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
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
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.
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)
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.
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.
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.
Discrete Systems
Discrete Channels
Channel
field
Evanescent fields
overlap  coupling
Compliments of George Stegeman - CREOL
Sample design
4.0µm
8.0µm
1.5µm
1.5µm
Al0.24Ga0.76As
Al0.18Ga0.82As
Al0.24Ga0.76As
4.8mm
@ 2.5 coupling
41 guides
length
Bandgap core semiconductor: lgap =
736nm
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, 817-23 (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
1 and 2 vs. 
1 .0
Bifurcation
1

2
2
0 .0
1
-1 .0
0 .0
0 .5

1 .0
1 .5
2 .0
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 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.
Low Amplitude Case
eta0 = 0.10, k0=0.158, E=0.710
eta0 = 0.10, k0=0.285, E=0.730
Medium Amplitude Case
eta0 = 0.30, k0=0.045, E=1.708
eta0 = 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
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
Variational Method:
•General approach
•Trial function
(Lowest level action-angle)
•Discrete NLS
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.
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