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

机译：双色散方程行波的不稳定性和稳定性

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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 u(tt) - u(xx) + au(xxxx) - bu(xxtt) = -(vertical bar u vertical bar(p-1)u)(xx) for p > 1, a > b > 0. The main characteristic of this equation is the existence of two sources of dispersion, characterized by the terms u(xxxx) and u(xxtt). We obtain an explicit condition in terms of a, b and p 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 = 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 analytical as well as numerical results on the variation of the stability region of wave velocities with a, b and p and then state explicitly the conditions under which the traveling waves are orbitally stable. (C) 2015 Elsevier Ltd. All rights reserved.
机译：在本文中，我们关注双色散方程u（tt）-u（xx）+ au（xxxx）-bu（xxtt）=-（竖线u竖线（p）的行波解的不稳定性和稳定性p> 1，a> b> 0时为-1）u）（xx）。该方程式的主要特征是存在两个色散源，其特征为u（xxxx）和u（xxtt）。我们根据a，b和p获得了一个明确的波速条件，以确保双频散方程的行波解因爆炸而非常不稳定。在Boussinesq方程的特殊情况下（b = 0），我们的条件简化为文献中给出的条件。对于双色散方程，我们还通过考虑标量函数的凸性来研究行波的轨道稳定性。我们提供了波速稳定区域随a，b和p变化的分析和数值结果，然后明确说明了行波在轨道上稳定的条件。 （C）2015 Elsevier Ltd.保留所有权利。

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