This paper sheds new light on the stability properties of solitary wavesolutions associated with models of Korteweg-de Vries andBenjamin\&Bona\&Mahoney type, when the dispersion is very lower. Via anapproach of compactness, analyticity and asymptotic perturbation theory, weestablish sufficient conditions for the existence of exponentially growingsolutions to the linearized problem and so a criterium of linear instability ofsolitary waves is obtained for both models. Moreover, the nonlinear stabilityand linear instability of the ground states solutions for both models isobtained for some specific regimen of parameters. Via a Lyapunov strategy and avariational analysis we obtain the stability of the blow-up of solitary wavesfor the critical fractional KdV equation. The arguments presented in this investigation has prospects for the study ofthe instability of traveling waves solutions of other nonlinear evolutionequations.
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机译：当色散非常低时，本文对与Korteweg-de Vries模型和Benjamin \ Bona \ Mahoney型模型相关的孤立波解的稳定性进行了研究。通过紧致性，解析性和渐近摄动理论，我们为线性化问题的指数增长解的存在建立了充分条件，因此对于这两个模型都获得了孤立波线性不稳定性的判据。此外，针对某些特定的参数方案，获得了两个模型的基态解的非线性稳定性和线性不稳定性。通过Lyapunov策略和变异分析，我们获得了临界分数KdV方程孤波爆炸的稳定性。这项研究提出的论点具有研究其他非线性演化方程行波解的不稳定性的前景。
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