µ-symmetry analysis, µ-conservation laws, and exact solutions of the modified equal width equation

Abstract

This study discusses the µ-symmetries, µ-conservation laws, and exact solutions of the modified equal width equation (MEWE). MEWE is used as a model in partial differential equations (PDE) to simulate one-dimensional wave transmission in nonlinear media with dispersion processes. First and foremost, we present some essential pieces of information about the offered techniques. In light of such information, we discover µ-symmetries. The essential idea behind the µ-symmetry approach is that it reduces one-independent variables in a system of PDEs by employing µ-symmetries and invariance surface conditions. The µ-symmetry method has been applied to MEWE and transformed into an ordinary differential equation (ODE). Then, we employ a modified version of the generalized exponential rational function method (mGERFM) to this reduced ODE to obtain soliton solutions. Thanks to the mGERFM, we discover unique wave solutions in the forms of exponential function solutions, combined periodic soliton solution, singular periodic wave solution, shock wave solutions, trigonometric function solutions, mixed-form soliton solution, hyperbolic solution in mixed form, and periodic soliton solution. Furthermore, by employing the variational problem procedure, we get the Lagrangian and the µ-conservation laws. The mGERFM, µ-symmetry analysis, and µ-conservation laws have not been discussed in previous investigations for the MEWE. We also demonstrate the properties with figures for these solutions. Here, we use Maple software to validate the complete outcomes of the study.