Truncated M-Fractional Exact Solutions, Stability Analysis, and Modulation Instability of the Classical Lonngren Wave Model

Many types of exact solutions to the truncated M-fractional classical Lonngren wave model are explored in this paper. The classical Lonngren wave model is a significant electronics equation. This model is used to explain the electronic signals within semiconductor materials, especially tunnel diodes. Through the application of a modified $(G^{\prime }/G^2)$-expansion technique and an extended sinh-Gordon equation expansion (EShGEE) method, we obtained various wave solutions, including periodic, kink, singular, dark, bright, and dark–bright types, among others. To ensure that the solutions in question are stable, linear stability analysis is also carried out. Moreover, the stationary solutions of the concerning equation are studied through modulation instability. The obtained results are useful in various areas, including electronic physics, soliton physics, plasma physics, nonlinear optics, acoustics, etc. Both techniques are useful for solving nonlinear partial fractional differential equations. Both techniques provide exact solutions, which can be important for understanding complex phenomena. Both techniques are reliable and yield distinct types of exact soliton solutions.