Study of Ion‐Acoustic Solitary Waves in a Magnetized Plasma Using the Three‐Dimensional Time‐Space Fractional Schamel‐KdV Equation

Abstract

The study of ion-acoustic solitary waves in a magnetized plasma has long been considered to be an important research subject and plays an increasingly important role in scientific research. Previous studies have focused on the integer-order models of ionacoustic solitary waves. With the development of theory and advancement of scientific research, fractional calculus has begun to be considered as a method for the study of physical systems. The study of fractional calculus has opened a new window for understanding the features of ion-acoustic solitary waves and can be a potentially valuable approach for investigations of magnetized plasma. In this paper, based on the basic system of equations for ion-acoustic solitary waves and using multi-scale analysis and the perturbation method, we have obtained a new model called the three-dimensional(3D) Schamel-KdV equation. Then, the integerorder 3D Schamel-KdV equation is transformed into the time-space fractional Schamel-KdV (TSF-Schamel-KdV) equation by using the semi-inverse method and the fractional variational principle. To study the properties of ion-acoustic solitary waves, we discuss the conservation laws of the new time-space fractional equation by applying Lie symmetry analysis and the RiemannLiouville fractional derivative. Furthermore, the multi-soliton solutions of the 3D TSF-Schamel-KdV equation are derived using the Hirota bilinear method. Finally, with the help of the multi-soliton solutions, we explore the characteristics of motion of ionacoustic solitary waves.

Long-Time Asymptotics of a Three-Component Coupled mKdV System

“…Connections between different approaches would be interesting. About coupled mKdV equations, there are many other studies such as integrable couplings [64], super hierarchies [65] and fractional analogous equations [66,67], and an important topic for further study is long-time asymptotics of those generalized integrable counterparts via the nonlinear steepest descent method. It is hoped that our result could be helpful in computing limiting behaviors of solutions incorporating features of other exact solutions, such as lumps [68,69], from the perspective of steepest descent based on RH problems.…”

N-Fold Darboux transformation of the discrete Ragnisco–Tu system

“…λ j and γ j (λ j = λ k , γ j = γ k , as k = j) are some parameters suitably chosen such that the determinants of coefficients for system (8) are nonzero. From (7)-(9), it is easy to see that…”

“…In recent years, a great deal of progress has been made on the theory of discrete integrable systems. Lots of important nonlinear integrable differential-difference equations have been obtained [1][2][3][4][5][6][7][8][9][10][11][12]. In particular, constructing exact solutions for a differential-difference equation is one of the most fundamental and significant topics.…”

The Finite Volume WENO with Lax–Wendroff Scheme for Nonlinear System of Euler Equations

“…It has been an important work to study nonlinear PDE [1] due to their rich mathematical structures and features [2][3][4][5] as well as important applications in fluid dynamics and plasma physics [6][7][8][9][10][11][12]. Although many theories and methods were proposed to discuss the PDE [13][14][15][16][17][18][19][20], however, most nonlinear PDE have no analytic solutions; numerical methods are necessary to study hydrodynamic characteristics of PDEs [21][22][23][24].…”

“…The same linear weights of U are used for U x , U xx . According to the definition of smooth, Equation (11) indicates constant β r and zero β r are used for U x and U xx , as…”