Reductions and Solutions of Two Types of Coupled Nonlinear Evolution Equations in Optical Fibers and Fluid Dynamics

2011 ◽  
Vol 66 (8-9) ◽  
pp. 552-558
Author(s):  
Li-Cai Liu ◽  
Bo Tian ◽  
Bo Qin ◽  
Xing Lü ◽  
Zhi-Qiang Lin ◽  
...  
Keyword(s):  
Fluid Dynamics ◽  
Raman Scattering ◽  
Optical Fibers ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Dark Soliton ◽  
Nonlinear Evolution Equations ◽  
The One ◽  
Gain Loss ◽  
Burgers Type Equations

Abstract Under investigation in this paper are the coupled nonlinear Schrödinger equations (CNLSEs) and coupled Burgers-type equations (CBEs), which are, respectively, a model for certain birefringent optical fibers Raman-scattering, Kerr and gain/loss effects, and a generalized model in fluid dynamics. Special attention should be paid to the existing claim that the solitons for the CNLSEs do not exist. Through certain dependent-variable transformations, the CNLSEs are reduced to a Manakov system and the CBEs are linearized. In that way, some new solutions of the CNLSEs and CBEs are obtained via symbolic computation. Especially the one-dark-soliton-like solutions for the CNLSEs have been found, against the existing claim.

scholarly journals Comment on “Conservation Laws of Two (2 + 1)-Dimensional Nonlinear Evolution Equations with Higher-Order Mixed Derivatives”

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
Long Wei ◽  
Yang Wang
Keyword(s):  
Conservation Laws ◽  
Nonlinear Equations ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Higher Order ◽  
Nonlinear Evolution Equations ◽  
Symmetric Form ◽  
Mixed Derivatives ◽  
The One

In a recent paper (Zhang (2013)), the author claims that he has proposed two rules to modify Ibragimov’s theorem on conservation laws to “ensure the theorem can be applied to nonlinear evolution equations with any mixed derivatives.” In this letter, we analysis the paper. Indeed, the so-called “modification rules” are needless and the theorem of Ibragimov can be applied to construct conservation laws directly for nonlinear equations with any mixed derivatives as long as the formal Lagrangian is rewritten in symmetric form. Moreover, the conservation laws obtained by the so-called “modification rules” in the paper under discussion are equivalent to the one obtained by Ibragimov’s theorem.

scholarly journals On a direct construction of inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension

2004 ◽  
Vol 2004 (58) ◽  
pp. 3117-3128
Author(s):  
H. H. Chen ◽  
J. E. Lin
Keyword(s):  
Inverse Scattering ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Spatial Dimension ◽  
Nonlinear Evolution Equations ◽  
Scattering Problems ◽  
Direct Construction ◽  
Inverse Scattering Problems ◽  
Petviashvili Equation ◽  
The One

We present a method to construct inverse scattering problems for integrable nonlinear evolution equations in the two-spatial dimension. The temporal component is the adjoint of the linearized equation and the spatial component is a partial differential equation with respect to the spatial variables. Although this idea has been known for the one-spatial dimension for some time, it is the first time that this method is presented for the case of the higher-spatial dimension. We present this method in detail for the Veselov-Novikov equation and the Kadomtsev-Petviashvili equation.

scholarly journals Three Different Methods for New Soliton Solutions of the Generalized NLS Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-8 ◽  
Author(s):  
Anwar Ja’afar Mohamad Jawad
Keyword(s):  
Optical Fibers ◽  
Evolution Equations ◽  
Pulse Propagation ◽  
Nonlinear Evolution ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Nls Equation ◽  
Trigonometric Function Solutions ◽  
Modified Simple Equation Method

Three different methods are applied to construct new types of solutions of nonlinear evolution equations. First, the Csch method is used to carry out the solutions; then the Extended Tanh-Coth method and the modified simple equation method are used to obtain the soliton solutions. The effectiveness of these methods is demonstrated by applications to the RKL model, the generalized derivative NLS equation. The solitary wave solutions and trigonometric function solutions are obtained. The obtained solutions are very useful in the nonlinear pulse propagation through optical fibers.

Some new dispersive dromions and integrability analysis for the Davey–Stewartson (DS-II) model in fluid dynamics

2021 ◽  
Author(s):  
Urooj Akram ◽  
Aly. R. Seadawy ◽  
Syed T. R. Rizvi ◽  
Muhammad Younis ◽  
Ali Althobaiti
Keyword(s):  
Fluid Dynamics ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Auxiliary Equation ◽  
Painlevé Test ◽  
Suggested Strategy

This paper focuses on the Davey–Stewartson (DS)-II equation, and the extended modified auxiliary equation mapping (EMAEM) architectonic is used to develop a new set of solutions such as kink, singular kink, rational, combined soliton-like solutions, bell-type solutions, trigonometric and hyperbolic solutions. Furthermore, this study reveals that the used technique is efficient for solving other nonlinear evolution equations (NLEEs). The Painleve test ([Formula: see text]-test) will also be used to examine the integrability of the DS-II equation. Finally, graphical simulations are designed to show the exact behavior of solutions as well as the efficacy of the suggested strategy.

Traveling wave solutions for the Couple Boiti-Leon-Pempinelli System by using extended Jacobian elliptic function expansion method

2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman
Keyword(s):  
Elliptic Function ◽  
Traveling Wave ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Traveling Wave Solutions ◽  
Expansion Method ◽  
Mathematical Physics ◽  
Nonlinear Evolution Equations ◽  
Jacobian Elliptic Function ◽  
Function Expansion

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.

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