Modeling: Making Mathematics Useful

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

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

3. 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).

4. 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.

5. MODELING CONSIDERATIONS One can never fully model any system: • 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. In order for us to optimumly model, we need: • the speed of computers, AND • the simplifications of analytics CLASSIC EXAMPLE: Solitons in optical fibers - theory is accurate across 12 orders of magnitude.

6. 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

7. 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?

8. 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”.

9. Solitons (Solitary Waves) There are many kinds of solitons, and many shapes. But each of them is characterized by only a few parameters. The major parameters are: • Amplitude * • Amplitude frequency (Breathers) • Phase • Phase oscillation frequency • Position • Velocity • Width* • Chirp 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.

10. Soliton Action-Angle Variables Consider an NLS-like system: Express in terms of an amplitude and a phase: Clearly, the momenta density of a is A2. Now, we want to expand in some way, so as to contain those major parameters, mentioned earlier.

11. Soliton Variational Action-Angle Variables Expand the phase as: Then the Lagrangian density becomes: 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.

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

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

14. 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, 817-23 (2003), and references therein.

15. Sample Stationary Solutions

16. 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.

17. Lagrangian & Averaged Lagrangian where

18. Variational Equations of Motion

19. Stationary Variational Singlets and Doublets Bifurcation

20. Variational Solution Results

21. Exact vs. Variational

22. Death of a Bifurcation

23. Moving Solitons • Can expand the equations for small amplitudes • (wide solitons – eta small – NLS limit). • There is a threshold of k2 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.

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

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

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

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

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

29. SUMMARY • Variational Method: • General approach • Trial function • (Lowest level action-angle) • Discrete NLS • Modeling: • Overview of approaches and purposes • Consideration of limitations • All simple systems done • Stationary Solitons: • Easily found and exists for all eta • Variational solutions quite accurate • Variational method uses bifurcation • 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.

30. 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