MATH 3078/3978 — Partial Differential Equations and Waves — Assignment 2 Assignment 2 is due on Wednesday, October 29, 4pm. Please drop it off in the locked collection boxes opposite the lift in the Carslaw building on Level 6. Your assignment, with a cover sheet, should be stapled to a manilla folder, on the cover of which you should write the initial of your family name as a LARGE letter. 1. (A warm-up problem - Solving a system of ODEs) Consider the initial value problem for a linear, 2-D system of ODEs 1 −1 0 u (t) = Au(t) , u(0) = =: u0 , A= 0 3 3 1 , (1) with a symmetric matrix A. Solve the problem using the following steps. (a) Use the ansatz u(t) = eλt v, v ∈ R2 , to get the eigenvalue problem λv = Av, and determine the eigenvalues λ1 , λ2 and the eigenvectors v1 , v2 of A. (b) Check that hAu, u˜i = hu, A˜ ui (with h·, ·i the standard inner product in R2 ) and verify directly that v1 , v2 are orthogonal. (Note that these properties are true in general for symmetric matrices.) (c) By using the representation u(t) = c1 (t)v1 + c2 (t)v2 , (2) of the solution u in terms of the orthogonal set {v1 , v2 }, derive that the general solution is u(t) = b1 eλ1 t v1 + b2 eλ2 t v2 , and determine the constants b1 , b2 ∈ R from the initial data. (d) (MATH3978) In order to see that the core mechanism of the solution procedure above relies on diagonalisation which decouples the 2-D problem in (1) into two decoupled scalar ODE problems, construct the matrix Q = (v1 |v2 ), write A = QDQ−1 and determine a change of variables which tranforms (1) into an equivalent system with D as matrix. How is this new system related to the scalar ODEs for c1 , c2 in (2) ? (e) (MATH3978) Using the proof technique of Proposition 6.2.6 in Chapter 6, show that, for symmetric matrices, eigenvectors belonging to different eigenvalues are orthogonal. 2. (The Schr¨ odinger equation) Consider the Schr¨odinger equation }2 i} ∂t ψ = − 4x +V (x) ψ , 2M for the wave function ψ = ψ(x, t) ∈ C, x ∈ Rn , n ∈ {1, 2, 3}, t ≥ 0, which is used to describe the probability of the position of a particle of mass M (with } = 2πh the Planck constant and V (x) the potential energy). Hence, we have the natural constraint that Z |ψ(x, t)|2 dx = 1 , t ≥ 0 . Rn (a) Using a separation of variables ansatz ψ(x, t) = f (t)φ(x) (with complex-valued f and realvalued φ) derive that f (t) = D exp(−i E} t) with some D ∈ C and }2 − 4x +V (x) φ(x) = Eφ(x) , (3) 2M with E a separation constant. (b) Consider for (3) the special case n = 1 and a potential 0 , x ∈ (0, a) V (x) = V0 , otherwise for some a > 0, and V0 → ∞. This scenario is referred to as the particle in a box. One has φ(x) = 0, x ≤ 0, x ≥ a, from this interpretation. Therefore, it remains to solve Z ∞ }2 00 |φ(x)|2 dx = 1 . (4) φ (x) = Eφ(x) , φ(0) = 0 , φ(a) = 0 , − 2M −∞ Compute the values of E for which (4) has a non-trivial solution and determine the corresponding solution. (c) Consider for (3) the special case n = 1 and V (x) = x2 , a scenario called the quantum harmonic oscillator. Bring the corresponding equation − }2 00 φ (x) + x2 φ(x) = Eφ(x) , 2M by the change of variables z = βx, β 4 = 2M ,λ }2 E = − 2M , y(z) = φ( βz ) to the form β2} y 00 (z) − z 2 y(z) = λy(z) . (d) Use another change of variables y(z) = e−z with ρ(z) = e−z 2 2 /2 u(z) to get (p(z) u0 (z))0 + q(z)u(z) = λρ(z)u(z) , (5) R∞ 2 and suitable p, q, the constraint −∞ e−z u(z)2 dz = 1, and, in particular, e−z 2 /2 u(z) → 0, z → ±∞ . (6) Does this Sturm-Liouville eigenvalue problem fit into the framework of Chapter 6? Why or why not? (e) Bring (5) into the equivalent formulation u00 (z) − 2zu0 (z) − u(z) = λu(z) , (7) expand the solution u into a power series u(z) = ∞ X ak z k , k=0 and derive a recurrence relation for the coefficients ak . (8) (f) Examine (8) for the case λ = −(2n + 1), n ∈ N0 . What do you observe? (g) It turns out that the eigenvalue problem (5) (or equiv. (7)) has eigenvalues λn = −(2n + 1), n ∈ N0 , with corresponding eigenfunctions Hn (z) called Hermite polynomials, that is, Hn00 (z) − 2zHn0 (z) = −2n Hn (z) . Derive the relation Z ∞ 2 00 0 e−z (Hn00 (z)Hm (z) − Hm (z)Hn (z) − 2z(Hn0 (z)Hm (z) − Hm (z)Hn (z))) dz −∞ Z ∞ 2 e−z Hn (z)Hm (z) dz = −2(n − m) −∞ and conclude the orthogonality property of Hermite polynomials by further manipulating the left-hand-side and using the additional decay constraints (6). (h) (MATH3978) Recall from Chapter 6 that one essentially concludes from Z r u(z)L[v](z) − v(z)L[u](z) dz = 0 , (9) l that an operator L (equipped with certain boundary conditions) is symmetric. Show that, for the case of the operator defined through (5) the respective formula with l = −∞, r = ∞ is fulfilled given the additional decay constraints (6). How is (9) related to the computation in 2.(g) ? 3. (Boussinesq model for water waves) Consider the improved Boussinesq model ∂t2 u = ∂x2 u + ∂x2 ∂t2 u − µ(∂x4 u + ∂x4 ∂t2 u) + ∂x2 (u2 ) , t > 0 , x ∈ R , u = u(t, x) ∈ R , that is used to describe the evolution of surface water waves taking into account the surface tension throught the parameter µ ∈ R (that is, µ = 0 gives a model without surface tension). (a) Compute the dispersion relation ω = ω(k) by inserting the ansatz u(t, x) = eikx+iωt into the linear part of the equation. Is the equation dispersive? (b) Make an ansatz of the form u(t, x) = ε2 A(ε(x − c0 t), |{z} ε3 t ) | {z } =z (c0 ∈ R , 0 < ε 1 ) (10) =τ assuming A(z, τ ) → 0, z → ±∞, τ ≥ 0, and derive that, for (10) to be a good approximation, we can set c20 = 1 and demand that A has to fulfill a Korteweg-deVries equation.