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

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.

Download Full-text

The stability of solitary waves

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1972.0074 ◽

1972 ◽

Vol 328 (1573) ◽

pp. 153-183 ◽

Cited By ~ 374

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.

Download Full-text

Stability and instability of some nonlinear dispersive solitary waves in higher dimension

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500030614 ◽

1996 ◽

Vol 126 (1) ◽

pp. 89-112 ◽

Cited By ~ 21

Author(s):

Anne de Bouard

Keyword(s):

Solitary Waves ◽

Acoustic Waves ◽

Vries Equation ◽

Three Dimensional ◽

Higher Dimension ◽

Ion Acoustic Waves ◽

Stability And Instability ◽

Ion Acoustic ◽

De Vries ◽

The Stability

We study the stability of positive radially symmetric solitary waves for a three dimensional generalisation of the Korteweg de Vries equation, which describes nonlinear ion-acoustic waves in a magnetised plasma, and for a generalisation in dimension two of the Benjamin–Bona–Mahony equation.

Download Full-text

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

Journal of Vibration and Acoustics ◽

10.1115/1.2838656 ◽

1995 ◽

Vol 117 (B) ◽

pp. 145-153 ◽

Cited By ~ 29

Author(s):

D. S. Bernstein ◽

S. P. Bhat

Keyword(s):

Asymptotic Stability ◽

Lyapunov Stability ◽

Sufficient Conditions ◽

Second Order ◽

Necessary And Sufficient Conditions ◽

Viscous Damping ◽

Stability And Instability ◽

The Stability ◽

Second Order Systems ◽

Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Download Full-text

Eigenvalue problems for the wave equation with strong damping

Proceedings of the Royal Society of Edinburgh Section A Mathematics ◽

10.1017/s0308210500023805 ◽

1997 ◽

Vol 127 (4) ◽

pp. 755-771 ◽

Cited By ~ 4

Author(s):

Pedro Freitas

Keyword(s):

Stationary Solution ◽

Wave Equations ◽

Eigenvalue Problems ◽

Sufficient Conditions ◽

Stationary Solutions ◽

Linear Operators ◽

Stability And Instability ◽

The Stability ◽

Necessary And Sufficient ◽

Strongly Damped

This paper presents a study of linear operators associated with the linearisation of general semilinear strongly damped wave equations around stationary solutions. The structure of the spectrum of such operators is considered in detail, with an emphasis on stability questions. Necessary and sufficient conditions for the stability of the trivial solution of the linear equation are given, together with conditions for this solution to become unstable. In the latter case, the mechanisms which are responsible for the change of stability are analysed. These results are then applied to obtain stability and instability conditions for the semilinear problem. In particular, a condition is given which ensures that the dimensions of the centre and unstable manifolds of a stationary solution are the same as when that solution is considered as a stationary solution of an associated parabolic problem.

Download Full-text

The stability of internal solitary waves

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100061193 ◽

1983 ◽

Vol 94 (2) ◽

pp. 351-379 ◽

Cited By ~ 32

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.

Download Full-text

Dynamics of Asymmetric Nonconservative Systems

Journal of Applied Mechanics ◽

10.1115/1.3166991 ◽

1983 ◽

Vol 50 (1) ◽

pp. 199-203 ◽

Cited By ~ 61

Author(s):

D. J. Inman

Keyword(s):

Sufficient Conditions ◽

Symmetric System ◽

Qualitative Behavior ◽

Parameter System ◽

Nonconservative Systems ◽

Asymmetric System ◽

Lumped Parameter ◽

Stability And Instability ◽

The Stability ◽

Necessary And Sufficient

This work examines a linear, asymmetric lumped parameter system. Results on the qualitative behavior of a certain subclass of such systems are presented. In particular, necessary and sufficient conditions for the existence of a linear transformation that transforms an asymmetric system into an equivalent symmetric system are derived. Results on the stability and instability of such systems are presented and stated in terms of the original asymmetric system’s coefficient matrices. This work is compared with that of other authors and numerical examples illustrating the utility and correctness of the results are presented.

Download Full-text

Some necessary and sufficient conditions for stability of three-layer difference schemes

Filomat ◽

10.2298/fil1802705v ◽

2018 ◽

Vol 32 (2) ◽

pp. 705-713

Author(s):

Vanja Vukoslavcevic

Keyword(s):

Sufficient Conditions ◽

Difference Schemes ◽

Necessary And Sufficient Conditions ◽

Special Cases ◽

The Stability ◽

Necessary And Sufficient

This paper investigates two classes of three-layer difference schemes with weights in the form ?y?t,n+??2y?tt,n+?1Ayn-1+(E-?1-?2)Ayn+?2Ayn+1 = ?n and ?y?t,n+??2y?tt,n+A(?1yn-1+(E-?1-?2)yn+?2yn+1) = ?n. It obtains some sufficient conditions for the stability in a defined norm and, also, in special cases we achieve conditions for stability which do not depend on the choice of norm.

Download Full-text

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

Journal of Mechanical Design ◽

10.1115/1.2836448 ◽

1995 ◽

Vol 117 (B) ◽

pp. 145-153 ◽

Cited By ~ 26

Author(s):

D. S. Bernstein ◽

S. P. Bhat

Keyword(s):

Asymptotic Stability ◽

Lyapunov Stability ◽

Sufficient Conditions ◽

Second Order ◽

Necessary And Sufficient Conditions ◽

Viscous Damping ◽

Stability And Instability ◽

The Stability ◽

Second Order Systems ◽

Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Download Full-text

On Solitary-Wave Solutions for the Coupled Korteweg – de Vries and Modified Korteweg – de Vries Equations and their Dynamics

Zeitschrift für Naturforschung A ◽

10.1515/zna-2006-3-403 ◽

2006 ◽

Vol 61 (3-4) ◽

pp. 125-132 ◽

Cited By ~ 2

Author(s):

Woo-Pyo Hong

Keyword(s):

Solitary Wave ◽

Numerical Simulations ◽

Solitary Waves ◽

Function Method ◽

Auxiliary Function ◽

Solitary Wave Solutions ◽

Wave Solutions ◽

De Vries ◽

Dynamical Properties ◽

52.35 Mw

Analytic sech4-type traveling solitary-wave solutions of the coupled Korteweg-de Vries and modified Korteweg-de Vries equations proposed by Kersten-Krasil’shchik, are found by applying the auxiliary function method. The dynamical properties of the solitary-waves are studied by numerical simulations. - PACS numbers: 03.40.Kf, 02.30.Jr, 47.20.Ky, 52.35.Mw

Download Full-text