Scattering and blowup problems for a class of nonlinear Schrödinger equations.
We study the scattering and blow-up problem for the equation \[ 2\text{i}\psi _t(x,t)+\Delta \psi (x,t)+f(\psi (x,t))=0,\quad x\in \mathbb {R}^d,\quad t\in \mathbb {R}, \] where \(f\: \mathbb {C}\rightarrow \mathbb {C}\) is a continuously differentiable function more general than pure power nonlinearity. We prove the existence of a standing-wave solution of minimal action (ground state) for any frequency \(\omega >0\). We define two invariant regions \(A_{\omega ,+}\) and \(A_{\omega ,-}\) for each \(\omega > 0\) in \(H^1(\mathbb {R}^d)\) such that any solution starting from \(A_{\omega ,+}\) behaves asymptotically free as \(t\rightarrow \pm \infty \), one from \(A_{\omega ,-}\) blows up or grows up, and the ground state belongs to \(\overline {A_{\omega ,+}}\bigcap \overline {A_{\omega ,-}}\).
scientific article; zbMATH DE number 6148068