For fourth-order NLS solitons, the Morse indices of the linearized operators equal the numbers of conjugate points, and this yields a Vakhitov-Kolokolov-type spectral stability criterion.
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Detecting eigenvalues in a fourth-order nonlinear Schr\"odinger equation with a non-regular Maslov box
For fourth-order NLS solitons, the Morse indices of the linearized operators equal the numbers of conjugate points, and this yields a Vakhitov-Kolokolov-type spectral stability criterion.