发表机构
University of Thessaly; University of Kansas; Victoria University of Wellington; State University of New York at Buffalo; University of Ioannina（塞萨洛尼基大学; 堪萨斯大学; 惠灵顿维多利亚大学; 纽约州立大学布法罗分校; 约阿尼纳大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该工作证明可积散焦Ablowitz-Ladik方程在无限和有限格上对单位圆外初始数据发生有限时间爆破，并构造了精确爆破时间及全局解，表明爆破是主导动力学。
AI 中文摘要
本工作建立了完全可积散焦Ablowitz-Ladik方程在无限格和有限格上的有限时间爆破。我们首先在一致有界序列的空间$\ell^\infty$中，对任意初始数据建立了局部适定性，无需衰减假设，且在一般边界条件下成立。然后我们证明单位圆作为格动力学的分界线：初始模在单位圆内、上或外的格点，只要解存在，其模就保持在各自区域内。因此，$\ell^\infty$单位球闭包中的初始数据产生全局时间解。另一方面，单位圆分界线开启了有限时间爆破的可能性，这仅限于初始模在单位圆外的格点。我们通过构造一类初始数据来建立这一情形，其对应解是奇异的，并显式给出精确的爆破时间。我们还构造了从单位圆外具有任意大单格点激发的初始数据出发的解，这些解在时间上全局存在，这一行为由相应模收敛到单位圆平衡态所驱动。在存在此类解以及其他重要的解析、全局时间解（这些解始终保持在闭$\ell^\infty$单位球之外）的情况下，我们考察了一般初始数据在闭单位球外的支配动力学这一基本问题。通过利用所构造的奇异解类作为一般问题的子解，我们证明了完全可积散焦Ablowitz-Ladik格上的动力学由有限时间爆破支配。最后，我们对一般初始数据的说明性例子进行了数值研究。
英文摘要
This work establishes finite-time blow-up for the completely integrable defocusing Ablowitz-Ladik equation on both infinite and finite lattices. We first establish local well-posedness for arbitrary initial data in the space $\ell^\infty$ of uniformly bounded sequences, with no decay assumptions and under general boundary conditions. We then show that the unit circle acts as a separatrix for the lattice dynamics: sites with an initial modulus inside, on, or outside the unit circle maintain their modulus within their respective region for as long as the solution exists. As a consequence, initial data in the closure of the $\ell^\infty$ unit ball generate global-in-time solutions. On the other hand, the unit circle separatrix opens up the possibility for finite-time blow-up, which is restricted to sites with initial modulus outside the unit circle. We establish this scenario by constructing a class of initial data whose corresponding solutions are singular, with explicitly given exact blow-up times. We also construct solutions emanating from initial data with an arbitrarily large single-site excitation outside the unit circle that exist globally in time, a behavior driven by the convergence of the corresponding modulus to the unit circle equilibrium. In the presence of such solutions, along with other important classes of analytical, global-in-time solutions that remain outside the closed $\ell^\infty$ unit ball for all times, we examine the fundamental question of the dominant dynamics for general initial data outside the closed unit ball. By utilizing the constructed class of singular solutions as subsolutions for the general problem, we prove that the dynamics for the completely integrable defocusing Ablowitz-Ladik lattice is dominated by finite-time blow-up. We conclude with numerical studies of illustrative examples of general initial data
