\delta }_0(t))}^2(-3+(bx\\ +{\delta }_0(t){)}^2+4{\tau }^2)+(-5+8t^2))/(3+18{(bx+{\delta }_0(t))}^2\\ +4{(bx+{\delta }_0(t))}^4+24{(bx+{\delta }_0(t))}^6\\ +4(33+4{(bx+{\delta }_0(t))}^2(-3+{(bx+{\delta }_0(t))}^2)){\tau }^2+144{\tau }^4\\ {\displaystyle +32(bx+{\delta }_0(t)){\tau }^6{)}^2]}\\ \times \text{ exp }\left[{\displaystyle {\int }_0^t}\right(\frac{\delta {(k)}_k}{\delta (k)}+2\gamma (k)\left)dk\right].& (31)\end{array}$

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:

$\begin{array}{ll}q(x,t)=\{\sqrt{P}+\chi (x,t)\}\times e^{(i{\phi }_{NL})},{\phi }_{NL}=\beta Px,& (32)\end{array}$

where χ < < P, P is the incident optical power, and φ_NL 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

$\begin{array}{l}i\frac{\partial \chi }{\partial t}+\frac{1}{2}\alpha (t)\frac{{\partial }^2\chi }{\partial x^2}+\beta (t)P(\chi +{\chi }^{*})\hfill \\ +(-i\gamma (t)+\frac{\omega (t)x^2}{2})\chi =0,\hfill & (33)\end{array}$

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

$\begin{array}{ll}\chi (x,t)={\eta }_1{e}^{i(kx-\nu t)}+{\eta }_2{e}^{-i(kx-\nu t)},& (34)\end{array}$

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:

$\begin{array}{l}-{\nu }^2+\alpha \nu k^2-2i\gamma \nu -\frac{{\alpha }^2}{4}{k}^4+i\alpha \gamma k^2\hfill \\ +\beta P\omega r^2+{\gamma }^2+\frac{{\omega }^2{r}^4}{4}=0.\hfill & (35)\end{array}$

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

$\begin{array}{l}\nu =\frac{1}{2}\alpha k^2\pm \frac{1}{2}\\ \sqrt{-4{\gamma }^2+{\omega }^2{r}^4+4\beta P{r}^2\omega \pm 4\sqrt{-{\gamma }^2{\omega }^2{r}^4-4\beta P{r}^2\omega {\gamma }^2}}.& (36)\end{array}$

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

$\begin{array}{ll}{\gamma }^2\omega r^2(\omega r^2+4\beta P)>0,& (37)\end{array}$

or

$\begin{array}{ll}(-4{\gamma }^2+{\omega }^2{r}^4+4\beta P{r}^2\omega \pm 4\sqrt{-{\gamma }^2{\omega }^2{r}^4-4\beta P{r}^2\omega {\gamma }^2})<0.& (38)\end{array}$

The MI gain spectrum is given as

$\begin{array}{l}g(\nu )=2Im(\nu )\\ =\sqrt{-4{\gamma }^2+{\omega }^2{r}^4+4\beta P{r}^2\omega \pm 4\sqrt{-{\gamma }^2{\omega }^2{r}^4-4\beta P{r}^2\omega {\gamma }^2}}.& (39)\end{array}$

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 $\delta (t)=0.7+0.9\text{ cos}(0.1t),{\delta }_0(t)=0.5t^2$, $\delta (t)=0.86+1.2\text{ cos}(0.1t),{\delta }_0(t)=0.35t^2$ and ${\delta }_0(t)=0.35t^2$, and δ(t) = 0.1 + 1.2 cos(0.1t) and ${\delta }_0(t)=0.35t^2$. The graphical representations are demonstrated in Figures 1C–E, respectively.

FIGURE 1

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) ${\delta }_0(t)=0.5t^2$ and δ(t) = 0.7 + 0.9 cos(0.1t), (D) ${\delta }_0(t)=0.35t^2$ and δ(t) = 0.86 + 1.2 cos(0.1t), and (E) ${\delta }_0(t)=0.35t^2$, δ(t) = 0.1 + 1.2 cos(0.1t), and ${\delta }_0(t)=0.35t^2$.

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

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 ${\delta }_0(t)=0.35t^2$, and (F) δ(t) = 0.1 + 1.2 cos(0.1t) and ${\delta }_0(t)=0.35t^2$.

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

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

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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