arXiv subjects

Horng-Tzer Yau

Publications and source records attributed to Horng-Tzer Yau.

93 records · Page 6Linked to original sources

arXiv2002

Stable Directions for Excited States of Nonlinear Schrödinger Equations

We consider nonlinear Schrödinger equations in $\R^3$. Assume that the linear Hamiltonians have two bound states. For certain finite codimension subset in the space of initial data, we construct solutions converging to the excited states in both non-resonant and resonant cases. In the resonant case, the linearized operators around the excited states are non-self adjoint perturbations to some linear Hamiltonians with embedded eigenvalues. Although self-adjoint perturbation turns embedded eigenvalues into resonances, this class of non-self adjoint perturbations turn an embedded eigenvalue into two eigenvalues with the distance to the continuous spectrum given to the leading order by the Fermi golden rule.

math-ph↗

arXiv2001

Relaxation of Excited States in Nonlinear Schrödinger Equations

We consider a nonlinear Schrödinger equation in $\R^3$ with a bounded local potential. The linear Hamiltonian is assumed to have two bound states with the eigenvalues satisfying some resonance condition. Suppose that the initial data is small and is near some nonlinear {\it excited} state. We give a sufficient condition on the initial data so that the solution to the nonlinear Schrödinger equation approaches to certain nonlinear {\it ground} state as the time tends to infinity.

math-ph↗

arXiv2001

Asymptotic Dynamics of Nonlinear Schrodinger Equations: Resonance Dominated and Radiation Dominated Solutions

We consider a linear Schrödinger equation with a small nonlinear perturbation in $R^3$. Assume that the linear Hamiltonian has exactly two bound states and its eigenvalues satisfy some resonance condition. We prove that if the initial data is near a nonlinear ground state, then the solution approaches to certain nonlinear ground state as the time tends to infinity. Furthermore, the difference between the wave function solving the nonlinear Schrödinger equation and its asymptotic profile can have two different types of decay: 1. The resonance dominated solutions decay as $t^{-1/2}$. 2. The radiation dominated solutions decay at least like $t^{-3/2}$.

math-ph↗