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The Sinh–Poisson equation and the RLC transmission line equation are important nonlinear model equations in the field of engineering and power transmission. The modified simple equation (MSE) procedure is a realistic, competent and efficient mathematical scheme to ascertain the analytic soliton solutions to nonlinear evolution equations (NLEEs). In the present article, the MSE approach is put forward and exploited to establish wave solutions to the previously referred NLEEs and accomplish analytical broad-ranging solutions associated with parameters. Whenever parameters are assigned definite values, diverse types of solitons originated from the general wave solutions. The solitons are explained by sketching three-dimensional and two-dimensional graphs, and their physical significance is clearly stated. The profiles of the attained solutions assimilate compacton, bell-shaped soliton, peakon, kink, singular periodic, periodic soliton and singular kink-type soliton. The outcomes assert that the MSE scheme is an advance, convincing and rigorous scheme to bring out soliton solutions. The solutions obtained may significantly contribute to the areas of science and engineering.

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