We study the existence of traveling wave solutions to a unidirectional shallow water model which incorporates the full linear dispersion relation for both gravitational and capillary restoring forces. Using functional analytic techniques, we show that for small surface tension (corresponding to Bond numbers between $0$ and ${1}/{3}$) there exists small amplitude solitary waves that decay to asymptotically small periodic waves at spatial infinity. The size of the oscillations in the far field are shown to be small beyond all algebraic orders in the amplitude of the wave. We also present numerical evidence, based on the recent analytical work of Hur \& Johnson, that the asymptotic end states are modulationally stable for all Bond numbers between $0$ and $1/3$.

中文翻译：

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重力-毛细管 Whitham 方程中的广义孤立波

我们研究了单向浅水模型的行波解的存在性，该模型结合了重力和毛细管恢复力的完整线性色散关系。我们使用泛函分析技术表明，对于小表面张力（对应于 $0$ 和 ${1}/{3}$ 之间的键数），存在小振幅孤立波，在空间无穷远处衰减为渐近小周期波。远场中振荡的大小显示为小，超出波幅的所有代数阶数。我们还提供了基于 Hur \& Johnson 近期分析工作的数值证据，即渐近终态对于 $0$ 和 $1/3$ 之间的所有债券数都是调制稳定的。