Spectral Analysis for Matrix Hamiltonian Operators
This work provides a rigorous verification of spectral stability for solitons in a specific nonlinear PDE, which is an incremental advance for researchers in nonlinear waves and spectral theory.
The authors prove the absence of embedded eigenvalues for matrix Hamiltonians arising from linearizing the 3D cubic nonlinear Schrödinger equation about solitons, using a numerically assisted proof. They also provide a new algorithm for verifying spectral properties relevant to soliton stability.
In this work, we study the spectral properties of matrix Hamiltonians generated by linearizing the nonlinear Schrödinger equation about soliton solutions. By a numerically assisted proof, we show that there are no embedded eigenvalues for the three dimensional cubic equation. Though we focus on a proof of the 3d cubic problem, this work presents a new algorithm for verifying certain spectral properties needed to study soliton stability. Source code for verification of our comptuations, and for further experimentation, are available at http://www.math.toronto.edu/simpson/files/spec_prop_code.tgz.