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 Systems
Evanescent fields overlap ? coupling
Compliments of George Stegeman - CREOL
13
Sample design
4.8mm _at_ 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, k00.158, E0.710
eta0 0.10, k00.285, E0.730
25
Medium Amplitude Case
eta0 0.30, k00.045, E1.708
eta0 0.30, k00.17, E1.746
26
Large Amplitude Case
eta0 1.00, k00.059, E4.67
27
Large Amplitude Case
eta0 1.00, k00.060, E4.7
28
Large Amplitude Case
eta0 1.00, k00.060, E4.50
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