Exploration of soliton solutions and modulation instability analysis for cold bosonic atoms in a zig-zag optical lattice in quantum physics

This paper investigates the widely used zigzag optical lattice prototype for cold bosonic atoms. This prototype generally represents nonlinear waves in quantum physics. To investigate the soliton solutions of the offered equation, we consider two analytical solution methods. The first is a new version of the generalized exponential rational function method (nGERFM), and the second is the G G2 -expansion function method. The nGERFM facilitates the generation of multiple solution types, including singular, shock, singular periodic, exponential, combo trigonometric, and hyperbolic solutions in mixed forms. Thanks to the G G2 - expansion function method, we obtain trigonometric, hyperbolic, and rational solutions. The modulation instability of the offered prototype is discussed, with numerical simulations complementing the analytical outcomes to better understand the solutions’ dynamic behavior. The findings are novel and can be useful for studying bosonic superfluidity, quantum magnetism, many-body spin dynamics, and Bose–Einstein condensation, among other studies involving ultracold atoms. These outcomes propose a foundation for future examination, making the solutions effective, manageable, and reliable for tackling complex nonlinear problems. The methodologies used in this study are robust, influential, and practicable for diverse nonlinear partial differential B. Kopçasız (B) Department of Computer Engineering, Faculty of Engineering and Architecture, Istanbul Gelisim University, Istanbul, Turkey e-mail: bkopcasiz@gelisim.edu.tr equations; to our knowledge, these methods of investigation have not been explored before for this equation.