Tikrit Journal of Pure Science Vol. 25 (2) 2020 124 Tikrit Journal of Pure Science ISSN: 1813 – 1662 (Print) --- E-ISSN: 2415 – 1726 (Online) Journal Homepage: http://tjps.tu.edu.iq/index.php/j New solitary Solution for the Kudryashov-Sinelshchikov (KS) equation by Modern Extension of the Hyperbolic Method Ali H. Hazza 1 , Wafaa M. Taha 2 , Raad A. Hameed 1 , Israa A. Ibrahim 1 1 Department of Math., College Education for Pure Sciences, Tikrit University, Iraq 2 Department of Math., College of Sciences, Kirkuk University, Iraq DOI: http://dx.doi.org/10.25130/tjps.25.2020.039 A r t i c l e i n f o. Article history: -Received: 30 / 9 / 2019 -Accepted: 7 / 11 / 2019 -Available online: / / 2020 Keywords: Modern extension of the hyperbolic method, analysis of hyperbolic tanh-function method, Kudryashov - Sinelshchikov (KS) equation, kawahara equation, travelling wave solution, standard hyperbolic tanh - function method, Corresponding Author: Name: Wafaa M. Taha E-mail: wafaa_y2005@yahoo.com Tel: ABSTRACT In the present paper, we apply the modern extension of the hyperbolic tanh function method of nonlinear partial differential equations (NLPDEs) of Kudryashov - Sinelshchikov (KS) equation for obtaining exact and solitary traveling wave solutions. Through our solutions, we gain various functions, such as, hyperbolic, trigonometric and rational functions. Additionally, we support our results by comparisons with other methods and painting 3D graphics of the exact solutions. It is shown that our method provides a powerful mathematical tool to find exact solutions for many other nonlinear equations in applied mathematics 1- Introduction The investigation of exact solutions of nonlinear Kudryashov-Sinelshchikov(KS) equation plays an important role in the study of nonlinear physical phenomena, Kudryashov and Sinelshchikov equation describe the motions of plasma waves, capillary- gravity water waves and water waves with surface tension[1], introduced the following equation:                       ….(1.1) Where       are positive constants, the Kudryashov-Sinelshchikov equation describe by the most famous model (KdV) equation[2], when       . Kudryashov-Sinelshchikov Equation also is the general form of Kawahara equation [3]. Modern extension of the hyperbolic tanh function method is used to obtain the general form of (KS) equation. In (1996), Malfliet and Hereman introduce the powerful tanh method for a reliable treatment of the nonlinear wave equations[4]. The tanh function method is widely used by many such as in[5][6][7][8][9][10]. The strategic of this research is marshaled as follows: In part 2, we list the essence of analysis modern extension of the hyperbolic tanh function method. In part 3, we implement our method to find new traveling wave solutions of Kudryashov- Sinelshchikov (KS) equation. in part 4, the traveling wave solutions for Kudryashov- Sinelshchikov equation by standard hyperbolic tanh function method. Finally, we present the conclusions in part 4. 2- Analysis of Modern Extension of the Hyperbolic Tanh Function Method We have a nonlinear evolution equation in two independent variables    in the following form:                ….(2.1) Where, G is a polynomial in    and      is an unknown function, the process of finding a traveling wave solution for the modern extension of the hyperbolic tanh function method, explain in the following five steps. Step1: We use the wave transformation          ….(2.2) Where c is a constant. Substituting (2.2) into (2.1), we obtain the following ordinary differential equation (ODE):