Peaked and rogue waves in nonlinear partial differential equations

我的研究计划是建立在非线性波传播的分析上，这在许多物理应用中很常见。这个应用数学和数学物理的经典领域与非线性偏微分方程和点阵微分方程理论的许多最新发展有关，并激发了谐波和泛函分析、动力系统理论、渐近方法和科学计算等新工具的发现。我的小组在分析非线性波的主要形式的存在和稳定性方面进行了前沿研究，包括孤立波、周期波、呼吸波、漩涡、畴壁和螺旋结构。这些非线性微分方程的解模拟了各种各样的物理现象，如海洋中的水波，流体流动中的相干结构，原子凝聚体中的捕获态以及分支光子晶体上的波传输。这个建议是基于最近在非常不同的问题上取得的突破，例如：1)证明存在旋转的峰值周期波的线性不稳定性；2)周期型背景下异常波的代数构造；3)大质量极限下量子图上驻波的完全分类；4)简并捕获态的非线性稳定性分析。我研究的长期目标是开发新的分析工具，用于解决非线性偏微分方程中涉及峰值波和异常波的数学问题。非线性方程在自由空间、约束势和表示薄波导的度量图上定义。这些问题是学术性的，但在对真实物理现象的建模中出现，并且可以通过物理实验在自然界中观察到。未来5年的短期目标将集中在以下主题上：1)峰值波的非线性不稳定性与峰值扰动的分析；2)非孤立异常波的构建与孤子气体湍流分析；3)无界量子图上驻波的振荡理论与渐近稳定性；4)多维谐波势中束缚态的临界维数研究。所提出的研究的预期影响和意义是在非线性数学和物理领域。流体中峰值波的不稳定性表征、海洋表面异常波的形成、分支波导中孤立波的传输以及多维势中的俘获原子态等问题的进展将有助于我们对世界的新认识，并将提供新的解决方法，开辟偏微分方程分析、动力系统和非线性波传播的新领域。我的研究和新的物理实验有望在水波和光脉冲在波导和激光领域的实际应用。

My research program is built on analysis of nonlinear wave propagation, which occurs commonly in many physical applications. This classical area of applied mathematics and mathematical physics is related to many recent developments in the theory of nonlinear partial and lattice differential equations and has inspired discoveries of new tools of harmonic and functional analysis, dynamical system theory, asymptotic methods and scientific computations. My group has developed cutting edge research on the analysis of the existence and stability of the principal forms of nonlinear waves including solitary waves, periodic waves, breathers, vortices, domain walls, and helical structures. These solutions of nonlinear differential equations model physical phenomena as diverse as water waves in oceans, coherent structures in fluid flows, trapped states in atomic condensates, and wave transmission over branched photonic crystals. The proposal is based on the recent breakthroughs in the very different problems such as 1) proof of linear instability of peaked periodic waves in the presence of rotation; 2) algebraic construction of rogue waves occurring on the background of periodic patterns; 3) complete classification of standing waves on quantum graphs in the large mass limit; 4) analysis of nonlinear stability of trapped states with degeneracy. The long term objective of my research is to develop new tools of analysis for solving mathematical problems involving peaked and rogue waves in nonlinear partial differential equations. The nonlinear equations are defined on free space, in confining potentials, and on metric graphs representing thin waveguides. These problems are of an academic nature but nevertheless have arisen in modeling of real physical phenomena and can be observed in nature with physical experiments. The short-term objectives in the next 5 years will be focused on the following main themes: 1) analysis of nonlinear instability of peaked waves with respect to peaked perturbations; 2) construction of non-isolated rogue waves  and analysis of soliton gas turbulence; 3) oscillation theory and asymptotic stability of standing waves on unbounded quantum graphs; 4) study of critical dimensions for bound states in multi-dimensional harmonic potentials. The anticipated impact and significance of the proposed research is in the areas of nonlinear mathematics and physics. Progress on the diverse problems such as characterizing instability of peaked waves in fluids, formation of rogue waves on the surface of ocean, transmission of solitary waves in branched waveguides, and trapped atomic states in multi-dimensional potentials will contribute to new knowledge about our world and will offer new methods of solutions to open new frontiers in analysis of PDEs, dynamical systems, and nonlinear wave propagation. Practical applications of my research and new physical experiments are expected in the area of water waves and optical pulses in waveguides and lasers.