Analytical Solution of the Complex Polymer Equation Systems via the Homogeneous Balance Method

Main Article Content

Aly M. Abourabia
Yasser A. Eldreeny


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.

Polymers, laser ablation, Doubly Dispersive Equation (DDE), travelling wave variable, normal dispersion, strain, homogeneous balance method, soliton solutions

Article Details

How to Cite
Abourabia, A. M., & Eldreeny, Y. A. (2019). Analytical Solution of the Complex Polymer Equation Systems via the Homogeneous Balance Method. Physical Science International Journal, 23(4), 1-8.
Original Research Article


Nariboli GA, Sedov A. Burgers’s-Korteweg-De Vries Equation for Viscoelastic Rods and Plates. J. Math. Anal. Appl. 1970;32:661-677.

Crighton DG. Applications of KdV in solids: Acta Applicandae Mathematicae. 1995;39: 39-67.

Ostrovskii LA, Sutin AM. Nonlinear elastic waves in rods.Appl. Math. Mech. 1977;41: 543.

Engelbrecht J. Nonlinear wave processes of deformation in solids. Pitman, London; 1981.

Samsonov AM. Soliton evolution in a rod with variable cross section. Sov. Phys. Dokl. 1984;29:586-588.

Dreiden GV, Ostrovsky YI, Samsonov AM, Semenova IV, Sokurinskaya EV. Formation and propagation of deformation solitons in a nonlinearly elastic solid, Sov. Phys. Tech. Phys. 1988;33:1237.

Semenova IV, Dreiden GV, Samsonov AM. Strain solitary waves in polymeric nanocomposites, Ioffe Physical Technical Institute of the Russian Academy of Sciences; 2010.

Karima R, Khusnutdinova AM. Samsonov. Fission of a longitudinal strain solitary wave in delaminated bar, Physical Reviewed 77, 066603; 2008.

Agranant MB, Krasyuk IK, Novikov NP, Perminov VP, Yu I, Yudin, Yampol'sskii PA. Destruction of transparent dielectrics by laser radiation. Zh. Eksp. Teor. Fiz. 1971;60:1748-1756.

Sandeep Ravi-Kumar, Benjamin Lies, Xiao Zhang , Hao Lyu , Hantang Qin. Laser ablation of polymers: A review. Polym. Int. 2019;68:1391-1401.
DOI: 10.1002/pi.5834

Samsonov AM. Strain solitons in solids and how to construct them. Chapman and Hall/CRC, Boca Raton, FL; 2001.

Born M, Wolf E. Principles of optics. 7th ed., Cambridge University Press, Cambridge, UK. 1999;Chap. 1.

Abdul-Majid Wazwaz, New solitons and kink solutions for the Gardner equation, Communications in Nonlinear Science and Numerical Simulation; 2006.

Fan EG, Zhang HQ. Solitary wave solutions of a class of nonlinear wave equations. Acta Physica Sinica. 1997; 46(7):1254–1258.

Dreiden GV, Samsonov AM, Semenova IV. Bulk elastic strain solitons in polycarbonate. Russian Academy of Sciences; 2011.

Kivshar Y. Compactons in discrete lattices. Nonlinear Coherent Struct Phys Biol. 1994; 329:255–8.

Dinda PT, Remoissenet M. Breather compactons in nonlinear Klein–Gordon systems. Phys Rev E. 1999;60(3):6218- 21.

Wolfram Mathmatica-9 is a trademark of Wolfram Research, inc; 2012.

Zhang ZM, Keunhan Park. On the group front and group velocity in a dispersive medium upon refraction from a nondispersive medium, Transactions of the ASME. 2004;244(126).