Novosibirsk State University, Novosibirsk, 630090, Russia and Institute of Thermophysics, SB RAS, Novosibirsk, 630090, Russia.
Phys Rev E. 2018 Feb;97(2-1):022208. doi: 10.1103/PhysRevE.97.022208.
The one-dimensional focusing nonlinear Schrödinger equation (NLSE) on an unstable condensate background is the fundamental physical model that can be applied to study the development of modulation instability (MI) and formation of rogue waves. The complete integrability of the NLSE via inverse scattering transform enables the decomposition of the initial conditions into elementary nonlinear modes: breathers and continuous spectrum waves. The small localized condensate perturbations (SLCP) that grow as a result of MI have been of fundamental interest in nonlinear physics for many years. Here, we demonstrate that Kuznetsov-Ma and superregular NLSE breathers play the key role in the dynamics of a wide class of SLCP. During the nonlinear stage of MI development, collisions of these breathers lead to the formation of rogue waves. We present new scenarios of rogue wave formation for randomly distributed breathers as well as for artificially prepared initial conditions. For the latter case, we present an analytical description based on the exact expressions found for the space-phase shifts that breathers acquire after collisions with each other. Finally, the presence of Kuznetsov-Ma and superregular breathers in arbitrary-type condensate perturbations is demonstrated by solving the Zakharov-Shabat eigenvalue problem with high numerical accuracy.
一维不稳定凝聚背景下的非线性薛定谔方程（NLSE）是可以应用于研究调制不稳定性（MI）发展和形成不规则波的基本物理模型。NLSE 通过逆散射变换的完全可积性使得初始条件可以分解为基本的非线性模式：呼吸子和连续谱波。MI 导致的小局域凝聚体微扰（SLCP）多年来一直是非线性物理中的基础研究热点。在这里，我们证明 Kuznetsov-Ma 和超正则 NLSE 呼吸子在各种 SLCP 动力学中起着关键作用。在 MI 发展的非线性阶段，这些呼吸子的碰撞导致不规则波的形成。我们提出了随机分布的呼吸子以及人为制备的初始条件下不规则波形成的新情景。对于后者，我们提出了一种基于在彼此碰撞后呼吸子获得的空间相位移动的精确表达式的解析描述。最后，通过以高精度数值求解 Zakharov-Shabat 特征值问题，证明了 Kuznetsov-Ma 和超正则呼吸子存在于任意类型的凝聚体微扰中。
