Instability of a class of dispersive solitary waves

1990 ◽  
Vol 114 (3-4) ◽  
pp. 195-212 ◽  
Author(s):  
P. E. Souganidis ◽  
W. A. Strauss
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Evolution Equations ◽  
Travelling Waves ◽  
Solitary Wave Solutions ◽  
Dispersive Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Stability And Instability ◽  
The Stability

SynopsisThis paper studies the stability and instability properties of solitary wave solutions φ(x – ct) of a general class of evolution equations of the form Muttf(u)x=0, which support weakly nonlinear dispersive waves. It turns out that, depending on their speed c and the relation between the dispersion (i.e. the order of the pseudodifferential operator) and the nonlinearity, travelling waves maybe stable or unstable. Sharp conditions to that effect are given.

Stability and instability of solitary waves of Korteweg-de Vries type

1987 ◽  
Vol 411 (1841) ◽  
pp. 395-412 ◽  
Keyword(s):  
Solitary Waves ◽  
Sufficient Conditions ◽  
Solitary Wave Solutions ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Stability And Instability ◽  
Special Cases ◽  
De Vries ◽  
The Stability ◽  
Necessary And Sufficient

Considered herein are the stability and instability properties of solitary-wave solutions of a general class of equations that arise as mathematical models for the unidirectional propagation of weakly nonlinear, dispersive long waves. Special cases for which our analysis is decisive include equations of the Korteweg-de Vries and Benjamin-Ono type. Necessary and sufficient conditions are formulated in terms of the linearized dispersion relation and the nonlinearity for the solitary waves to be stable.

scholarly journals Existence and linearized stability of solitary waves for a quasilinear Benney system

2016 ◽  
Vol 146 (3) ◽  
pp. 547-564 ◽  
Author(s):  
João-Paulo Dias ◽  
Mário Figueira ◽  
Filipe Oliveira
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Compact Support ◽  
Travelling Waves ◽  
Standing Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Linearized Stability ◽  
Stability Of Solitary Waves ◽  
Image Position

We prove the existence of solitary wave solutions to the quasilinear Benney systemwhere , –1 < p < +∞ and a, γ > 0. We establish, in particular, the existence of travelling waves with speed arbitrarily large if p < 0 and arbitrarily close to 0 if . We also show the existence of standing waves in the case , with compact support if – 1 < p < 0. Finally, we obtain, under certain conditions, the linearized stability of such solutions.

scholarly journals The stability of solitary waves

1972 ◽  
Vol 328 (1573) ◽  
pp. 153-183 ◽  
Keyword(s):  
Solitary Wave ◽  
Spectral Theory ◽  
Solitary Waves ◽  
Vries Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
De Vries ◽  
Long Time ◽  
Stability Of Solitary Waves ◽  
The Stability

The Korteweg-de Vries equation, which describes the unidirectional propagation of long waves in a wide class of nonlinear dispersive systems, is well known to have solutions representing solitary waves. The present analysis establishes that these solutions are stable, confirming a property that has for a long time been presumed. The demonstration of stability hinges on two nonlinear functionals which for solutions of the Korteweg-de Vries equation are invariant with time: these are introduced in § 2, where it is recalled that Boussinesq recognized their significance in relation to the stability of solitary waves. The principles upon which the stability theory is based are explained in § 3, being supported by a few elementary ideas from functional analysis. A proof that solitary wave solutions are stable is completed in § 4, the most exacting steps of which are accomplished by means of spectral theory. In appendix A a method deriving from the calculus of variations is presented, whereby results needed for the proof of stability may be obtained independently of spectral theory as used in § 4. In appendix B it is shown how the stability analysis may readily be adapted to solitary-wave solutions of the ‘regularized long-wave equation’ that has recently been advocated by Benjamin, Bona & Mahony as an alternative to the Korteweg-de Vries equation. In appendix C a variational principle is demonstrated relating to the exact boundaryvalue problem for solitary waves in water: this is a counterpart to a principle used in the present work (introduced in §2) and offers some prospect of proving the stability of exact solitary waves.

The stability of internal solitary waves

1983 ◽  
Vol 94 (2) ◽  
pp. 351-379 ◽  
Author(s):  
D. P. Bennett ◽  
R. W. Brown ◽  
S. E. Stansfield ◽  
J. D. Stroughair ◽  
J. L. Bona
Keyword(s):  
Solitary Wave ◽  
Internal Waves ◽  
Solitary Waves ◽  
Model Equation ◽  
Focus Of Attention ◽  
Internal Solitary Waves ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability ◽  
The One

A theory is developed relating to the stability of solitary-wave solutions of the so-called Benjamin-Ono equation. This equation was derived by Benjamin (5) as a model for the propagation of internal waves in an incompressible non-diffusive heterogeneous fluid for which the density is non-constant only within a layer whose thickness is much smaller than the total depth. In his article, Benjamin wrote in closed form the one-parameter family of solitary-wave solutions of his model equation whose stability will be the focus of attention presently.

Construction of the numerical and analytical wave solutions of the Joseph–Egri dynamical equation for the long waves in nonlinear dispersive systems

2020 ◽  
Vol 34 (30) ◽  
pp. 2050289
Author(s):  
Abdulghani R. Alharbi ◽  
M. B. Almatrafi ◽  
Aly R. Seadawy
Keyword(s):  
Solitary Wave ◽  
Numerical Solution ◽  
Evolution Equations ◽  
Dynamical Equation ◽  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Nonlinear Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
The Stability

The Kudryashov technique is employed to extract several classes of solitary wave solutions for the Joseph–Egri equation. The stability of the achieved solutions is tested. The numerical solution of this equation is also investigated. We also present the accuracy and the stability of the numerical schemes. Some two- and three-dimensional figures are shown to present the solutions on some specific domains. The used methods are found useful to be applied on other nonlinear evolution equations.

Quasi-soliton and other behaviour of the nonlinear cubic-quintic Schrödinger equation

1986 ◽  
Vol 64 (3) ◽  
pp. 311-315 ◽  
Author(s):  
Stuart Cowan ◽  
R. H. Enns ◽  
S. S. Rangnekar ◽  
Sukhpal S. Sanghera
Keyword(s):  
Solitary Wave ◽  
Schrödinger Equation ◽  
Parameter Space ◽  
Schrodinger Equation ◽  
Solitary Wave Solutions ◽  
Wave Interactions ◽  
Wave Solutions ◽  
The Stability

The stability of the solitary-wave solutions of the nonlinear cubic–quintic Schrödinger equation (NLCQSE) is examined numerically. The solutions are found not to be solitons, but quasi-soliton behaviour is found to persist over wide regions of parameter space. Outside these regions dispersive and explosive behaviour is observed in solitary-wave interactions.

A class of nondispersive evolution equations with solitary wave solutions

1982 ◽  
Vol 6 (1) ◽  
pp. 43-47
Author(s):  
A. Abbas ◽  
A. C. Bryan ◽  
A. E. G. Stuart
Keyword(s):  
Solitary Wave ◽  
Evolution Equations ◽  
Solitary Wave Solutions ◽  
Wave Solutions
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