Executive Summary : | Nonlinearity is fundamental and inevitable in almost all branch of science and engineering. The physical mechanism of phenomena in nature which are described by nonlinear partial differential equation (PDEs) can be understood by the means of exact solutions of the nonlinear PDEs. In that case, the methods for exploiting exact solutions of the governing equations are extremely important and and has become hot research topic in mathematical physics. In physics, integrability is a leading subject which have great importance as almost every physical principle can be expressed mathematically in terms of system of differential equations. In last few decades, integrability aspects of nonlinear PDEs are being investigated by many research groups throughout the world but still there is no perfect definition for integrability. A model integrable under one significance may not be integrable under different implications. However indicators like Hirota bilinear form, Inverse scattering transformation, Painlev`e analysis, infinite conservation laws, N- soliton solutions, bilinear B¨acklund transformation and Lax pair etc. can characterize the integrability of nonlinear system. According to the Painleve analysis method, a NLEE possesses Painlev`e property if its solutions are single-valued about a movable singularity manifold. On the other hand, the Hirota bilinear method can be taken as the most straightforward and effective tool to check integrability aspect of a nonlinear evolution equation. It converts a nonlinear equation into a bilinear form via a dependent variable transformation and produces quasi periodic wave solutions, lump solutions, multi soliton solutions, rational solutions and other exact solutions via the bilinear form . Again, a concise application of the Bell polynomial method obtains bilinear Backlund transformation and corresponding Lax pairs of different nonlinear evolution equations. The existence of infinite conservation laws can be considered as another powerful indicator to check integrability of a nonlinear equation. Soliton solutions exist analytically for all integrable equations which are exponentially localized in certain directions. In contrast with soliton solutions, a large interest has been grown up in lump or lump-type solutions which are rationally localized in almost all directions in the space. A number of possibilities are present in the theory of lump and multi-lump to be executed by the help of Hirota bilinear form. The motivation behind obtaining exact soliton solutions this lies in the fact that, new integrable systems may give rise to new solutions. The investigation of integrable reductions of nonlinear evolution equations having nonlocal property and their exact closed form soliton solutions-their interactions provides new ideas to move further research and disclose interesting new results in the field and also open new possibility in the respective fields. |