An Investigation of 2 plus 1 Dimensional Nonlinear Equations Exploring LSRI and Maccari s Systems Through the Truncated Painleve Approach

Abstract

This thesis focuses on the application of the Truncated Painlevé Approach (TPA) to newlinea specific class of (2+1) dimensional nonlinear Partial Differential Equations (PDEs). newlineIn Chapter 1, the field of nonlinear dynamics is introduced, highlighting the study of newline(2+1) dimensional nonlinear PDEs and discussing their historical development. Also, newlineessential mathematical tools like the first integral method, sine-Gordon equation method, newlineG /G expansion scheme, Hirota bilinear method and Bäcklund transformations are explored. newlineThe chapter provides an overview of the Painlevé analysis, which is employed newlineto analyze the integrability of the nonlinear PDEs. Also, it explores a unique mathematical newlinetool so called Truncated Painlevé approach (TPA) which is used to derive solutions newlinein the closed form for nonlinear PDEs. newlineThe Chapter 2 presents a comprehensive study of the (2+1) dimensional Long wave newline- Short wave Resonance Interaction equation (LSRI), which is a complex nonlinear system newlineof partial differential equations. The LSRI system describes the two-dimensional newlineLSRI system, describing the resonant interaction between a gravity wave and surface newlinegravity packets. Previous analysis utilizing Bell polynomials have identified soliton newlinesolutions and added intrigue to the system. Integrability properties are explored using newlinethe Painlevé analysis, and the Truncated Painlevé Approach is employed to obtain newlineclosed-form localized solutions. The solutions obtained will be expressed in terms of newlinelower-dimensional arbitrary functions, including rogue waves, lump, one-dromion, and newlinetwo-dromion wave patterns. The above study will improve our understanding of the newlinedynamics of the LSRI system newline