Analytical study of nonlinear models using a modified Schrödinger’s equation and logarithmic transformation

Abstract

Optical solitons are special waves that maintain their shape while traveling. Hence, solitary optical waves have great significance when it comes to representing pulse propagation through nonlinear partial differential equations. In our research, we have uncovered unique solutions for the formation of solitary wave patterns in the equation of the Heisenberg ferromagnetic spin chain (HFSC). This equation aptly describes the behavior of electromagnetic waves in contemporary magnetism. By employing innovative techniques based on logarithmic transformations and analytical methods, we have obtained various solution forms expressed in a concise manner using elementary functions. We have tested the correctness of these solutions by substituting them directly into the main equation. These recent discoveries provide novel perspectives on the complex realm of solitons as depicted by this model. Moreover, since the model finds applications in fiber optic communications, fluid dynamics, and other fields, the implications of these discoveries are broad. The utilized methods stand out for being simple, reliable, and capable of creating fresh solutions for nonlinear partial differential equations in the realm of mathematical physics. The research findings presented here showcase the effectiveness of the applied methods in reliably studying nonlinear phenomena in the HFSC equation and other nonlinear problems in mathematical physics. By utilizing these tools, scholars have the opportunity to expand and enhance their understanding of the complex mathematical frameworks underlying real-world problems.