The study aims at different families of analytical solutions and their dynamics for the ( \(2+1\) )-dimensional extended Boiti–Leon–Manna–Pempinelli (eBLMP) problem, which is widely used in the fields of physics such as non-linear optics, fluid dynamics, mathematical physics, plasma physics and quantum mechanics. The paper utilises two recently developed efficient mathematical methods: the generalised exponential rational function (GERF) method and the generalised Kudryashov (gK) method. These two methods are versatile, simply applicable to enlighten the new non-linear waveforms. Consequently, these discoveries enhance our understanding of complex systems like ( \(2+1\) )-dimensional eBLMP in the realm of non-linear science.
