ISSN 1749-3889 (print), 1749-3897 (online) International Journal of Nonlinear Science Vol.9(2010) No.4,pp.430-443 The Derivation and Study of the Nonlinear Schr¨ odinger Equation for Long Waves in Shallow Water Using the Reductive Perturbation and Complex Ansatz Methods A. M. Abourabia 1 ∗ , K. M. Hassan 2 , E. S. Selima 3 1, 2, 3 Department of Mathematics, Faculty of Science, Menoufiya University, Shebin El-koom 32511, Egypt (Received 13 November 2009, accepted 10 June 2010) Abstract: In this paper, the water wave flow problem for an incompressible and inviscid fluid of constant depth is studied under the influence of acceleration of gravity and surface tension. The nonlinear Schr¨ odinger (NLS) equation and the dispersion relation are derived from the nonlinear shallow water equations by using the reductive perturbation technique, which differs from the derivations of the same problem illustrated in previous works. The complex ansatz method is presented for constructing exact traveling wave solutions of NLS equation and from them the physical variables of the water wave problem are obtained. Depending on the Ursell parameter, the diagrams are drawn to illustrate the behavior of the solutions of NLS equation, free surface elevation and velocity of the model. The results indicate that the solutions of this problem are in the form of envelope traveling solitary waves where the Ursell parameter affects the wave profile significantly. It is concluded the chosen methods of solutions are sufficiently accurate to demonstrate that the conservation of power is satisfied except at very few spots where it fluctuates about zero by the orders of 10 −14 − 10 −18 depending on the values of the Ursell parameter. Keywords: shallow water equations; reductive perturbation method; nonlinear Schr¨ odinger equation; com- plex ansatz method 1 Introduction Many phenomena in physics and other applied fields are described by nonlinear partial differential equations (PDEs). To understand the physical situation of these phenomena in nature, we need to find the exact solutions of the PDEs, which have become one of the most important topics in mathematical physics. In the study of equations modeling wave phenomena, one of the fundamental objects of study is the traveling wave solution, meaning a solution of constant form moving with a fixedvelocity. Of particular interest are three types of traveling waves: solitary waves, periodic waves and kink waves [1-9]. The problem of nonlinear waves propagating in plasma and fluid can be described by the KdV, nonlinear Schr¨ odinger, Davy-Stewartson equations and others. These last equations are derived by multiple scale, reductive perturbation methods and other asymptotic methods [10-19]. Due to its central importance to the theory of quantum mechanics, the nonlinear equation of Schr¨ odinger type has a great interest. They arise in many physical problems, including nonlinear water waves, ocean waves, waves in plasma, propagation of heat pulses in a solid self trapping phenomenon in nonlinear optics, nonlinear waves in a fluid-filled vis- coelastic tube, and various nonlinear instability phenomena in fluids and plasma, and are of importance in the development of solitons and inverse scatting transform theory [1-4]. In the past the exact solution of Schrodinger type was obtained by converting these equations into real forms through some transformations and then using some methods such as Jacobi elliptic expansion, tanh-function method, Cole-Hopf transformation, Hirota bilinear method, inverse scattering method and so on. Recently, direct methods are proposed to obtain the exact wave solutions of these equations such as complex tanh-function method, complex hyperbolic-function method, sub-ODE method, complex ansatz method, complex Jacobi elliptic method and others [1, 2, 20-30]. ∗ Corresponding author. E-mail address: am abourabia@yahoo.com. Tel.: +2010-1382826; fax: +2 048-2235689. dr Kawser99@yahoo.com(K. Hassan), es.selima@yahoo.com(E. Selima). Copyright c ⃝World Academic Press, World Academic Union IJNS.2010.06.30/370