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，这通常发生在许多 物理应用。应用数学和数学物理学的这个经典领域与 非线性偏微分方程和格点微分方程理论的许多最新进展， 启发发现新的工具，谐波和功能分析，动力系统理论，渐近 方法和科学计算。 我的团队已经开发出了 委托人存在性与稳定性分析研究 非线性波的形式包括孤立波、周期波、呼吸子、涡旋、畴壁和螺旋波 结构.这些非线性微分方程的解模拟了各种物理现象，如海洋中的水波，流体流动中的相干结构， 原子凝聚体中的陷阱态，以及分支光子晶体中的波传输。 该建议是基于最近在非常不同的问题上取得的突破， 1)存在时峰值周期波线性不稳定性的证明 旋转; 2)在周期性图案背景上发生的异常波的代数构造; 3)大质量量子图上驻波的完全分类 限制; 4)具有简并的囚禁态的非线性稳定性分析 长期 我的研究目标是开发新的分析工具来解决数学问题。 非线性偏微分中的尖峰波和异常波问题 方程非线性方程组定义在自由空间，在封闭势，和度量图表示薄 波导管.这些问题是学术性质的，但在真实的建模中已经出现。 物理现象，可以通过物理实验在自然界中观察到。 今后五年的短期目标将侧重于以下主题： 1)峰波对峰扰动的非线性不稳定性分析; 2)非孤立流氓波的构造和孤立子气体湍流的分析; 3)无界量子图上驻波的振荡理论和渐近稳定性; 4)研究了多维简谐势中束缚态的临界维数。 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。各种问题的进展，如流体中尖峰波的不稳定性，海洋表面的流氓波的形成，孤立波在分支波导中的传输，以及多维势中的原子态，将有助于对我们的世界的新知识，并将提供新的解决方法，开辟新的前沿分析偏微分方程，动力系统，和非线性波传播。我的研究和新的物理实验的实际应用领域的水波和光脉冲在波导和激光器。

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.