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 difficult (indeed, much more so) as evaluating a general Fourier integral.
The full asymptotic expansion of general Fourier integrals in physically interesting asymptotic limits was made possible by the method of stationary phase/steepest descent, attributed to Lord Kelvin. One encounters such asymptotic limits in calculations of long-time system behaviors, as well as semi-classical (large frequency or small Planck constant) calculations. 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 or "model" RHP, which determines the main contribution to the solution has as contour a union of intervals or arcs in the complex plane, (b) we introduced the "g-function mechanism", 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 function and established that the waveform is mostly a modulated quasi-periodic nonlinear wave. This work was done in the context of the celebrated Korteweg de Vries equation (KdV). In joint work, with Deift, Kriecherbauer, McLaughlin (Ken) and Zhou, we implemented this approach to prove an important universality result in the theory of random matrices of the unitary ensemble.
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 solution till the second break in the presence of a soliton content. In both cases, it is analytic 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. Again with Tovbis, we revealed the deeper structure of the modulation equations by bringing them into a form that involves determinants. We also analyzed the limit of the inverse scattering transform in the asymptotic limit.
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 evolution of multiphase modes. The analogue of caustics appears in space-time along the boundaries at which the number of participating modes jumps. We have already shown that, with our intial data, there is only one break in the pure radiation case. The local asymptotic analysis of the first break was performed by Bertola and Tovbis.
In collaboration with former student Belov, we are working to understand the second breaking of the solution of the NLS equation in the presence of solitons, as well as possible subsequent successive breaks that are suggested by numerics. The challenge is that, for a fixed spatial position, we reach a point in time, at which there is an obstacle to our systematic advance of the solution in time. We have made a rigorous asymptotic calculation of the curve in space-time, along which this difficulty presents itself. We suspect that overcoming this obstacle involves a transformational advance in the asymptotic method itself and we are working in this direction.
Wave Propagation in complex media
In earlier work with Bonilla and Higuera, we 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 analyzed the generation and propagation of traveling fronts in semiconductor superlattices.
More recently, in collaboration with Haider and Shipman, we studied the scattering of plane waves off a photonic crystal slab, composed of two dielectrics that are distributed periodically along the slab with different refractive indices. We discovered anomalous transmission behavior, as the angle of incidence is varied from normal. With Shipman, we showed that the anomaly is mediated by resonance in the system, in which the incident wave excites a mode along the slab, and we derived an asymptotic formula for the anomalous transmission near the resonant frequency. The formula has very good agreement with the results of simulation. Significantly, the derived profile depends only on a small number of parameters. These few parameters encapsulate all the possible geometric configurations of the photonic crystal.
Most of the materials used in practice, are either linear or weakly nonlinear. However, fully nonlinear phenomena occur near the above resonance, due to the large magnitude of the resonant fields. With Shipman, we constructed and solved a fully nonlinear model displaying such phenomena. The model involves a linear transmission line in which an incident plane wave scatters off a point defect, that is coupled to a nonlinear oscillator. As the coupling increases from zero, a frequency band emerges near the resonant frequency of the defect, in which three (as opposed to one) harmonic solutions are possible. Three solutions also appear at all frequencies beyond a high frequency threshold. As the coupling constant is further increased (but is still quite small) the band grows, the threshold frequency diminishes, until the two 3-solution regions touch and merge to one. It was pointed out to us by physicist and applied mathematician, S. Komineas, that our line/oscillator model is of interest in the Bose- Einstein condensation community, for being a simplification of a model for the onset of vortex-antivortex pairs in polariton superfluids in the optical parametric oscillator (OPO) regime. Our current effort, with Shipman and Komineas, is to develop the mathematical tools for solving this broader model.
Several years ago, I joined the "laser group" of colleagues Dan Kiehart (group leader, Biology) and Glenn Edwards (Physics). The group studies the drosophila dorsal closure (see below) and derives its name from experiments involving laser ablations of the drosophila embryo. The group includes postdocs and graduate students and works through weekly meetings. My interest 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 and is only covered by an extra-embryonic, epithelial tissue, the amnioserosa; during closure the opposite flanks are "zipped" together at the canthi ("eye" edges). The challenge is to understand the nature of the forces, how they affect the kinetics and their biological and physical origin.
We developed a mathematical model that connects the empirical kinematic observations with contributing tissue forces, that affect the morphology of the dorsal surface and, in particular, the movements of the purse string and of the canthi. We model the coordinated elastic and contractile motor forces, attributed to the action of actin and myosin that drive DC, by introducing a unit that satisfies a law similar to the law derived by Hill in the early modeling of muscle dynamics. We model the zipping process through a phenomenological law that summarizes the complicated processes of 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.) The current goal is a transformational extension of the model that (a) will allow deformations that are not restricted to ones that are traverse to the dorsal midline, (b) will introduce the individuality of the amnioserosa cells and (c) will account for small time-scale oscillations observed in the amnioserosa. The scantness of the understanding of the underlying biological mechanism makes this effort quite challenging.