verifying a solution of $i\partial_tu=(-\Delta)^{\beta/2}u$.

Question

My goal is to check that $$w(x,t)=\lambda^\frac{n}{2\beta}u(\lambda^{1/\beta}x-2\lambda^{1/\beta}x_0t, \lambda t)e^{i(x\cdot x_0-|x_0|^\beta t)},\quad n\in\mathbb{Z}^+$$ is solution of $$\tag{1} i\partial_t u=(-\Delta)^{\beta/2}u. $$ Here $\beta>0$ and $u=u(x,t)$. On the one hand we know that it is defined $$\mathcal{F}\left((-\Delta)^{\beta/2}u(x)\right)=|\xi|^\beta \widehat{u(\xi)}.$$ Hence, taking Fourier transform on both sides of $(1)$, it results $$i\partial_t \widehat{u(\xi)}=|\xi|^\beta\widehat{u(\xi)}.$$ From now on I don't know how to check what I need.