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.

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 Numerical approximation to Benjamin type equations. Generation and stability of solitary waves

2020 ◽  
Author(s):  
VA Dougalis ◽  
A Duran ◽  
Dimitrios Mitsotakis
Keyword(s):  
Solitary Wave ◽  
Solitary Waves ◽  
Numerical Approximation ◽  
Solitary Wave Solutions ◽  
Time Stepping ◽  
Runge Kutta Method ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Numerical Generation ◽  
The Waves

© 2018 Elsevier B.V. This paper is concerned with the study, by computational means, of the generation and stability of solitary-wave solutions of generalized versions of the Benjamin equation. The numerical generation of the solitary-wave profiles is accurately performed with a modified Petviashvili method which includes extrapolation to accelerate the convergence. In order to study the dynamics of the solitary waves the equations are discretized in space with a Fourier pseudospectral collocation method and a fourth-order, diagonally implicit Runge–Kutta method of composition type as time-stepping integrator. The stability of the waves is numerically studied by performing experiments with small and large perturbations of the solitary pulses as well as interactions of solitary waves.

Analytic Solitary-wave Solutions for Modified Korteweg - de Vries Equation with t-dependent Coefficients

2001 ◽  
Vol 56 (5) ◽  
pp. 366-370 ◽  
Author(s):  
Woo-Pyo Hong ◽  
Myung-Sang Yoona
Keyword(s):  
Solitary Wave ◽  
Symbolic Computation ◽  
Vries Equation ◽  
Bäcklund Transformation ◽  
Kdv Equation ◽  
Solitary Wave Solutions ◽  
Wave Solutions ◽  
Modified Kdv Equation ◽  
De Vries ◽  
Truncated Painlevé Expansion

Abstract We find analytic solitary wave solutions for a modified KdV equation with t-dependent coefficients of the form ut - 6α(t)uux + ß (t) uxxx -6γu2ux = 0. We make use of both the application of the truncated Painleve expansion and symbolic computation to obtain an auto-Bäcklund transformation. We show that kink-type analytic solitary-wave solutions exist under some constraints on α (t), ß (t) and γ.

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.

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