543 research outputs found

    An energy-based stability criterion for solitary traveling waves in Hamiltonian lattices

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    In this work, we revisit a criterion, originally proposed in [Nonlinearity {\bf 17}, 207 (2004)], for the stability of solitary traveling waves in Hamiltonian, infinite-dimensional lattice dynamical systems. We discuss the implications of this criterion from the point of view of stability theory, both at the level of the spectral analysis of the advance-delay differential equations in the co-traveling frame, as well as at that of the Floquet problem arising when considering the traveling wave as a periodic orbit modulo a shift. We establish the correspondence of these perspectives for the pertinent eigenvalue and Floquet multiplier and provide explicit expressions for their dependence on the velocity of the traveling wave in the vicinity of the critical point. Numerical results are used to corroborate the relevant predictions in two different models, where the stability may change twice. Some extensions, generalizations and future directions of this investigation are also discussed

    On the Stability of Periodic Solutions of the Generalized Benjamin-Bona-Mahony Equation

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    We study the stability of a four parameter family of spatially periodic traveling wave solutions of the generalized Benjamin-Bona-Mahony equation to two classes of perturbations: periodic perturbations with the same periodic structure as the underlying wave, and long-wavelength localized perturbations. In particular, we derive necessary conditions for spectral instability to perturbations to both classes of perturbations by deriving appropriate asymptotic expansions of the periodic Evans function, and we outline a nonlinear stability theory to periodic perturbations based on variational methods which effectively extends our periodic spectral stability results.Comment: 27 pages, 3 figure

    Transverse Instability of Periodic Traveling Waves in the Generalized Kadomtsev-Petviashvili Equation

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    In this paper, we investigate the spectral instability of periodic traveling wave solutions of the generalized Korteweg-de Vries equation to long wavelength transverse perturbations in the generalized Kadomtsev-Petviashvili equation. By analyzing high and low frequency limits of the appropriate periodic Evans function, we derive an orientation index which yields sufficient conditions for such an instability to occur. This index is geometric in nature and applies to arbitrary periodic traveling waves with minor smoothness and convexity assumptions on the nonlinearity. Using the integrable structure of the ordinary differential equation governing the traveling wave profiles, we are then able to calculate the resulting orientation index for the elliptic function solutions of the Korteweg-de Vries and modified Korteweg-de Vries equations.Comment: 26 pages. Sign error corrected in Lemma 3. Statement of main theorem corrected. Exposition updated and references added

    Nonlinear Stability of Periodic Traveling Wave Solutions of the Generalized Korteweg-de Vries Equation

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    In this paper, we study the orbital stability for a four-parameter family of periodic stationary traveling wave solutions to the generalized Korteweg-de Vries equation. In particular, we derive sufficient conditions for such a solution to be orbitally stable in terms of the Hessian of the classical action of the corresponding traveling wave ordinary differential equation restricted to the manifold of periodic traveling wave solution. We show this condition is equivalent to the solution being spectrally stable with respect to perturbations of the same period in the case of the Korteweg-de Vries equation, and in neighborhoods of the homoclinic and equilibrium solutions in the case of a power-law nonlinearity.Comment: 24 page

    Instability and stability properties of traveling waves for the double dispersion equation

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    In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation  utt−uxx+auxxxx−buxxtt=−(∣u∣p−1u)xx ~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~ for  p>1~p>1,  a≥b>0~a\geq b>0. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms uxxxxu_{xxxx} and uxxttu_{xxtt}. We obtain an explicit condition in terms of aa, bb and pp on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation (b=0b=0), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with aa, bb and pp and then state explicitly the conditions under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure

    Instability and stability properties of traveling waves for the double dispersion equation

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    In this article we are concerned with the instability and stability properties of traveling wave solutions of the double dispersion equation  utt−uxx+auxxxx−buxxtt=−(∣u∣p−1u)xx ~u_{tt} -u_{xx}+a u_{xxxx}-bu_{xxtt} = - (|u|^{p-1}u)_{xx}~ for  p>1~p>1,  a≥b>0~a\geq b>0. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms uxxxxu_{xxxx} and uxxttu_{xxtt}. We obtain an explicit condition in terms of aa, bb and pp on wave velocities ensuring that traveling wave solutions of the double dispersion equation are strongly unstable by blow up. In the special case of the Boussinesq equation (b=0b=0), our condition reduces to the one given in the literature. For the double dispersion equation, we also investigate orbital stability of traveling waves by considering the convexity of a scalar function. We provide both analytical and numerical results on the variation of the stability region of wave velocities with aa, bb and pp and then state explicitly the conditions under which the traveling waves are orbitally stable.Comment: 16 pages, 4 figure
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