Extended (G'/G) Method Applied to the Modified Non-Linear Schrodinger Equation in the Case of Ocean Rogue Waves

Abstract

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.

Accurate perceptions of the bright and dark soliton solutions to the modified nonlinear Schrödinger equation

“…The first one which is regrestsed with the name the SWAM which achieves new perceptions of the soliton solution to the suggested equation figures (1)(2)(3)(4).While the second one is the ESEM which has been applied effectively to establish other new visions to the soliton solutions to the suggested equation figures (5-7). Our achieved solutions are new and new distinct perceptions to the soliton solutions of this model compared with that obtained previously [32][33][34][35][36] who applied different techniques.…”

“…Specially, few tries were constructed through some authors to demonstrate the soliton solutions to the MNLSE using different methods namely, Stéphane et al [32] who apply the extended (G'/G) method to the modified nonlinear Schrodinger equation in the case of ocean rogue waves, Chan et al [33] who calculated the rogue waves of a derivative non-linear Schrödinger equation as a long-wave limit of a breather ( a pulsing mode) which widely occurs in fluid dynamics and optical waveguides that have unexpected large displacements, Yu and Yan [34] who constructed explicit rouge wave solutions and dark-bright solutions for the inhomogeneous coupled nonlinear Schrödinger equation with variable coefficients by means of similarity transformations and Younis et al [35] who used the extended Fan subequation method with five parameters to achieve new families of exact traveling wave solutions for the modified nonlinear Schrödinger equation.…”

Gravity waves’ modulational instability under the effect of drag coefficient in the ocean

“…The evolution of a waveform produced by a group unstable wave only on the nonlinear Schrodinger equation which can be obtained from the fully nonlinear potential theory by using the Zakharov's Integral equation [51] is:…”

Bifurcation analysis on ion sound and Langmuir solitary waves solutions to the stochastic models with multiplicative noises

“…Yajima et al [ 13 ] was formed solitonic interaction solutions of the Sonic-Langmuir wave model with ion-acoustic waves through the inverse scattering approach. However, there are huge new techniques to derive various exact dynamical solutions to the nonlinear complex models literally, in particular, the Jacobi elliptic function expansion [ 14 ], the Improved Kudryashov [ 15 ], the modified simple equation [ 16 ], the variational iteration [ 17 ], the modified piecewise variational iteration [ 18 ], the tan (φ/2)-expansion [ 19 ], the Riccati-Bernoulli sub-ODE function [ 20 ], the Petrov-Kudrin-Xiong [ 21 ], the first integral method [ 22 ], the finite difference [ 23 ], the -expansion [ 24 ], the Hirota bilinear [ [25] , [26] , [27] ], the sumudu homotopy perturbation [ 28 ], Adomian's decomposition [ 29 ], the (G′/G)-expansion [ 30 , 31 ], the unified [ 32 , 33 ] and the bifurcation scheme with theory of dynamic systems approach [ 34 , 35 ], which have been effectively employed to obtain new travelling brandish solutions of complex nonlinear precise models. Among these methods, the bifurcation as well as theory of dynamic structures approach are one most qualitative as this can derive the exact solutions according to the energy orbits of their phase portraits.…”