Stability and instability of standing waves for one dimensional nonlinear Schrödinger equations with double power nonlinearity (Q1902051)

The author considers the stability and instability of standing waves for the following nonlinear Schrödinger equation \[ iu_t+ u_{xx}+ f(u)=0,\quad t\geq 0,\quad x\in\mathbb{R}, \] where \(f(u)= a|u|^{p-1}u+ b|u|^{q-1}\) with \(a,b\in\mathbb{R}\) and \(1<p<q<\infty\). Basically, the author applies the abstract theory of stability and instability by Grillakis, Shatah and Strauss. The result in the paper is interesting.