Soliton solutions for the fifth-order KdV equation and the Kawahara equation with time-dependent coefficients

Abstract

In this work, a generalized fifth-order Korteweg–de Vries (KdV) equation will be examined. The study will also be carried out to the fifth-order Kawahara equation. Both equations are characterized by time-dependent coefficients. The wave soliton ansatz will be used to obtain soliton solutions for these equations. The conditions for the existence of the derived solitons will be established.

Group classification of variable coefficient generalized Kawahara equations

“…In recent papers [30,52] different techniques for finding exact solutions were applied to construct exact solutions of Kawahara equations with time-dependent coefficients. In both papers exact solutions were derived for equations whose coefficients obey additional constraints, namely, when all the coefficients are proportional to each other.…”

Exact Solutions Expressible in Hyperbolic and Jacobi Elliptic Functions of Some Important Equations of Ion-Acoustic Waves

“…The generalized ZK equation was first derived for describing weakly nonlinear ion-acoustic waves in strongly magnetized lossless plasma in two dimensions and governs the behavior of weakly nonlinear ion-acoustic waves in plasma comprising cold ions and This equation appears in the theory of shallow water waves with surface tension and the theory of magneto-acoustic waves in plasmas [9]. Wazwaz [55] studied soliton solutions of fifth-order KdV equation. We can use a suitable method to construct the exact solutions of some special types of nonlinear evolution equations aries in plasma physics such as Liouville, sine-Gordon and sinh-Poisson equations.…”

A new numerical approach to Gardner Kawahara equation in magneto‐acoustic waves in plasma physics

“…u x (a, t) = g 3 (t), u x (b, t) = g 4 (t), t ≥ 0, (4) u xx (a, t) = g 5 (t), u xx (b, t) = g 6 (t), t ≥ 0, (5) in which f (x) is a given smooth function, g 𝓁 (t), (𝓁 = 1, 2, 3, 4, 5, 6) are prescribed functions to be given in numerical computation and analysis section. First of all, by using the conversion u xxx = v in the equation, it will convert the equation given in (1) into a coupled system of equations.…”