NLS型方程中超呼吸子的态转换、可积湍流及调制不稳定性非线性阶段特征的研究

Nonlinear waves and modulation instability (MI) are important research branch in nonlinear physics. The theoretical framework combining a statistical approach of random waves together with the property of integrability of the nonlinear Schrödinger equation (NLS) is known as integrable turbulence. In the framework of higher-order, nonlocal, nonautonomous and multi-component NLS equations, the main contents and innovation points of this project are as the following: (1) we study the caculation and dynamics of the superregular breathers (SRBs), state transition and interactions among different types of nonlinear waves on consatant backgrounds; (2) we investigate the characteristics of the nonlinear stage and asymptotic stage of MI; (3) we study the dynamics and control of the (SR) breather turbulence, soliton turbulence and mixed turbulence; (4) Using the IST and numerical simulation and considering the higher-order, nonautonomous and coupling effects, we investigate the effects of the non-ideal intitial pertubations on the eigenvalues of the IST, transition between different turbulence states, probability density function and physical spectra of the chaotic wave fields; (5) we study the internal relationship among different types of nonlinear modes in the nonlinear stage of MI. We hope our work will clarify the new physical mechanism of the above problems which are affected by various physical effects, and provide some important theoretical reference for the related experiments in hydrodynamics and optics.

非线性波和调制不稳定性的研究一直是非线性物理领域重要分支。可积湍流研究涉及非线性随机波的统计描述并结合非线性Schrödinger模型可积性的理论方法。基于高阶、非局域、非自治和多元NLS型方程的框架，本课题主要研究内容和创新点包括：(1)研究NLS型方程及梯队中超常呼吸子解的计算和动力学性质，态转换机制和非零背景非线性波相互作用；(2)研究调制不稳定性非线性阶段新的非线性模式特征和渐进态性质；(3)研究（超常）呼吸子湍流、孤子湍流和混合湍流的动力学机制与控制；(4)研究高阶、耦合和非自治等效应影响下非理想初始扰动对散射谱的影响，不同可积湍流态之间的转换机制，概率密度函数以及混沌波场的物理谱性质；(5)研究调制不稳定性非线性阶段不同非线性模式之间的内在联系。本课题的研究将期望阐明各类物理效应影响下上述问题中新的物理机制，并为相关的流体力学和光学实验提供理论参考。