| Abstract: |
| We study higher-order rogue waves and multi-generation cascades on the Salerno lattice. The model interpolates between integrable Ablowitz-Ladik and DNLS, and we generate cascades from Thomas-Fermi quench data. In the integrable periodic case, rogue waves come from spectral degeneracies at band endpoints on modulationally unstable arcs. We ask whether similar degeneracies continue to organize cascade formation once integrability is broken.
The work has two parts. On the lattice, we compute Floquet and Lax spectral portraits of TF data along the Salerno deformation, looking for branch crossings and arc reconnections that precede focusing events. In the continuum limit, we use the Madelung transform and Whitham modulation to track dispersionless focusing, gradient catastrophe, and the onset of oscillations. The goal is a single picture connecting continuum modulation to lattice spectral geometry. |
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