Asymptotics for small data solutions of the Ablowitz-Ladik equation

Stewart, Gavin

doi:10.48550/arxiv.2303.10496

Cited by 1 publication

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“…Therefore the AL equations, constructed as integrable lattice generalization of the NLS equations and reducing to them in the limit (2), possess different interesting continuous limits, like equations (46), making even more clear that the case η = −1 is defocusing only if the amplitudes are less than 1; otherwise it is focusing, and this focusing effect is so strong (now the dispersion is nonlinear) to lead to the blow up of the solution at finite time. Another interesting continuous limit is in [70].…”

Section: Another Continuous Limit Of the Al Equationsmentioning

confidence: 99%

Modulation instability, periodic anomalous wave recurrence, and blow up in the Ablowitz–Ladik lattices

Coppini,

Santini

2023

J. Phys. A: Math. Theor.

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The Ablowitz-Ladik equations, hereafter called AL+ and AL− , are distinguished integrable discretizations of respectively the focusing and defocusing nonlinear Schrödinger (NLS) equations. In this paper we first study the modulation instability of the homogeneous background solutions of AL± in the periodic setting, showing in particular that the background solution of AL− is unstable under a monochromatic perturbation of any wave number if the amplitude of the background is greater than 1, unlike its continuous limit, the defocusing NLS. Then we use Darboux transformations to construct the exact periodic solutions of AL± describing such instabilities, in the case of one and two unstable modes, and we show that the solutions of AL− are always singular on curves of spacetime. At last, using matched asymptotic expansion techniques, we describe in terms of elementary functions how a generic periodic perturbation of the background solution i) evolves according to AL+ into a recurrence of the above exact solutions, in the case of one and two unstable modes, and ii) evolves according to AL− into a singularity in finite time if the amplitude of the background is greater than 1. The quantitative agreement between the analytic formulas of this paper and numerical experiments is perfect.

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Section: Another Continuous Limit Of the Al Equationsmentioning

confidence: 99%