Quantifying Variational Solutions†

University of Central FloridaInstitute for Simulation & Training

Department of Mathematics

and

D.J. Kaup and Thomas K. Vogel

† Research supported in part by NSF and AFOSR

and

CREOL

(Preprint available at http://gauss.math.ucf.edu/~kaup/)

OUTLINE

• History of Variation Methods• Uses and Variational Approach• Derivation of Variational Corrections• Linear Example• Nonlinear Example• Summary

History of Variational Methods

• 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

= ?

Expansion

Zeroth order is the VA:

Calculate Action and Expand:

Next order is (vary u1):

0 • 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

Variation of and u results in Euler- Lagrange equation.

``Action” for eigenvalue problems is eigenvalue itself.

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:

Then the action is:

Take the Lagrangian and Ansatz to be:

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