In this paper we consider the decay rate of solitary-wave solutions to some classes of non-linear and non-local dispersive equations, including for example the Whitham equation and a Whitham–Boussinesq system. The dispersive term is represented by a Fourier multiplier operator that has a real analytic symbol that either decays/grows, and we show that all supercritical/subcritical solitary-wave solutions decay exponentially, and moreover provide the exact decay rate, which in general will depend on the speed of the wave. We also prove that solitary waves have only one crest and are symmetric for some class of equations.

中文翻译：

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孤立波的衰减和对称性

在本文中，我们考虑了某些类别的非线性和非局部色散方程的孤立波解的衰减率，包括例如 Whitham 方程和 Whitham-Boussinesq 系统。色散项由傅立叶乘子算子表示，该算子具有衰减/增长的实解析符号，我们表明所有超临界/亚临界孤立波解都呈指数衰减，此外还提供精确的衰减率，这通常取决于在波的速度上。我们还证明了孤立波只有一个波峰，并且对于某类方程是对称的。