Copyright © Salam Subhaschandra Singh. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. International Journal of Physical Research, 8 (2) (2020) 40-44 International Journal of Physical Research Website: www.sciencepubco.com/index.php/IJPR Research paper Dark soliton solutions to (2 + 1)-dimensional Kundu-Mukherjee-Naskar equation via the first integral method Salam Subhaschandra Singh * Department of Physics, Imphal College, Imphal, 795001, Manipur, India *Corresponding author E-mail: subhasic@yahoo.co.in Abstract In the present work, the First Integral Method is being applied in finding a non-soliton as well as a soliton solution of the ( 2 + 1 ) di- mensional Kundu-Mukherjee-Naskar (KMN) equation which is a variant of the well-known Nonlinear Schrodinger ( NLS ) equation. Using the method, a dark optical soliton solution and a periodic trigonometric solution to the KMN equation have been suggested and the relevant conditions which guarantee the existence of such solutions are also indicated therein. Keywords: Kundu-Mukherjee-Naskar Equation; First Integral Method. 1. Introduction Optical solitons are pulses which compose the basic fabric of signal transmission across trans-continental and trans-oceanic distances in telecommunication engineering. In the present decade, the study of the dynamics of optical soliton propagation through optical fibers yields many promising results in the research of optical communication systems. Several equations / models have been proposed so far in the past a few decades to describe such physical phenomena and the Kundu-Mukherjee-Naskar (KMN ) equation [1- 6] is one of them. This equation was first proposed in the year 2014 by three Indian physicists namely Anjan Kundu, Abhik Mukherjee and Tapan Naskar for modelling the dynamics of two-dimensional rogue waves in ocean water and also two-dimensional ion-acoustic waves in magnetized plasmas [7,8]. We can consider this equation as an extension of the well-known Nonlinear Schrodinger (NLS) equation. It can also be used in describing propagation of optical wave through coherently excited resonant wave-guides in particular in the theory of bending of light beams. Various methods have been proposed so far by various authors for solving Nonlinear Evolution Equations (NLEEs) and the First Integral Method [ 9 - 16] is picked up here to find a soliton solution of KMN equation. 2. Kundu-Mukherjee-Naskar (KMN) equation in (2 + 1) dimensions The KMN equation is written in a dimensionless form as (1) where x and y are spatial variables, t is the temporal variable, is a dependent variable depending on and giv- ing the profile of the optical soliton or nonlinear wave envelope, the asterisk denotes complex conjugation and the subscripts indicate derivatives of the dependent variable with respect to them; the first term ensures a temporal evolution, the second term with a coefficient ‘a’ represents a dispersion term, the final term on the left hand side with a coefficient ‘b’ is the nonlinear term that is different from the conventional Kerr-type or any known non-Kerr type nonlinearity arising in the celebrated Nonlinear Schrodinger ( NLS ) equation and other earlier generalized models. Although the equation was proposed by Kundu, Mukherjee and Naskar to describe oceanic rogue waves as well as hole waves, it may also be used in describing optical waves or soliton propagation through Erbium doped coherently excited resonant wave-guides. Further, this equation can also be used in the study of the phenomenon of bending of light beams. Due to these reasons, many researchers turned to the investigation of this equation and subsequently several findings had been published elsewhere. In the following sections, we de- scribe an algorithm of the First Integral Method and it is applied in solving the KMN equation.