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In this paper, with the aid of Truncated Painlev´e Approach, (2+1) dimensional Coupled Integrable Maccari’s System is investigated. The obtained result contains some arbitrary functions which can be properly selected to study the significance of the mathematical problem. Various kinds of localized solutions such as dromion triplet pairs, dromions, and rogue waves are derived from the obtained solution by means of appropriate arbitrary functions. Using suitable initial parameters, arbitrary functions are chosen to investigate the collisional behavior of the dromion triplet pairs in the two-dimensional plane. We graphically illustrated the nonlinear wave structures with the aid of 3D plots. It is worth noting that these localized nonlinear waves are unstable under various situations.

This paper investigates the most straightforward extension of the (2+1) dimensional NLS equation, termed the Fokas system. The evolution equation is trilinearized, employing a unique method called Truncated Painlev\'{e} Approach for the (2+1) dimensional Fokas system. In terms of arbitrary functions, this method finds relatively extensive classes of solutions. Localized solutions, including dromion triplet, lump, multi-compacton and multi-rogue wave are generated by efficiently utilizing arbitrary functions. The analysis reveals that the localized solutions evolved do not move in space and only their amplitude changes with time.

Using the Truncated Painlevѐ Approach, this work investigates the (2+1) dimensional coupled integrable Maccari’s system. As a result, the solutions are constructed in terms of arbitrary functions. Utilizing the arbitrary functions present in the solution, a variety of localized solutions such as dromion triplet pairs, dromions and rogue waves are generated. The dromion pairs in the two dimensional plane were constructed and their collisional behaviors were explored by selecting the arbitrary functions with adequate initial parameters. In addition to the dromion triplet pairs, dromions and rogue wave solutions were also generated. It is observed that the dromions and rogue waves are unstable and stationary. Keywords: Dromion triplet pairs, Rogue wave, Truncated Painlevѐ Approach