University of Central Florida
Institute for Simulation & Training
and
Department of Mathematics
and
CREOL
D.J. Kaup and Thomas K. Vogel
Quantifying Variational Solutions†
(Preprint available at http://gauss.math.ucf.edu/~kaup/)
† Research supported in part by NSF and AFOSR
OUTLINE
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History of Variation Methods
Uses and Variational Approach
Derivation of Variational Corrections
Linear Example
Nonlinear Example
Summary
History of Variational Methods
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Early Greeks – Max. area/perimeter
Hero of Alexandria – Equal angles of incidence /reflection
Fermat - least time principle (Early 17th Century)
Newton and Leibniz – Calculus (Mid 17th Century)
Johann Bernoulli - brachistochrone problem (1696)
Euler - calculus of variations (1744)
Joseph-Louis Lagrange – Euler-Lagrange Equations (??)
William Hamilton – Hamilton's Principle (1835)
Raleigh-Ritz method – VA for linear eigenvalue problems (late 19th
Century)
Quantum Mechanics - Computational methods – (early 20th Century)
Morse & Feshbach – technology of variational methods (1953)
Solid State Physics, Chemistry, Engineering – (mid-late 20th Century)
Personal computers – new computational power (1980’s)
Technology of variational methods essentially lost (1980-2000)
D. Anderson – VA for perturbations of solitons (1979)
Malomed, Kaup – VA for solitary wave solutions (1994 – present)
Why Use Variational Methods?
•Linear problems are very well understood.
•Nonlinear problems are very different.
•Nonlinear waves have solitary wave (soliton) solutions.
•They exist in a limited parameter space.
•Where should one look?
•Amplitude=?, width=?, phase=?, etc.
•Equation coefficients for solitons=?
•These Q’s mostly irrelevant for linear systems.
•VA for nonlinear system is same as for linear system.
•Simple ansatzes point to regions where solitons are.
•Basic functional relations found from ansatzes.
•No need to search entire parameter space.
•Each parameter in ansatz reduces parameter space.
•Cascading knowledge.
Variational Approximations
• Is based on a Minimization Principle
• Solution = path that extremizes an “Action”
• Action = time-integral over a Lagrangian
• Lagrangian is specified by the system
• By freezing out specific modes, one can
obtain reduced systems
• The method will still find the path which
is closest to the actual solution
• Definite need for quantitative measure
Variational Corrections
Definition of Action is:
Definition of variational derivative is:
Euler – Lagrange Equations are:
Now consider Variational Perturbations about an ansatz:
Ansatz
Variational Parameters
Corrections
e=?
Expansion
Calculate Action and Expand:
Zeroth order is the VA:
Next order is (vary u1):
0
• e is determined by E-L Eq.
• R is thereby defined
Equation for Correction
Perturbed Euler-Lagrange Equation with Source
SUMMARY:
• Drop Ansatz into Action
• Calculate new E-L equations to determine q’s
• Drop Ansatz plus correction into the full E-L equations
• Solve for u1
• Determine quantitative accuracy
Vibrating String Eigenmodes
Examples -- two different Ansatzes:
•Only need fundamental mode
•Will normalize intensity to unity
Variational Eigenvalues
``Action” for eigenvalue problems is eigenvalue itself.
Variation of l and u results in Euler- Lagrange equation.
For our models:
Ansatzes and Corrections
where:
Quantitative Estimates
Eigenvalues and corrections:
RMS measure:
which gives:
KdV Example
Look for soliton solution and integrate once:
Take the Lagrangian and Ansatz to be:
Then the action is:
With the variational solution:
KdV correction
The correction equation can be scaled:
In which case, it reduces to:
where
Ansatz and Correction
Erms = 0.038
Soliton separation
Want soliton separation such that tails = 0.001
Ansatz = 1.56; plus correction = 2.1
Ratio = 2.1 / 1.56 = 1.35; whence 35% error
Quantitative Variational
•Can calculate variational corrections
•Can quantify variational approximations
•Do not need exact solutions
•Only need to solve linear equation
•Quantitative estimate depends on what use is
•Most VA’s will be poor/excellent depending on
use