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The Landau-Ginzburg-Higgs equation and the modified equal width wave equation (MEWE) underscore to describe superconductivity and unidirectional wave propagation in nonlinear media with dispersion systems. In the present study, the improved Bernoulli sub-equation function method (IBSEFM) has been introduced to accomplish applicable soliton solutions to the above stated wave equations. We ascertain adequate soliton solutions, videlicet the combination of hyperbolic function, exponential function etc. and speculate the physical significance of the obtained solutions for the definite values of the included parameters through depicting graphs and interpreted the physical phenomena. It is shown that the IBSEFM is powerful, suitable, direct and provide general wave solutions to NLEEs in mathematical physics.