For the Cauchy problem of the half-wave equation with $L^2$-critical combined nonlinearities $\begin{array}{l}\mathrm{i}{\psi }_t=\sqrt{-\mathrm{\Delta }}\psi -|\psi {|}^{2/N}\psi -|\psi {|}^p\psi ,0<p<\frac{2}{N}(N\ge 2),\\ \psi (0,x)={\psi }_0\left(x\right),\end{array}$ we obtain the existence of traveling wave solution of the form $\psi (t,x)={\mathrm{e}}^{\mathrm{i}t\mu }{u}_v(x-vt)$ with $0<|v|<1$. If $\parallel {\psi }_0{\parallel }_2<\parallel Q_v{\parallel }_2$, where $Q_v$ is a solution of the equation $\sqrt{-\mathrm{\Delta }}{u}_v+\mathrm{i}(v\cdot \mathrm{\nabla })u_v-|u_v{|}^{2/N}{u}_v=-u_v,$ we prove that the traveling wave is stable.

中文翻译：

---

L2临界组合非线性半波方程稳定行波的存在性

对于半波方程的柯西问题${\mathrm{大号}}^2$-临界组合非线性$\begin{array}{l}\mathrm{一世}{\psi }_{吨}=\sqrt{-\mathrm{\Delta }}\psi -|\psi {|}^{2/ñ}\psi -|\psi {|}^p\psi ,0<p<\frac{2}{ñ}(ñ\ge 2),\\ \psi (0,X)={\psi }_0\left(X\right),\end{array}$我们得到以下形式的行波解的存在性$\psi (吨,X)={\mathrm{e}}^{\mathrm{一世}吨\mu }{你}_v(X-v吨)$和$0<|v|<1$. 如果$\parallel {\psi }_0{\parallel }_2<\parallel {问}_v{\parallel }_2$， 在哪里${问}_v$是方程的解$\sqrt{-\mathrm{\Delta }}{你}_v+\mathrm{一世}(v\cdot \mathrm{\nabla }){你}_v-|{你}_v{|}^{2/ñ}{你}_v=-{你}_v,$我们证明行波是稳定的。