Existence of the stable traveling wave for half-wave equation with L²-critical combined nonlinearities

Abstract

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.

Keywords:

• Half-wave equation
• $L^2$-mass critical nonlinearity
• traveling wave
• concentration compactness
• orbital stability

2010 Mathematics Subject Classifications:

• 35Q41
• 35Q55
• 35J20

Acknowledgments

The authors are very grateful to the referee for his (her) valuable suggestions and helps for improving the representation of the paper. This work was supported by the NSFC under grant numbers 11971212, 11475073 and 11801519.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by the National Natural Science Foundation of China (NSFC) [grant numbers 11971212, 11475073, 11801519].