In this paper, we prove the non-existence and existence of normalized traveling solitary waves for a class of half-wave equations with nonlinear term in the $L^2$-supercritical regime. Our results are based on Riesz transforms, minimax approach and Concentration–compactness Lemma. In addition, we also obtain the Cauchy problem of the focusing half-wave equations does not have small data scattering.

中文翻译：

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一类非线性半波方程的归一化孤行

在本文中，我们证明了在非线性条件下，一类带有非线性项的半波方程的归一化行波孤波的不存在和存在。 ${\mathrm{大号}}^{\mathrm{2个}}$-超临界状态。我们的结果基于Riesz变换，最小极大值方法和浓度-紧凑度引理。另外，我们还得到了聚焦半波方程的柯西问题，该散射不具有较小的数据散射。