SPECTRAL STABILITY OF ELLIPTIC SOLUTIONS TO THE DEFOCUSING LAKSHMANAN-PORSEZIAN-DANIEL EQUATION

In this paper, by exploiting the integrability of the defocusing Lakshmanan-Porsezian-Daniel (LPD) equation, we establish the spectral stability for the elliptic solutions with respect to subharmonic perturbations. We achieve this goal by constructing explicitly the squared-eigenfunction connection between the linear stability problem and its Lax pair. Furthermore, based on the spectral stability results, the linear stability for subharmonic perturbations is obtained by applying directly the Skew-symmetric Composed with Self-adjoint (SCS) basis lemma. Although it is challenging to determine analytically the spectra for all operators except the simplest ones, this research provides a detailed analytical description of the stable Lax spectrum and the stability spectrum.