Extreme Focusing and Spectral Collisions on the Salerno Lattice

The nonlinear Schrödinger (NLS) equation and its integrable discretization, the Ablowitz–Ladik (AL) system, provide closely related settings for studying coherent focusing. I will explain how AL arises from discretizing the NLS Lax pair, then use WKB and Bohr–Sommerfeld quantization to construct exact multisoliton states with extreme focusing and compare their dynamics with continuum PDE models. I will then use the Salerno Hamiltonian to deform from the integrable AL system to the physical but nonintegrable discrete NLS model, tracking the frozen AL Lax spectrum with an Evans function and an exact soliton transport law. For two-soliton data, real Lax eigenvalues can collide and become temporarily complex, and trace identities together with Salerno conservation laws give a field-level description of that collision. The broader goal of this talk is to understand how far integrable structure remains useful after integrability is broken, particularly for extreme focusing in more physically relevant nonlinear wave models.