I'm reading about the stability around the solitons(or kinks) in several dispersive PDE's. At some point in the argument, the authors take time to explicit the continuous spectrum and counting the negative eigenvalues of the linearized operator around soliton. However, I don't understand why is so important the continuous spectrum. Then, I wonder if there is some reference where explains the relevance of the continuous spectrum in the stability problem. Also, I want to clarify the concept of resonance and internal modes.
Thank you in advance!