On the Exact Soliton Solutions of Some Nonlinear PDE by Tan-Cot Method

2018 ◽  
Vol 5 (1) ◽  
pp. 31-36
Author(s):  
Md Monirul Islam ◽  
Muztuba Ahbab ◽  
Md Robiul Islam ◽  
Md Humayun Kabir
Keyword(s):  
Solitary Wave ◽  
Acoustic Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Rectangular Channel ◽  
Soliton Solutions ◽  
Boussinesq Equations ◽  
Travelling Wave ◽  
Ion Acoustic Waves ◽  
Analytical System

For many solitary wave applications, various approximate models have been proposed. Certainly, the most famous solitary wave equations are the K-dV, BBM and Boussinesq equations. The K-dV equation was originally derived to describe shallow water waves in a rectangular channel. Surprisingly, the equation also models ion-acoustic waves and magneto-hydrodynamic waves in plasmas, waves in elastic rods, equatorial planetary waves, acoustic waves on a crystal lattice, and more. If we describe all of the above situation, we must be needed a solution function of their governing equations. The Tan-cot method is applied to obtain exact travelling wave solutions to the generalized Korteweg-de Vries (gK-dV) equation and generalized Benjamin-Bona- Mahony (BBM) equation which are important equations to evaluate wide variety of physical applications. In this paper we described the soliton behavior of gK-dV and BBM equations by analytical system especially using Tan-cot method and shown in graphically. GUB JOURNAL OF SCIENCE AND ENGINEERING, Vol 5(1), Dec 2018 P 31-36

The effect of the induced mean flow on solitary waves in deep water

1998 ◽  
Vol 355 ◽  
pp. 317-328 ◽  
Author(s):  
T. R. AKYLAS ◽  
F. DIAS ◽  
R. H. J. GRIMSHAW
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Deep Water ◽  
Water Waves ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Phase Speed ◽  
Mean Flow ◽  
Nls Equation ◽  
Envelope Soliton

Two branches of gravity–capillary solitary water waves are known to bifurcate from a train of infinitesimal periodic waves at the minimum value of the phase speed. In general, these solitary waves feature oscillatory tails with exponentially decaying amplitude and, in the small-amplitude limit, they may be interpreted as envelope-soliton solutions of the nonlinear Schrödinger (NLS) equation such that the envelope travels at the same speed as the carrier oscillations. On water of infinite depth, however, based on the fourth-order envelope equation derived by Hogan (1985), it is shown that the profile of these gravity–capillary solitary waves actually decays algebraically (like 1/x2) at infinity owing to the induced mean flow that is not accounted for in the NLS equation. The algebraic decay of the solitary-wave tails in deep water is confirmed by numerical computations based on the full water-wave equations. Moreover, the same behaviour is found at the tails of solitary-wave solutions of the model equation proposed by Benjamin (1992) for interfacial waves in a two-fluid system.

Analysis of the generalized (2+1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium

2017 ◽  
Vol 31 (22) ◽  
pp. 1750135
Author(s):  
Han-Peng Chai ◽  
Bo Tian ◽  
Hui-Ling Zhen ◽  
Jun Chai ◽  
Yue-Yang Guan
Keyword(s):  
Inhomogeneous Medium ◽  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Variable Coefficients ◽  
Bilinear Method ◽  
Ion Acoustic Waves ◽  
Dispersion Terms ◽  
Kdv Type Equations ◽  
Higher Order Dispersion

Korteweg-de Vries (KdV)-type equations are seen to describe the shallow-water waves, lattice structures and ion-acoustic waves in plasmas. Hereby, we consider an extension of the KdV-type equations called the generalized (2[Formula: see text]+[Formula: see text]1)-dimensional Nizhnik–Novikov–Veselov equations with variable coefficients in an inhomogeneous medium. Via the Hirota bilinear method and symbolic computation, we derive the bilinear forms, N-soliton solutions and Bäcklund transformation. Effects of the first- and higher-order dispersion terms are investigated. Soliton evolution and interaction are graphically presented and analyzed: Both the propagation velocity and direction of the soliton change when the dispersion terms are time-dependent; The interactions between/among the solitons are elastic, independent of the forms of the coefficients in the equations.

Multi-Soliton Solutions of the Generalized Sawada–Kotera Equation

2016 ◽  
Vol 71 (4) ◽  
pp. 305-309 ◽  
Author(s):  
Da-Wei Zuo ◽  
Hui-Xia Mo ◽  
Hui-Ping Zhou
Keyword(s):  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Elastic Interaction ◽  
Bell Polynomials ◽  
Nonlinear Phenomena ◽  
Shallow Water Waves ◽  
Ion Acoustic Waves ◽  
Soliton Propagation ◽  
Kdv Type Equations

AbstractKorteweg–de Vries (KdV)-type equations can describe the nonlinear phenomena in shallow water waves, stratified internal waves, and ion-acoustic waves in plasmas. In this article, the two-dimensional generalization of the Sawada–Kotera equation, one of the KdV-type equations, is discussed by virtue of the Bell polynomials and Hirota method. The results show that there exist multi-soliton solutions for such an equation. Relations between the direction of the soliton propagation and coordinate axes are shown. Elastic interaction with the multi-soliton solutions are analysed.

Multi-soliton Collisions and Bäcklund Transformations for the (2+1)-dimensional Modified Nizhnik–Novikov–Vesselov Equations

2015 ◽  
Vol 70 (8) ◽  
pp. 629-635
Author(s):  
Xi-Yang Xie ◽  
Bo Tian ◽  
Yu-Feng Wang ◽  
Wen-Rong Sun ◽  
Ya Sun
Keyword(s):  
Acoustic Waves ◽  
Water Waves ◽  
Soliton Solutions ◽  
Bäcklund Transformations ◽  
Bell Polynomials ◽  
Ion Acoustic Waves ◽  
Backlund Transformations ◽  
Kdv Type Equations ◽  
Binary Bell Polynomials ◽  
Soliton Collisions

AbstractThe Korteweg–de Vries (KdV)-type equations can describe the shallow water waves, stratified internal waves, ion-acoustic waves, plasma physics and lattice dynamics, while the (2+1)-dimensional Nizhnik–Novikov–Vesselov equations are the isotropic extensions of KdV-type equations. In this paper, we investigate the (2+1)-dimensional modified Nizhnik–Novikov–Vesselov equations. By virtue of the binary Bell polynomials, bilinear forms, multi-soliton solutions and Bäcklund transformations are derived. Effects of some parameters on the solitons and monotonic function are graphically illustrated. We can observe the coalescence of the two solitons in their collision region, where their shapes change after the collision.

Interaction of a Second-Order Solitary Wave With a Vertical Permeable Plane Breakwater

2004 ◽  
Author(s):  
A. Basmat
Keyword(s):  
Solitary Wave ◽  
Water Waves ◽  
Boussinesq Equations ◽  
Second Order ◽  
Wave Loading ◽  
Coastal Structures ◽  
Transformation Techniques ◽  
Incident Plane ◽  
Impermeable Wall ◽  
Plane Barrier

The purpose of this paper is to develop mathematical models to investigate the interaction between long non-linear water waves and dissipative/absorbing coastal structures. The diffraction of a plane second-order solitary wave at a vertical permeable plane barrier standing in front of an impermeable wall, with calculation of the second-order wave loading is investigated. An incident plane second-order solitary wave is the Laitone solution of Boussinesq equations. The analytical solution is obtained by means of a small parameter development and Fourier transformation techniques. Computational results were performed using the software MATHEMATICA version 4.0.1.0.

Growth rates of bending KdV solitons

1982 ◽  
Vol 28 (3) ◽  
pp. 469-484 ◽  
Author(s):  
E. W. Laedke ◽  
K. H. Spatschek
Keyword(s):  
Growth Rate ◽  
Lower Bounds ◽  
Acoustic Waves ◽  
Variational Principles ◽  
Soliton Solutions ◽  
Type Equation ◽  
Maximum Growth ◽  
Maximum Growth Rate ◽  
Upper And Lower Bounds ◽  
Ion Acoustic Waves

Nonlinear ion-acoustic waves in magnetized plasmas are investigated. In strong magnetic fields they can be described by a Korteweg-de Vries (KdV) type equation. It is shown here that these plane soliton solutions become unstable with respect to bending distortions. Variational principles are derived for the maximum growth rate γ as a function of the transverse wavenumber k of the perturbations. Since the variational principles are formulated in complementary form, the numerical evaluation yields upper and lower bounds for γ. Choosing appropriate test functions and increasing the accuracy of the computations we find very close upper and lower bounds for the γ(k) curve. The results show that the growth rate peaks at a certain value of k and a cut-off kc exists. In the region where the γ(k) curve was not predicted numerically with high accuracy, i.e. near the cut-off, we find very precise analytical estimates. These findings are compared with previous results. For k≥kc, stability with respect to transverse perturbations is proved.

scholarly journals New multi-soliton solutions of Whitham-Broer-Kaup shallow-water-wave equations

2017 ◽  
Vol 21 (suppl. 1) ◽  
pp. 137-144 ◽  
Author(s):  
Sheng Zhang ◽  
Mingying Liu ◽  
Bo Xu
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Equations ◽  
Soliton Solutions ◽  
Bilinear Forms ◽  
Shallow Water Waves ◽  
Bilinear Method ◽  
Shallow Water Wave ◽  
Shallow Water Wave Equations

In this paper, new and more general Whitham-Broer-Kaup equations which can describe the propagation of shallow-water waves are exactly solved in the framework of Hirota?s bilinear method and new multi-soliton solutions are obtained. To be specific, the Whitham-Broer-Kaup equations are first reduced into Ablowitz- Kaup-Newell-Segur equations. With the help of this equations, bilinear forms of the Whitham-Broer-Kaup equations are then derived. Based on the derived bilinear forms, new one-soliton solutions, two-soliton solutions, three-soliton solutions, and the uniform formulae of n-soliton solutions are finally obtained. It is shown that adopting the bilinear forms without loss of generality play a key role in obtaining these new multi-soliton solutions.

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