Quantifying Variational Solutions †

1. University of Central Florida Institute for Simulation & Training and Department of Mathematics and CREOL D.J. Kaupand Thomas K. Vogel Quantifying Variational Solutions† (Preprint available at http://gauss.math.ucf.edu/~kaup/) † Research supported in part by NSF and AFOSR

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

3. 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)

4. 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.

5. 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

6. Variational Corrections Definition of Action is: Definition of variational derivative is: Euler – Lagrange Equations are: Now consider Variational Perturbations about an ansatz: Ansatz Corrections e = ? Variational Parameters

7. Expansion Calculate Action and Expand: Zeroth order is the VA: Next order is (vary u1): • eis determined by E-L Eq. • R is thereby defined 0

8. 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

9. Vibrating String Eigenmodes Examples -- two different Ansatzes: • Only need fundamental mode • Will normalize intensity to unity

10. Variational Eigenvalues ``Action” for eigenvalue problems is eigenvalue itself. Variation of l and u results in Euler- Lagrange equation. For our models:

11. Ansatzes and Corrections where:

12. Quantitative Estimates Eigenvalues and corrections: RMS measure: which gives:

13. KdV Example Look for soliton solution and integrate once: Take the Lagrangian and Ansatz to be: Then the action is: With the variational solution:

14. KdV correction The correction equation can be scaled: In which case, it reduces to: where

15. Ansatz and Correction Erms = 0.038

16. 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

17. 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