The stability of solitary waves

1986 ◽  
Vol 29 (3) ◽  
pp. 650 ◽  
Author(s):  
Mitsuhiro Tanaka
Keyword(s):  
Solitary Waves ◽  
Stability Of Solitary Waves ◽  
The Stability

scholarly journals Orbital Stability of Solitary Waves to Double Dispersion Equations with Combined Power-Type Nonlinearity

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1398
Author(s):  
Natalia Kolkovska ◽  
Milena Dimova ◽  
Nikolai Kutev
Keyword(s):  
Solitary Waves ◽  
Orbital Stability ◽  
Second Derivative ◽  
Cubic Nonlinearity ◽  
Abstract Theory ◽  
Power Type ◽  
Analytical Formulas ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Double Dispersion

We consider the orbital stability of solitary waves to the double dispersion equation utt−uxx+h1uxxxx−h2uttxx+f(u)xx=0,h1>0,h2>0 with combined power-type nonlinearity f(u)=a|u|pu+b|u|2pu,p>0,a∈R,b∈R,b≠0. The stability of solitary waves with velocity c, c2<1 is proved by means of the Grillakis, Shatah, and Strauss abstract theory and the convexity of the function d(c), related to some conservation laws. We derive explicit analytical formulas for the function d(c) and its second derivative for quadratic-cubic nonlinearity f(u)=au2+bu3 and parameters b>0, c2∈0,min1,h1h2. As a consequence, the orbital stability of solitary waves is analyzed depending on the parameters of the problem. Well-known results are generalized in the case of a single cubic nonlinearity f(u)=bu3.

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.

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 Characterization and dynamical stability of fully nonlinear strain solitary waves in a fluid-filled hyperelastic membrane tube

2020 ◽  
Vol 231 (10) ◽  
pp. 4095-4110 ◽  
Author(s):  
A. T. Il’íchev ◽  
V. A. Shargatov ◽  
Y. B. Fu
Keyword(s):  
Solitary Waves ◽  
Wave Speed ◽  
Finite Amplitude ◽  
Quiescent State ◽  
Governing Equations ◽  
Membrane Tube ◽  
Stability Of Solitary Waves ◽  
The Stability ◽  
Intermediate Value

Abstract We first characterize strain solitary waves propagating in a fluid-filled membrane tube when the fluid is stationary prior to wave propagation and the tube is also subjected to a finite stretch. We consider the parameter regime where all traveling waves admitted by the linearized governing equations have nonzero speed. Solitary waves are viewed as waves of finite amplitude that bifurcate from the quiescent state of the system with the wave speed playing the role of the bifurcation parameter. Evolution of the bifurcation diagram with respect to the pre-stretch is clarified. We then study the stability of solitary waves for a representative case that is likely of most interest in applications, the case in which solitary waves exist with speed c lying in the interval $$[0, c_1)$$ [ 0 , c 1 ) where $$c_1$$ c 1 is the bifurcation value of c, and the wave amplitude is a decreasing function of speed. It is shown that there exists an intermediate value $$c_0$$ c 0 in the above interval such that solitary waves are spectrally stable if their speed is greater than $$c_0$$ c 0 and unstable otherwise.

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