2020/06/01 by J. Sullivan, Sullivan, J., E. G. Charalampidis +9 · 3 citations
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Nonlinear Photonic Systems #Nonlinear Waves and Solitons #Numerical methods for differential equations #Pattern Formation and Solitons (nlin.PS) #nlin.PS #nlin.SI
paper · pdf · doi:10.48550/arxiv.2006.00958
9 pages, 4 figures
arxiv created 2020/06/01 · openalex publication_date 2020/06/01 · arxiv updated 2020/06/02 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
The Salerno model is a discrete variant of the celebrated nonlinear Schrödinger (NLS) equation interpolating between the discrete NLS (DNLS) equation and completely integrable Ablowitz-Ladik (AL) model by appropriately tuning the relevant homotopy parameter. Although the AL model possesses an explicit time-periodic solution known as the Kuznetsov-Ma (KM) breather, the existence of time-periodic solutions away from the integrable limit has not been studied as of yet. It is thus the purpose of this work to shed light on the existence and stability of time-periodic solutions of the Salerno model. In particular, we vary the homotopy parameter of the model by employing a pseudo-arclength continuation algorithm where time-periodic solutions are identified via fixed-point iterations. We show that the solutions transform into time-periodic patterns featuring small, yet non-decaying far-field oscillations. Remarkably, our numerical results support the existence of previously unknown time-periodic solutions \it even at the integrable case whose stability is explored by using Floquet theory. A continuation of these patterns towards the DNLS limit is also discussed.