The aim of the present paper is to extract the exact traveling wave solutions of the chiral nonlinear Schr\”{o}dinger’s equation (cNLSE). The $(\frac{G’}{G^{2}})-$expansion method and the first integral method along with symbolic computation package has been exerted to celebrate the exact solutions. As a consequence, the obtained solutions can be categorized into trigonometric, hyperbolic and rational with some free parameters of the problem studied. In addition, these types of the solutions lead to understand the physical phenomena of the problem such as solitary, periodic, complex function, singular optical solitons and dark-singular combo solitons.

In this manuscript, some novel exact traveling wave solutions are constructed for $(2+1)$-dimensional Boiti-Leon-Manna-Pempinelli(BLMP) equation. The analytical techniques, namely extended rational sine-cosine method and extended rational sinh-cosh method are utilized for constructing the new solitary wave solutions of BLMP equation. The proposed techniques provides different types of solutions which are expressed in terms of singular periodic wave, solitary waves, bright solitons, dark solitons, periodic wave and kink wave solutions with specific values of parameters.