Title: Infinite-order rogue waves that are small (but not small in $L^2$) Show Chinese title

Title: 无限阶奇异波，其在$L^2$中是小的（但不是很小）

Abstract: General rogue waves of infinite order constitute a family of solutions of the focusing nonlinear Schr\"odinger equation that have recently been identified in a variety of asymptotic limits such as high-order iteration of B\"acklund transformations and semiclassical focusing of pulses with specific amplitude profiles. These solutions have compelling properties such as finite $L^2$-norm contrasted with anomalously slow temporal decay in the absence of coherent structures. In this paper we investigate the asymptotic behavior of general rogue waves of infinite order in a parametric limit in which the solution becomes small uniformly on compact sets while the $L^2$-norm remains fixed. We show that the solution is primarily concentrated on one side of a specific curve in logarithmically rescaled space-time coordinates, and we obtain the leading-order asymptotic behavior of the solution in this region in terms of elliptic functions as well as near the boundary curve in terms of modulated solitons. The asymptotic formula captures the fixed $L^2$-norm even as the solution becomes uniformly small. Show Chinese abstract

Abstract: 一般无限阶怪波构成聚焦非线性薛定谔方程的一族解，最近在各种渐近极限中被识别出来，例如Bäcklund变换的高阶迭代和具有特定振幅轮廓的脉冲的半经典聚焦。 这些解具有引人注目的性质，如有限的$L^2$-范数，而在没有相干结构的情况下表现出异常缓慢的时间衰减。 在本文中，我们研究了一般无限阶怪波在参数极限下的渐近行为，在该极限下，解在紧集上均匀变小，而$L^2$-范数保持固定。 我们证明了解主要集中在对数缩放时空坐标中的特定曲线的一侧，并在此区域内以椭圆函数的形式以及在边界曲线附近以调制孤子的形式获得了解的主导渐近行为。 渐近公式即使在解均匀变小时也能捕捉到固定的$L^2$-范数。