Rigorous validation of the conjectured eigenvalue-counting instability for onsite dark solitons in the defocusing DNLS

Establish rigorously the quartet eigenvalue–induced oscillatory instability of onsite dark solitons in the defocusing discrete nonlinear Schrödinger equation in the strong coupling regime (C → ∞) by proving the validity of the conjectured eigenvalue-counting argument despite the continuous spectrum of the linearized operator covering the entire imaginary axis.

Background

The paper applies exponential asymptotics to the discrete nonlinear Schrödinger equation (DNLS) to construct onsite and intersite solitons and to analyze their stability. For dark solitons (defocusing nonlinearity), the method successfully captures the exponentially small real eigenvalues responsible for the instability of intersite solitons, and it provides existence and structure results for both soliton types.

However, for onsite dark solitons, the linearized operator’s continuous spectrum spans the entire imaginary axis, preventing the exponential asymptotics method from directly establishing the oscillatory (quartet) instability. To address this, the authors propose a conjectured eigenvalue-counting argument that suggests instability, but a rigorous proof is not provided, leaving the validation of this argument as an open problem.