In this paper, we consider the half-wave equations with combined power nonlinearities i∂tu=-Δu-μuq-1u-up-1u,t,x∈R×Rd,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} i\partial _tu=\sqrt{-\Delta }u-\mu \left| u \right| ^{q-1}u-\left| u \right| ^{p-1}u,~~~~\left( t,x \right) \in \mathbb {R}\times \mathbb {R}^{d}, \end{aligned}$$\end{document}where d≥2\documentclass[12pt]{minimal}
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\begin{document}$$d\ge 2$$\end{document}, μ∈R\documentclass[12pt]{minimal}
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\begin{document}$$\mu \in \mathbb {R}$$\end{document} and 1<q<p<1+2d-1\documentclass[12pt]{minimal}
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\begin{document}$$1<q<p< 1+\frac{2}{d-1}$$\end{document}. We study traveling solitary waves of the form u(x,t)=eiωtQ(x-vt),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} u(x,t)=e^{i\omega t}Q(x-vt), \end{aligned}$$\end{document}with frequency w∈R\documentclass[12pt]{minimal}
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\begin{document}$$w \in \mathbb {R}$$\end{document}, and velocity v∈Rd\documentclass[12pt]{minimal}
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\begin{document}$$v\in \mathbb {R}^{d}$$\end{document}. As v⩾1\documentclass[12pt]{minimal}
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\begin{document}$$\left| v \right| \geqslant 1$$\end{document}, we establish a general nonexistence of traveling solitary waves by using Riesz transforms and a virial-type identity. As 0<v<1\documentclass[12pt]{minimal}
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\begin{document}$$0< \left| v \right| < 1$$\end{document}, under different assumptions on q<p\documentclass[12pt]{minimal}
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\begin{document}$$q< p$$\end{document}, we prove several existence results for traveling solitary waves. In particular, we consider cases when 1<q<1+2d<p<1+2d-1,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} 1< q< 1+\frac{2}{d}< p< 1+\frac{2}{d-1}, \end{aligned}$$\end{document}i.e., the two nonlinearities have different character with respect to the L2\documentclass[12pt]{minimal}
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\begin{document}$$L^{2}$$\end{document}-critical exponent. Note that such traveling solitary waves Q is not radially symmetric in x∈Rd\documentclass[12pt]{minimal}
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\begin{document}$$x \in \mathbb {R}^{d}$$\end{document}, and we need to overcome the lack of compactness and obtain the existence of mountain-pass-type solution and saddle-type solution. In addition, based on the existence and properties of traveling solitary waves, we also prove that small data scattering fails to hold for the nonlinear half-wave equations.
