Peaked and rogue waves in nonlinear partial differential equations

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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.