Modeling: Making Mathematics Useful University of Central Florida Institute for Simulation & Training Department of Mathematics and D.J. Kaup Research supported in part by NSF and the Simulation Technology Center, Orlando, FL and CREOL

OUTLINE Modeling Considerations Purposes and Mathematics How to Model Nonsimple Systems Variational Approach DNLS Stationary Solitons Moving Solitons Summary

MODELING Approaches: Experimental direct measurements Numerical Computations number-crunch fundamental and basic laws Curve Fitting looking for mathematical approximations Mathematical Modeling analytically massages fundamental equations, reduction of complexity to simplicities. Simulations crude, but accurate approximations, avoid actual experiment (if dangerous).

PURPOSES To be able to predict an experimental result, To obtain an understanding of something unknown, To represent in a realistic fashion, To test new ideas, postulates and hypotheses, To reduce the complexities, To find simpler representations. There are different levels of approaches for each one of these purposes. One needs to choose a level of approach consistent with the purpose.

MODELING CONSIDERATIONS In order for us to optimumly model, we need: Real physical systems do not need to solve our versions of the physical laws, in order to do just what they do. They just do it. They themselves ARE the embodiment of the physical laws. In order to predict what they do do, WE have to add other actions on TOP of what they do. Any of our laws will always find higher level forms. CLASSIC EXAMPLE: Solitons in optical fibers - theory is accurate across 12 orders of magnitude. One can never fully model any system: the speed of computers, AND the simplifications of analytics

MATHEMATICAL MODELING Purpose is to: predict, simplify, and/or obtain an understanding. METHODS: Analytical solution of simplified models Perturbation expansions about small parameters Series expansions (Fourier, etc.) Variational approximations Large-scale numerical computations of full equations Hybrid methods

Questions: (that an experimentalist might ask) Given a physical system, how can one determine if it will contain solitons? What physical systems are most likely, or more likely, to contain solitons, of whatever breed (pure, embedded, breathers, virtual)? What properties might these solitons have that would be of interest, or of use, to me? Where in the parameter space should I look?

Comments on the questions: One can find solitons with experimentation, numerics, and theory. Each has been successful. The properties of solitons in simple physical systems (NLS, Manakov, KdV, sine-Gordon, SIT, SHG, 3WRI), and their requirements, are well known and DONE. As a system becomes more complex, the possibilities grow exponentially - (consider the GL system). On the other hand, the more complex a system is, the more constraints are required to make it useful.

Solitons (Solitary Waves) Amplitude * Amplitude frequency (Breathers) Phase Phase oscillation frequency Position Velocity Width * Chirp There are many kinds of solitons, and many shapes. But each of them is characterized by only a few parameters. The major parameters are: If you know these parameters, then you know the major features of any soliton, and regardless of the exact shape, you still can make intelligent predictions about its interactions.

Soliton Action-Angle Variables Consider an NLS-like system: Clearly, the momenta density of is A 2. Express in terms of an amplitude and a phase: Now, we want to expand in some way, so as to contain those major parameters, mentioned earlier.

Soliton Variational Action-Angle Variables Then the Lagrangian density becomes: Expand the phase as: We integrate this over x, and see that the resulting momenta are simply the first three moments of the number density, and: These six parameters gives us a model accurate through the first three taylor terms of the phase, and the first three moments of the number density.

Discrete Systems Discrete Channels Evanescent fields overlap coupling Channel field Compliments of George Stegeman - CREOL

Sample design Bandgap core semiconductor: gap = 736nm 4.8mm 2.5 coupling length Compliments of George Stegeman - CREOL

Discrete Nonlinear Schroedinger Equation Consider a set of parallel channels: nearest neighbor interactions (diffraction) interacting linearly Kerr nonlinearity Propagates in z-direction Reference: Discretizing Light in Linear and Nonlinear Waveguide Lattices, Demetrios N. Christodoulides, Falk Lederer and Yaron Silberberg Nature 24, (2003), and references therein.

Sample Stationary Solutions

Variational Approximation Action –angle variables A, alpha – amplitude and phase k, n-sub-0 – velocity and position beta, eta – chirp and width Will take limit of beta vanishing.

Lagrangian & Averaged Lagrangian where

Variational Equations of Motion

Stationary Variational Singlets and Doublets Bifurcation

Variational Solution Results

Exact vs. Variational

Death of a Bifurcation

Moving Solitons Can expand the equations for small amplitudes (wide solitons – eta small – NLS limit). There is a threshold of k before the soliton will move. Below this value, the soliton rocks back and forth. Above this value, it moves as though it was on a washboard. If E is not the correct value, the chirp grows (creation of radiation - reshaping). As eta approaches unity: collapses can occur, reversals can occur, solutions become very sensitive. Above features have been seen in other simulations and experiments. Contrast this with the Ablowitz-Ladik model: In that model, the nonlinearity is nonlocal, no thresholds, but fully integrable.

Low Amplitude Case eta0 = 0.10, k0=0.158, E=0.710eta0 = 0.10, k0=0.285, E=0.730

Medium Amplitude Case eta0 = 0.30, k0=0.045, E=1.708eta0 = 0.30, k0=0.17, E=1.746

Large Amplitude Case eta0 = 1.00, k0=0.059, E=4.67

Large Amplitude Case eta0 = 1.00, k0=0.060, E=4.7

Large Amplitude Case eta0 = 1.00, k0=0.060, E=4.50

SUMMARY Moving Solitons: Threshold required for motion Low and medium amplitudes stable (Analytical expansions exist) High amplitudes very unstable - chaotic (stability basin small, if there at all) Very different from AL case Consequences for numerical methods. Stationary Solitons: Easily found and exists for all eta Variational solutions quite accurate Variational method uses bifurcation Modeling: Overview of approaches and purposes Consideration of limitations All simple systems done Variational Method: General approach Trial function (Lowest level action-angle) Discrete NLS

SUMMARY Pure analytics are insufficient Pure numerics are insufficient Computer algebra necessary to extend analytics Numerics needed in order to expose whatever is contained in the analytics Hybrid methods useful for understanding