Existence and stability of one-dimensional nonlinear topological edge states

The uploaded paper (Ma & Susanto, 2020) studies the semi-infinite SSH chain with on-site Kerr nonlinearity and reports two core results: (i) existence of a nonlinear edge soliton for every positive energy E in the topological band gap, derived via a phase-space (reversible 2D map) “sandwiching” argument; and (ii) a stability dichotomy: for fixed t′=1, all edge solitons are spectrally stable for t below a critical tc≈0.35, while for t>tc there exist energy intervals with oscillatory (Hamiltonian–Hopf-type) instability. These claims are stated clearly and demonstrated numerically, with the existence criterion explained via geometry of stable manifolds in the (A,B)-plane and the stability chart shown in their Fig. 4. The paper’s abstract, spatial-dynamics setup, and figures explicitly support these conclusions, including the governing equations (their Eq. (2)), the stationary map (Eq. (3)), the phase-portrait mechanism for edge-soliton existence for E>0, and the numerically observed oscillatory instability arising via collisions of imaginary eigenvalues bifurcating from band edges. See the abstract and conclusion for the headline claims, the spatial-dynamics construction and phase portraits for existence, and the (t,E)-stability diagram and spectra for the instability mechanism. The candidate solution reaches essentially the same conclusions by a different mathematical route. For existence, it uses Crandall–Rabinowitz local bifurcation from the linear zero-energy edge state and then a global continuation/Fredholm alternative argument to conclude that the connected branch covers every E∈(0,t′−t). For stability, it frames the linearization as a Hamiltonian eigenvalue problem JLξ=λξ, identifies the essential spectrum on the imaginary axis, invokes a VK/Hamiltonian–Krein index count to rule out real (exponential) instabilities on the fundamental branch, proves stability in the strongly dimerized regime (small r=t/t′) by perturbation from the decoupled-dimer limit, and explains the onset of oscillatory instability near r↗1 via a Krein-signature collision. This aligns with the paper’s observations: stable for small t (all E), unstable intervals for larger t, and oscillatory instability. The only notable discrepancy is mechanistic wording: the paper emphasizes a collision between two discrete imaginary eigenvalues that bifurcate from adjacent band edges, while the model’s narrative at one point phrases the event as a collision with the essential spectrum; practically both describe the same Hamiltonian–Hopf onset seen in the numerics. The paper does not provide a fully rigorous proof of global existence/uniqueness or a VK-based count (it relies on topology and numerics, with a perturbative stability remark in the Λ→0 limit), whereas the model outlines a rigorous pathway but leaves several technical steps (Fredholm compactness at blow-up, negativity count of L+, and transversality) as assumptions. Overall, the conclusions agree and the proofs are different.