New Bright and Kink Soliton Solutions for Fractional Complex Ginzburg–Landau Equation with Non-Local Nonlinearity Term

Abstract

In this paper, we aim to discuss a fractional complex Ginzburg–Landau equation by using the parabolic law and the law of weak non-local nonlinearity. Then, we derive dynamic behaviors of the given model under certain parameter regions by employing the planar dynamical system theory. Further, we apply the ansatz method to derive soliton, bright and kinked solitons and verify their existence by imposing certain conditions. In addition, we integrate our solutions in appropriate dimensions to explain their behavior at various groups of parameters. Moreover, we compare the graphical representations of the established solutions at different fractional derivatives and illustrate the impact of the fractional derivative on the investigated soliton solutions as well.

Editorial for the Special Issue “Operators of Fractional Calculus and Their Multidisciplinary Applications”

“…The geographical distribution of the contributors to this Special Issue is remarkably widely-scattered. Their contributions (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) originated in many different countries on every continent of the world. The subject matter of these nineteen publications (see [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]) deals extensively with such topics as fractional-order complex Ginzburg-Landau equations, fractional modeling for the treatment of cancer by using radiotherapy, fractional-order fuzzy complex-valued neural networks, the fractal-fractional Michaelis-Menten enzymatic reaction model, fractional-calculus operators involving the (p, q)-extended Bessel and Bessel-Wright functions, fractional-order diffusion-wave equations, Abel integral equations and their fractional-order analogues, nonlinear integro-differential equations, fractionalorder investigations of a number of celebrated integral inequalities, such as those that are popularly called the Pólya-Szegö inequality, the Grüss inequality, the Hermite-Hadamard inequality, and so on.…”

“…We will now briefly describe the developments which are reported in this Special Issue. The authors of [1] have studied a fractional-order complex Ginzburg-Landau equation by using the parabolic law and the law of weak non-local non-linearity. The study presented in [2] is based upon a modification of a well-known predator-prey equation or the Lotka-Volterra competition model.…”

Traveling Wave Solutions for Time-Fractional mKdV-ZK Equation of Weakly Nonlinear Ion-Acoustic Waves in Magnetized Electron–Positron Plasma

“…In recent decades, the study of fractional differential equations (FDEs) became an attractive field due to its significant and notable application in several areas of science. Consequently, it has become necessary to develop and present new methods and approaches to derive numerical and analytical solutions for this type of equation (see, e.g., [18][19][20][21][22][23][24][25][26][27][28][29][30]). This paper deals with the time-fractional mKdV-ZK equation:…”