Collision properties of nonlinear waves in the modified Gardner equation

Abstract

Small-amplitude ion-acoustic waves near supercritical plasma compositions are described by the modified Gardner (mG) equation. In this dissertation, we analyze solutions of the mG equation and simulate collision of soliton-type solutions. The presence of nonlinear waves is strongly influenced by the coefficients of the mG equation. First of all, we conduct an analytical analysis of the parameter space of the mG equation in order to identify coefficients where interesting nonlinear solutions such as kinks, the coexistence of positive and negative solitons, and supersolitons occur. We begin by introducing a moving frame analysis, which reduces the mG equation to a second order ordinary differential equation (ODE). This is evaluated using roots of a related cubic polynomial. Using a cubic polynomial analysis, we can determine the plasma compositional requirements for nonlinear waves, and the solutions are graphically represented. The existence of nonlinear waves and supercritical compositions are thus characterised by our research. In addition, we use numerical simulation to study the collision properties between different nonlinear waves and with regular solitons. To construct the initial conditions on sufficiently large spatial intervals, we determine the tails through an asymptotic analysis. This is accomplished by expanding the Sagdeev pseudopotential function using a Taylor series expansion. After constructing the initial condition, a MATLABcode is developed to numerically integrate the mG equation. In this code, the spatial component is discretized using the finite difference method, and the time component is integrated using the fourth-order Runge-Kutta method. We performed numerical simulations of overtaking collisions between regular solitons and other nonlinear structures. These results may be useful in deepening the understanding of nonlinear waves in space plasmas.