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

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.


INTRODUCTION
One of the interesting known models with a time-dependent coefficient is the nonautonomous NLSE with a harmonic potential. This is expressed as: 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)r 2 /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][2][3][4][5][6][7][8], super fluids, and so on [9][10][11][12][13][14][15][16][17][18]. In recent times, the theoretical study of these kinds of waves has become an interesting part of the field of nonlinear sciences [19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34]. The following section deals with the extraction of wave solutions with ST.

ROGUE WAVE SOLUTIONS
The envelope field q is considered in the following form [33]: where q R , q I , q, and φ are all dependent functions of x and t, while the intensity is defined by: 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: where λ is a constant. Substituting this new transformation into the real and imaginary part equations, the following equations are obtained: Simplifying the above equations, we perform the similarity reduction in the following way.

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: 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 where " * " denotes a complex conjugate. Consider that the solution of Equation (33) has of the form 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: 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 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 or − 4γ 2 + ω 2 r 4 + 4βPr 2 ω ± 4 −γ 2 ω 2 r 4 − 4βPr 2 ωγ 2 < 0.
The MI gain spectrum is given as The MI is significantly affected by P. If P is increased, the growth rate of MI will appear to disperse.

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.