_____________________________________________________________________________________________________ *Corresponding author: E-mail: aly.abourabia@science.menofia.edu.eg, am_abourabia@yahoo.com; Physical Science International Journal 23(4): 1-8, 2019; Article no.PSIJ.53244 ISSN: 2348-0130 Analytical Solution of the Complex Polymer Equation Systems via the Homogeneous Balance Method Aly M. Abourabia 1* and Yasser A. Eldreeny 1 1 Department of Mathematics, Faculty of Science, Menoufiya University, Shebin Elkom 32511, Egypt. Authors’ contributions This work was carried out in collaboration between both authors. Both authors read and approved the final manuscript. Article Information DOI: 10.9734/PSIJ/2019/v23i430164 Editor(s): (1) Dr. Smain Femmam, Strasbourg University of Haute Alsace, France and Safety Systems of Polytechnic School of Engineering “L3S”, France. (2) Dr. Shi-Hai Dong, Professor, Department of Physics School of Physics and Mathematics, National Polytechnic Institute, Mexico. Reviewers: (1) Yanxia Hu, North China Electric Power University, China. (2) Adel H. Phillips, Ain Shams University, Egypt. Complete Peer review History: http://www.sdiarticle4.com/review-history/53244 Received 03 October 2019 Accepted 10 December 2019 Published 19 December 2019 ABSTRACT In this article, we solve analytically the nonlinear Doubly Dispersive Equation (DDE) in (1+1)-D by the homogeneous balance method, introduced to investigate the strain waves propagating in a cylindrical rod in complex polymer systems. The linear dispersion relation plays important role in connecting the frequency of the emitted nonlinear waves with the wave number of the ablating laser beam affecting the polymers with their characteristic parameters. In accordance with the normal dispersion conditions, the resulting solitary wave solutions show the compression characters in the nonlinearly elastic materials namely Polystyrene (PS) and PolyMethylMethAcrylate (PMMA). The ratio between the estimated potential and kinetic energies shows good agreement with the physical situation, and as well in making comparisons with the bell- shaped model conducted in the literature. Keywords: Polymers; laser ablation; Doubly Dispersive Equation (DDE); travelling wave variable; normal dispersion; strain; homogeneous balance method; soliton solutions. Pacs Numbers: 61.41.+e; 02.30.Jr; 02.30Hq; 62.20.-x; 62.20.Dc; 05.45.Yv. Original Research Article