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Nonlinear differential equations, both ordinary and partial, provide mathematical descriptions of many dynamical processes in physical, biological and social sciences. My research involves the investigation of detailed properties of solutions to these equations to provide insights into physical processes crucial to climate change, and formation mechanisms of natural patterns. |
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Climate Dynamics A central theme of my postdoctoral research is to model climate components that have been under-represented in current climate models. The projects that I am currently working on include supraglacial lake patterns in Antarctica, simple ODE models of sea ice retreat, and water vapor dynamics in the atmosphere. I also participate in the online meetings of two working groups, one on mathematics of climate tipping points, and the other on water cycle and atmospheric processes. More details about individual topics will be posted later. 
Aerial view of supraglacial lakes on the George VI Ice Shelf in west Antarctica. Photograph: Dr. Dominic A. Hodgson, British Antarctic Survey. |
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Localized States An important discovery in twentieth century mathematics is that certain nonlinear wave equations, such as the KdV and NLS equations describing water waves, are integrable and possess soliton solutions. More recently, there has been considerable interest in localized states (sometimes referred to as dissipative solitons) in pattern forming systems. A motivating experimental discovery is the observation of oscillons in vertically vibrated granular media and colloidal suspensions. These localized oscillations are capable of forming isolated or linked dipoles, triangles or even square lattices. Another example with complicated spatiotemporal dynamics is turbulent spots invading the surrounding laminar region in plane Couette flow. In common with nonlinear wave systems, pattern forming systems are often described by nonlinear partial differential equations (PDEs) frequently in the form of evolution equations. However in contrast to nonlinear wave equations, pattern forming PDEs are typically not integrable. As a result, their solutions can exhibit complicated dynamical properties that vastly generalize such phenomena as bifurcation and chaos that prevail in nonlinear ordinary differential equations (ODEs). My dissertation research focuses on exploring localized states in the 1:1 forced complex Ginzburg-Landau equation (FCGLE)  The FCGLE is the normal form for forced Hopf bifurcations in spatially extended systems. Among other applications, the 1:1 FCGLE may be regarded as a natural generalization of the Lugiato-Lefever equation modeling cavity solitons in nonlinear optics. The localized states in 1D may be fruitfully studied with tools borrowed from dynamical systems theory. The basic strategy is to study the steady state ODE as a dynamical system in the space variable x using numerical continuation techniques and perform time evolution of these steady solutions using spectral methods for PDEs. A class of localized states of particular interest consists of a patterned state P of arbitrary length embedded in a featureless background A. These localized states exist because of the bistability between P and A. In pattern forming PDEs this patterned state P is often spontaneously formed from a featureless state B through a generic mechanism known as the Turing instability. The corresponding transition that connects B to P in the steady state ODEs is referred to as the Turing bifurcation. Two central questions emerge:
These localized states have been previously studied in the quadratic-cubic Swift-Hohenberg equation (SH23). In this equation the featureless states A and B coincide, and the bistability between P and A results from a subcritical Turing bifurcation that creates P from A. In this case localized states grow at the locations of the fronts between P and A in a process known as standard homoclinic snaking, and outside the snaking region they evolve by creation or destruction of P also at the locations of the fronts. In the 1:1 FCGLE, however, the featureless states A and B are separated by two saddle-node bifurcations on an S-shaped bifurcation curve, and the bistability between P and A results from the existing bistability between A and B interacting with a supercritical Turing bifurcation that creates P from B. In this case localized states grow at the center of the wavetrain of P in a process known as defect-mediated snaking (shown in the figure below), and outside the snaking region they evolve by creation or destruction of P in the interior of the wavetrain mediated by successive phase slips. In the following papers these novel phenomena are described in detail, and the associated bifurcation structures and temporal dynamics are understood using asymptotic and semi-analytic theories.
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Pattern Equations Among the most important examples of natural patterns are convection patterns in fluids. In the classical Rayleigh-Bénard convection, a horizontal layer of liquid is heated from below and cooled from above. When the temperature gradient is small, the fluid simply conducts heat without flowing. But as the temperature gradient increases, there exists a critical point where the fluid becomes unstable to arbitrarily small perturbations. In experiments, it is right beyond this point that the fluid starts to flow and eventually settles into a stable convection pattern. Near criticality, one can seek a weakly nonlinear theory for the convection system. When the aspect ratio is large, this problem belongs to the wide class of bifurcation problems with continuous spectra. In the following paper pattern equations describing weakly nonlinear Rayleigh-Bénard convection are derived using center manifold reduction in Fourier space. This derivation is inspired by the Bogoliubov approach to kinetic theory, and terms in the perturbation series are intuitively represented by diagrams (shown in the figure below).
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