2024/11/25 by Mitchell Curran, Curran, Mitchell, Robert Marangell +1
Physics and Astronomy · #34L05 #35B35 #37K40 #37K45 #47A75 #53D12 #FOS: Mathematics #Nonlinear Photonic Systems #Quantum Mechanics and Non-Hermitian Physics #Quantum optics and atomic interactions #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2411.16903
openalex publication_date 2024/11/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We use the Maslov index to study the eigenvalue problem arising from the linearisation about solitons in the fourth-order cubic nonlinear Schrödinger equation (NLSE). Our analysis is motivated by recent work by Bandara et al., in which the fourth-order cubic NLSE was shown to support infinite families of multipulse solitons. Using a homotopy argument, we prove that the Morse indices of two selfadjoint fourth-order operators appearing in the linearisation may be computed by counting conjugate points, as well as a lower bound for the number of real unstable eigenvalues of the linearisation. We also give a Vakhitov-Kolokolov type stability criterion. The interesting aspects of this problem as an application of the Maslov index are the instances of non-regular crossings, which feature crossing forms with varying ranks of degeneracy. We handle such degeneracies directly via higher order crossing forms, using a definition of the Maslov index developed by Piccione and Tausk.