基于混沌控制的随机NLS方程的机理分析及相关系统的应用研究

This project takes stochastic nonlinear Schrödinger (SNLS) equation(1) as the research object. The project study the model and sub-model of the evolution mechanism,random solutions. Simulating the rogue wave transient evolution of emergence, development and annihilation, as well as the evolution law of the nonlinear characteristics wave..First, based on the symmetry reduction, branches and chaos theory, we seek or modify nonlinear wave models of deep water like the nonlinear Schrödinger system (SNLS). Which solve and verify dynamics characteristics of the model.Such as amplitude, phase, velocity and other parameters and so on. And we explore the rouge wave formation the mechanism and temporal evolution..Second,the combination of qualitative and quantitative, we determine the structure of the model and the corresponding soliton parameters area, a novel soliton solutions description of the new model is given by convergent theorem in the vicinity of the critical value, such as ring solitons, soliton peak, P solitons, etc., and we list the existence proof of stochastic solution and graphical analysis..Finally, we obtain the bifurcation equations with the hamilton structure, turbulence ( spatiotemporal chaos ) and spiral waves with feedback control, sum up the general rule and the formation of abnormal wave theory of early warning, while avoiding disadvantages.

本项目以非线性随机薛定谔方程(SNLS)为研究对象, 研究模型及导出族的演化机理与随机解，模拟畸波发生、发展及湮灭的瞬态过程,以及变量演化的规律..首先，基于对称约化、分支和混沌理论，寻求或修正随机非线性深水波模型（SNLS）,求解并验证模型的动力学，如：振幅、相位、波速等参量，进而探讨巨幅畸波形成的机理和时空演化..其次，定性与定量相结合，确定模型中孤子结构及相应的参数区域，在临界值附近，通过收敛衰减分析，求解描述模型演化机理的新型随机孤子解（如圈孤子、尖峰孤子、P孤子等），并给出相应解存在的证明及图形分析..最后，给出具有Hamilton结构的导出族方程的分支分析、湍流（时空混沌）和螺旋波的反馈优控，归纳出一般畸波形成的规律及其先期预警理论，趋利避害.

项目执行期间，我们主要研究了KD方程、SNLS方程、粘弹方程及流体力学相关方程解的存在性，解的传播相互作用、复杂机理分支现象及衰减行为等演化机理的动力学问题，模拟畸波发生、发展及湮灭的瞬态过程,以及变量演化的规律.有关研究成果，分别发表在《Nonlinear Dynamics》,《 J. Differential Equations》等期刊上.另外，我们还研究了微分方程在图像处理、光纤通信与信息安全、混沌控制与同步中实际应用. . 到目前为止，相关研究结果累计在SCI收录期刊发表论文12 篇，其中发表在一区的期刊4篇，第一标注12篇；在省级期刊发表论文2篇等.. 项目执行期间，我们多次参加了学术研讨会，并访问其他专家.我们亦指导了多名本科生的毕业论文，与同事开展合作研究.