Verifications of the physical validation of the solutions of the perturbed KdV equation for convective fluids
Department of Mathematics, Faculty of Science, Menoufiya University, 32511, Shebeen El-Koom, Egypt
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Revised: 22 October 2014
Accepted: 7 February 2015
Published online: 2 April 2015
Various techniques are applied to solve the perturbed KdV (PKdV) equation, which describes the evolution of surface waves velocities in convecting fluids. Under certain conditions use is made of the characteristic Galileo and Prandtl numbers of water to plot the resulting solutions, by which a variety of pattern formations for the wave velocities (in mm/s) at different temperatures are illustrated. Some solutions resulted by applying factorization technique to represent bright solitons, others give a combination of bright and dark solitons. A comparison is made with the solution of the same problem tackled in another paper (Cornejo-Perez and Rosu, Cent. Eur. J. Phys., 8, 523 (2009).). The Hamiltonian method of solution gives solitary wave behaviors. Kink solutions emerged through the application of Painlevé analysis. The resulting nonlinear second-order differential equation is dealt with in the phase portrait, which reveals the stability of the system by demonstrating that the corresponding eigenvalues indicate stable saddles and centers.
© Società Italiana di Fisica and Springer-Verlag Berlin Heidelberg, 2015