Qualitative Aspects of Nonlinear Wave Motion: Complexity and Simplicity

1993 ◽  
Vol 46 (12) ◽  
pp. 509-518 ◽  
Author(s):  
J. Engelbrecht
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Phase Changes ◽  
Mathematical Tool ◽  
Qualitative Properties ◽  
Wave Processes ◽  
Spectral Changes ◽  
Nonlinear Mathematical Models ◽  
Nonlinear Wave Motion ◽  
Physical Effects

The nonlinear wave processes possess many qualitative properties which cannot be described by linear theories. In this presentation, an attempt is made to systematize the main aspects of this fascinating area. The sources of nonlinearities are analyzed in order to understand why and how the nonlinear mathematical models are formulated. The technique of evolution equations is discussed then as a main mathematical tool to separate multiwave processes into single waves. The evolution equations give concise but in many cases sufficient description of wave processes in solids permitting to analyze spectral changes, phase changes and velocities, coupling of waves, and interaction of nonlinearities with other physical effects of the same order. Several new problems are listed. Knowing the reasons, the seemingly complex problems can be effectively analyzed.

scholarly journals Soliton solutions of nonlinear wave equation in finite de-formation elastic cylindrical rod by solitary wave ansatz method

2016 ◽  
Vol 4 (1) ◽  
pp. 12
Author(s):  
Salam Subhaschandra Singh
Keyword(s):  
Wave Equation ◽  
Solitary Wave ◽  
Nonlinear Wave ◽  
Nonlinear Wave Equation ◽  
Finite Deformation ◽  
Soliton Solutions ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
Ansatz Method ◽  
Solitary Wave Ansatz Method

<p>In this paper, we consider nonlinear wave equation in finite deformation elastic cylindrical rod and obtain soliton solutions by Solitary Wave Ansatz method. It is shown that the ansatz method provides a very effective and powerful mathematical tool for obtaining solutions for Nonlinear Evolution Equations (NLEEs) in nonlinear Science.</p><div style="mso-element: para-border-div; border: none; border-bottom: solid windowtext 1.0pt; mso-border-bottom-alt: solid windowtext .25pt; padding: 0cm 0cm 1.0pt 0cm;"><p class="IJOPCMKeywards" style="margin-bottom: 0.0001pt; text-align: justify; border: none; padding: 0cm;"><span style="font-size: 8.0pt; mso-fareast-language: EN-US;">Elastic Rod; Finite Deformation; Nonlinear Wave Equation; Solitary Wave Ansatz Method; Soliton.</span></p></div>

scholarly journals Hamiltonian description of wave dynamics in nonequilibrium media

1994 ◽  
Vol 1 (4) ◽  
pp. 234-248 ◽  
Author(s):  
N. N. Romanova
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Normal Modes ◽  
Weak Nonlinearity ◽  
Hamiltonian Description ◽  
Weakly Nonlinear ◽  
Wave Dynamics ◽  
Canonical Variables ◽  
Stable Points ◽  
Stable Area

Abstract. We consider Hamiltonian description of weakly nonlinear wave dynamics in unstable and nonequilibrium media. We construct the appropriate canonical variables in the whole wavenumber space. The essentially new element is the construction of canonical variables in a vicinity of marginally stable points where two normal modes coalesce. The commonly used normal variables are not appropriate in this domain. The mater is that the approximation of weak nonlinearity breaks down when the dynamical system is written in terms of these variables. In this case we introduce the canonical variables based on the linear combination of modes belonging to the two different branches of dispersion curve. As an example of one of the possible applications of presented results the evolution equations for weakly nonlinear wave packets in the marginally stable area are derived. These equations cannot be derived if we deal with the commonly used normal variables.

scholarly journals David J. Benney: Nonlinear Wave and Instability Processes in Fluid Flows

2020 ◽  
Vol 52 (1) ◽  
pp. 21-36 ◽  
Author(s):  
T.R. Akylas
Keyword(s):  
Nonlinear Wave ◽  
Evolution Equations ◽  
Flow Instability ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Nonlinear Evolution Equations ◽  
Nonlinear Phenomena ◽  
New Paradigm ◽  
Applied Mathematician ◽  
Modulated Wave Trains

David J. Benney (1930–2015) was an applied mathematician and fluid dynamicist whose highly original work has shaped our understanding of nonlinear wave and instability processes in fluid flows. This article discusses the new paradigm he pioneered in the study of nonlinear phenomena, which transcends fluid mechanics, and it highlights the common threads of his research contributions, namely, resonant nonlinear wave interactions; the derivation of nonlinear evolution equations, including the celebrated nonlinear Schrödinger equation for modulated wave trains; and the significance of three-dimensional disturbances in shear flow instability and transition.

Nonlinear wave motion in a strong potential

Wave Motion ◽  
1991 ◽  
Vol 13 (3) ◽  
pp. 291-302 ◽  
Author(s):  
Joseph B. Keller ◽  
Jacob Rubinstein
Keyword(s):  
Nonlinear Wave ◽  
Wave Motion ◽  
Strong Potential ◽  
Nonlinear Wave Motion

Application of the Exp-Functionmethod to the Riccati Equation and New Exact Solutions with Three Arbitrary Functions of Quantum Zakharov Equations

2008 ◽  
Vol 63 (10-11) ◽  
pp. 646-652 ◽  
Author(s):  
Mohamed A Abdou ◽  
Essam M. Abulwafa
Keyword(s):  
Exact Solutions ◽  
Riccati Equation ◽  
Function Method ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Nonlinear Evolution Equations ◽  
Mathematical Tool ◽  
New Exact Solutions ◽  
Generalized Riccati Equation ◽  
Solitary Solutions

The Exp-function method with the aid of the symbolic computational system is used for constructing generalized solitary solutions of the generalized Riccati equation. Based on the Riccati equation and its generalized solitary solutions, new exact solutions with three arbitrary functions of quantum Zakharov equations are obtained. It is shown that the Exp-function method provides a straightforward and important mathematical tool for nonlinear evolution equations in mathematical physics.

Nonlinear wave motion in magnetoelasticity

1974 ◽  
Vol 55 (2) ◽  
pp. 124-192 ◽  
Author(s):  
J. Bazer ◽  
W. B. Ericson
Keyword(s):  
Nonlinear Wave ◽  
Wave Motion ◽  
Nonlinear Wave Motion
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