University of Central Florida. Quantifying Variational Solutions

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1 University of Central Florida Institute for Simulation & Training and epartment of Mathematics and CREOL.J. Kaup and Thomas K. Vogel Quantifying Variational Solutions (Preprint available at Research supported in part by NSF and AFOSR

2 OUTLINE History of Variation Methods Uses and Variational Approach erivation 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 17 th 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 19 th Century) Quantum Mechanics - Computational methods (early 20 th Century) Morse & Feshbach technology of variational methods (1953) Solid State Physics, Chemistry, Engineering (mid-late 20 th Century) Personal computers new computational power (1980 s) Technology of variational methods essentially lost ( ). 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 efinite need for quantitative measure

6 Variational Corrections efinition of Action is: S = L[u]dt efinition of variational derivative is: lim ǫ 0 S[u(t) + ǫδu(t)] S[u(t)] ǫ = δl [u(t)] δu(t)dt = 0 δu Euler Lagrange Equations are: δl[u] δu = 0 Now consider Variational Perturbations about an ansatz: u e = u 0 (t; q) + ǫu 1 (t) + ǫ 2 u 2 (t) +... Ansatz Variational Parameters Corrections ε =?

7 Zeroth order is the VA: Expansion Calculate Action and Expand: S = = S q = q +ǫ L[u e ]dt = L[u 0 ]dt + ǫ ( L[u 0 + ǫu 1 + ǫ 2 u ]dt ( ) δl δu 0 δ 2 L δu(t)δu(t ) L[u 0 ]dt +... = 0 (u 1 + ǫu 2 )dt ) 0 u 1 (t)u 1 (t )dtdt + O(ǫ 3 ) Next order is (vary u 1 ): ( ) δl δu 0 = ǫr(t; 0 q)[q] ε is determined by E-L Eq. R is thereby defined

8 Equation for Correction ( δ 2 L ) δu(t)δu(t ) 0 u 1 (t )dt = R(t)[q] Perturbed Euler-Lagrange Equation with Source SUMMARY: rop Ansatz into Action Calculate new E-L equations to determine q s rop Ansatz plus correction into the full E-L equations Solve for u 1 etermine quantitative accuracy

9 Vibrating String Eigenmodes L(u, u x ) = 1 2 ( ) 2 du 1 d 2 u dx 2 k2 u 2 dx 2 + k2 u = 0 Examples -- two different Ansatzes: u (sawtooth) 0 (x; A, l) = { 2A l x 0 x < l 2, A 2A l (x l 2 ) l 2 x < l u (parabolic) 0 (x; A, l) = A[1 ( 2x l 1)2 ], 0 x < l Only need fundamental mode Will normalize intensity to unity A (sawtooth) = 3 l, A(parabolic) = 15 8l

10 Variational Eigenvalues ``Action for eigenvalue problems is eigenvalue itself. k 2 = λ = l 0 ( du dx) 2 dx l 0 u2 dx Variation of λ and u results in Euler- Lagrange equation. For our models: λ (sawtooth) 0 = 12 l 2 = π2 /l 2, λ (parabolic) 0 = 10 l 2 = π2 /l 2

11 ( Ansatzes and Corrections δ 2 L δu(x)δu(x ) ) 0 u 1 (x )dx + R(x) + δλ u 0 (x) = 0 where: R(x) = d2 u 0 dx 2 + λ 0 u Sawtooth Ansatz 2.0 Parabolic Ansatz 1.5 u u 0 +u 1 u u 0 +u 1 u 1 u u 0 u η η

12 Quantitative Estimates Eigenvalues and corrections: δλ (sawtooth) = π 2 /l 2, δλ (parabolic) = π 2 /l 2 (λ 0 + δλ) (sawtooth) = π 2 /l 2, (λ 0 + δλ) (parabolic) = π 2 /l 2 RMS measure: E 2 rms == l 0 (u 1) 2 dx l 0 (u 0) 2 dx which gives: E (sawtooth) rms = , E (parabolic) rms =

13 KdV Example u t + 6uu x + u xxx = 0 Look for soliton solution and integrate once: cu + 3U 2 + U ξξ = 0, ξ = x c t Take the Lagrangian and Ansatz to be: L = 1 2 cu2 + U (U ξ) 2 Then the action is: S = + L[u 0 (ξ)]dξ = 2π 4 U 0 (ξ) = Aexp( ξ 2 /ρ 2 ) (ca 2 ρ 6 4 ) A 3 ρ + A2 ρ With the variational solution: δs δa = 0 = cρ2 6Aρ δs δρ = 0 = cρ2 4 Aρ A = 6 5 c, ρ2 = 5 c

14 KdV correction The correction equation can be scaled: ξ = ρy, H 0 (y) = c U 0 (ξ), H 1 (y) = c U 1 (ξ) In which case, it reduces to: d 2 H 1 dy 2 + (6 6e y2 5)H 1 = R (y) where R = 6 5 e y2 [ (4y 2 7) + 3 6e y2]

15 Ansatz and Correction 0.6 H 0 and H 0 + H 1 vs. y H 0.3 E rms = H 0 + H H y

16 Soliton separation 1 H 0 and H 0 + H 1 vs. w 1/ H w 1/2 Want soliton separation such that tails = 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 o 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