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In this investigation, the nonplanar (spherical and cylindrical) modified fifth-order Korteweg–de Vries (nmKdV5) equation, otherwise known as the nonplanar modified Kawahara equation (nmKE), is solved using the ansatz approach. Two general formulas for the semi-analytical symmetric approximations are derived using the recommended methodology. Using the obtained approximations, the nonplanar modified Kawahara (mK) symmetric solitary waves (SWs) and cnoidal waves (CWs) are obtained. The fluid equations for the electronegative plasmas are reduced to the nmKE as a practical application for the obtained solutions. Using the obtained solutions, the characteristic features of both the cylindrical and spherical mK-SWs and -CWs are studied. All obtained solutions are compared with each other, and the maximum residual errors for these approximations are estimated. Numerous researchers that are interested in studying the complicated nonlinear phenomena in plasma physics can use the obtained approximations to interpret their experimental and observational findings.
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Alharbey et al. (2022) studied this question.
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