Integrable systems mostly consist of families of nonlinear differential equations (ordinary and partial) that can be solved (integrated) in explicit ways through the general principle of the Lax pair, named after its discoverer, Peter Lax. The process of solution has conceptual similarities with the method of the Fourier transform used in the solution of linear differential equations. As in the Fourier transform, there is a spectral variable at hand. While the solution of linear equations is given by a Fourier integral in the spectral variable along a certain contour, the nonlinear case is more complicated: The initial data are used to specify (a) an oriented contour on the plane of the complex spectral variable and (b) a square "jump" matrix at each point of the contour. To find the solution to the differential equation, one has to derive a matrix that (a) is an anlytic function of the spectral variable off the contour, (b) jumps across the contour, the left limit being equal to the right limit multiplied by the jump matrix, and (c) has a certain normalization at the infinity point of the spectral variable. Such a problem is known as a Riemann-Hilbert problem (RHP). Solving such a problem in the general case is as dificult (indeed, much more so) as evaluating a general Fourier integral.
Yet, since the advent of Lord Kelvin's method of stationary phase/steepest descent, the full asymptotic expansion of general Fourier integrals is possible in asymptotic limits. Physically important one include long time limits as well as semiclassical (large frequency or small Planck constant) limits. The foundation of this approach is that the main contribution from the integral arises from the neighborhood of points of the contour of integration where the fast growing exponent under the integral is stationary. Properly restricted to these neighborhoods, the integral reduces asymptotically to a Gaussian integral, hence it is readily computable. The situation is analogous in the nonlinear case. Through a procedure introduced by Deift and Zhou in the case of long time limits, factorization of the jump matrix coupled with contour deformations allows the localization of the contour, the simplification of the jump matrix and the rigorous asymptotic reduction to a solvable RHP. The procedure is known as steepest descent for RHP, arising from the "pushing" of parts of the contour to regions where it is exponentially close to the identity and can be thus neglected.
In dispersive equations involving oscillations, the method was readily applicable when the asymptotic oscillation was weakly nonlinear i.e. consisted of modulated plane wave solutions. In the presence of fully nonlinear oscillations simply finding the stationary points of a scalar function was not appropriate. In collaboration with Deift and Zhou, (a) we found that the reduced RHP lives on a union of intervals of order 1 length in the complex plane, (b) we introduced the "g-function mechanism", namely a procedure that led to a system of transcendental equations and inequalities that the endpoints of the intervals satisfy and from which they are identified uniquely when they exist. (c) having identified these points, we solved the reduced RHP through a Riemann theta functon and established that the waveform is mostly a modulated quasiperiodic nonlinear wave. This work was done in th econtext of the celebrated Korteweg de Vries equation (KdV).
In collaboration with Tovbis and Zhou, we then tackled the problem of the nonlinear focusing Schroedinger (NLS) equation that is known to be modulationally unstable (KdV is stable) and thus presented a further difficulty. We have succeeded in obtaining the global space-time solution to the initial value problem for special data that contain only radiation and the soluiton till the second break in the presence of a soliton content. In both cases, it is analyticity properties of the spectral data (jump matrix) that save us from the instability. Spectral data NLS calculations are delicate when possible; it required special work in collaboration with Tovbis to calculate the data in the above cases.
What one learns from these theories is that as waveforms evolve, they break into more complicated waveforms or relax to simpler ones. Multiple theta functions in the formulae describe the evoluiton of multiphase modes. The analogue of caustics appears in space-time along the boundaries at which the number of participating modes jumps. Still in collaboration with Tovbis and Zhou, we are working to understand the successive NLS breaking of the solution in the presence of solitons. We have already shown that, with our intial data, there is only one break in the pure radiation case. We are also working to find how to study how the modulational instability manifests itself in our theory.
Wave Propagation in complex media
In earlier work with Bonilla and Higuera, we have studied the breakdown of the stability of the steady state in a Gunn semiconductor, that leads to the generation of a time periodic pulse train that is commonly used as a microwave source. With Bonilla Kindelan and Moscoso we have studied the generation and propagation of travelling fronts in semiconductor superlattices.
In collaboration with V. Papanicolaou, Haider and Shipman we have studied optical wave propagation in a medium composed of two dielectrics that are distributed in space periodically (photonic crystal) or randomlly or as a combination (periodic medium with randomly distributed defects). In recent work with Shipman, we have explained the role of anomalous transmission behavior mediated by resonances in the system. We have made advances in the optimization of the quality factor of certain resonances.
Most of the materials used in practice as well as in most theory in this domain are either linear or weakly nonlinear. My current direction is towards introducing strong nonlinearity in the above media in a way that is physically realizable. Success in this would be interesting physically and mathematically; it can be better achieved with collaborations that cut across disciplinary lines.
In recent years I have joined the drosophila dorsal closure group of colleagues at Duke. The group was started by Dan Kiehart (Biology) then Glenn Edwards (Physics) joined, then myself and recently Anita Layton (Mathematics). It includes postdos and graduate students and works through regular meetings. My interest here is the modeling of the closure of the dorsal opening of the drosophila embryo in the process of morphogenesis. The dorsal opening has the shape of a human eye; during closure the opposite flanks are "zipped" together at the canthi. The challenge is to understand the nature of the forces, how they affect the kinetics and their biological and physical origin.
We are developing a quantitative model that connects the empirical kinematic observations with contributing tissue forces. We explicitly model the coordination of the elastic and active contractile forces by introducing a unit that consists of an elastic subunit serially connected with a contractile subunit in the major drivers of DC. The morphology of the dorsal surface, particularly, the shape change of the purse string and the movement of the canthi, is dynamically described through balance of forces. We address the zipping process by attributing zipping to force and deriving a function that summarizes the complications in the canthus. Our model recapitulates the experimental observations of wild type native, laser perturbed and mutant native closure made in earlier work of the group (Hutson et.al.)
A remarkable feature is our employment of a force velocity-law to model active contractility in the actin/myosin complex. In coordination with the elasticity of actin, this leads to a type of early equation introduced by Hill in his muscle model that preceded the detailed knowledge of the interaction of actin and myosin through crossbridges. I believe that this law is intrinsically appropriate for description of motor activity.