The Derivation and Study of the Nonlinear Schrödinger Equation for Long Waves in Shallow Water Using the Reductive Perturbation and Complex Ansatz Methods (original) (raw)
Related papers
Gazi University Journal of Science
In this work, we use two different analytic schemes which are the Sine-Gordon expansion technique and the modified exp -expansion function technique to construct novel exact solutions of the non-linear Schrödinger equation, describing gravity waves in infinite deep water, in the sense of conformable derivative. After getting various travelling wave solutions, we plot 3D, 2D and contour surfaces to present behaviours obtained exact solutions.
Deep-Water Waves: on the Nonlinear Schrödinger Equation and its Solutions
Journal of Theoretical and Applied Mechanics, 2013
We present a brief discussion on the nonlinear Schrödinger equation for modelling the propagation of the deep-water wavetrains and a discussion on its doubly-localized breather solutions, that can be connected to the sudden formation of extreme waves, also known as rogue waves or freak waves.
A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation
Journal of Fluid Mechanics, 1985
In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrodinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.
Solutions of Differential Equations in Nonlinear Water Waves
2015
This book is concerned with the study of nonlinear water waves, which is one of the important observable phenomena in Nature. This study is related to the fluid dynamics, in general, and to the oceans dynamics in particular. The solutions of nonlinear PDEs with constant and variable coefficients, which describe the wave motion of undulant bores in shallow water, are investigated by using various analytical methods to illustrate the relation between solitary and water waves. The important ideas and results for nonlinear dispersive properties and solitons, which originated from the investigations of water waves, are discussed. The stability analysis for the second order system of PDEs is studied by using the phase plane method. In addition, we use perturbation methods to study the water wave problems for an incompressible fluid under the acceleration gravity and surface tension. The conservation laws of some PDEs are established. We illustrate the resulting solutions in several 3D-gra...
Open Journal of Marine Science, 2014
The existence of rogue (or freak) waves is now universally recognized and material proofs on the extent of damage caused by these ocean's phenomena are available. Marine observations as well as laboratory experiments show exactly that rogue waves occur in deep and shallow water. To study the behavior of freak waves in terms of their space and time evolution, that is, their motion and also in terms of mechanical transformations that these systems may suffer in their dealings with other systems, we derive a modified nonlinear Schrödinger equation modeling the propagation of rogue waves in deep water in order to seek analytic solutions of this nonlinear partial differential equation by using generalized extended G'/G-expansion method with the aid of mathematica. Particular attentions have been paid to the behavior of rogue wave's amplitude which highlights rogue wave's destructive power.
Soliton solutions of shallow water wave equations by means of G′/G expansion method
Journal of Applied Analysis and Computation, 2014
In this work, we investigate the traveling wave solutions for some generalized nonlinear equations: The generalized shallow water wave equation and the Whitham-Broer-Kaup model for dispersive long waves in the shallow water small-amplitude regime. We use the G ′ /G expansion method to determine different soliton solutions of these models. The conditions of existence and uniqueness of exact solutions are also presented.
A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity
Physics of Fluids, 2012
A nonlinear Schrödinger equation for the envelope of two dimensional surface water waves on finite depth with non zero constant vorticity is derived, and the influence of this constant vorticity on the well known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth.
Exact solutions of the nonlinear generalized shallow water wave equation
2013
In this article, we have employed an enhanced (G′/G)-expansion method to find the exact solutions first and then the solitary wave solutions of the nonlinear generalized shallow water wave equation. Here we have derived solitons, singular solitons and periodic wave solutions through the enhanced (G′/G)-expansion method. The solutions obtained hereby reveal the richness of explicit solitons and periodic solutions to the applied equation. It has been shown that the enhanced (G′/G)-expansion method is effective and can be used for many other nonlinear evolution equations (NLEEs) in mathematical physics. Mathematics subject classification: 35K99 • 35P05 • 35P99 Key words: Enhanced (G′/G)-expansion method • nonlinear generalized shallow water wave equation • solitary waves • soliton • exact solution • NLEEs