Novel exact and approximate solutions to the family of the forced damped Kawahara equation and modeling strong nonlinear waves in a plasma

Abstract

In this investigation, a set of novel exact and approximate analytic solutions to the family of the forced damped Kawahara equation (KE) are derived in detail. This work is divided into four main parts: in the first part, the ansatz method is employed to obtain some novel exact solutions to the integrable Extended KE (EKE). In this part, we use some different hypotheses to derive many exact traveling wave solutions (including solitary waves (SWs) and cnoidal waves (CWs)). In the second part, the ansatz method is devoted for deriving some exact SWs and CWs solutions to the forced KE. In the same part, two different general formulas to the analytical approximations to the forced damped KE are obtained. In the third part, the family of forced damped modified KE (mKE) is analyzed comprehensively via several different techniques in order to obtain several approximations. At the beginning, two different general formulas with different accuracy to the forced damped mKE are derived. Also, a particular soliton solution to the forced damped mKE is obtained using a direct ansatz. After that, a general formula for the approximate solution to the forced undamped mKE is obtained. In the fourth part, we focus our efforts on obtaining some approximations to forced EKE. The validity of all exact solutions is checked. Moreover, the maximum global residual error to the analytical approximations is estimated. This investigation can help all researchers interested in studying the characteristics of many nonlinear phenomena in various fields of science such as physics of plasmas, physics of fluids, nonlinear optics, Veins and capillaries, Oceans and seas, etc.