2. The Klein-Gordon equation Utt Uxx+u=0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞< x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| →∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. = dw dk (b) Establish that the Fourier transform solution is given by 1 u(x,t) = 2/7/7 √∞ = ∞ (A(k)e' (k+t√k²+1) + B(k)e³(k=−t√k²+1)) dk. 2πT Further, show that, in view of the fact that u(x, 0) and ut(x, 0) are real we indeed have 1 u(x,t) = 2πT ·S∞ (A(k)ei(kz+t√k²+1)) dk + (*), where (*) denotes the complex conjugate of the first integral. (c) Show that as t∞ with x/t fixed, |x/t| < 1, we have 1 u(x, t) ~ t -x A √√2π (t²x²)3/4 ei√t²−x²+i¹ + (*). +2 x2

2. The Klein-Gordon equation Utt Uxx+u=0 arises in many areas of physics, including relativistic quantum mechanics. Let -∞< x < ∞, t > 0, and consider the following initial value problem: the given initial values u(x, 0) and ut(x, 0) are real, and we assume that u → 0 sufficiently rapidly as |x| →∞. (a) Show that the dispersion relation for the wave solution u = ei(kx-wt) is given by w² = 1+k². Further, show that the phase speed c(k) = w/k satisfies |c(k)| ≥ 1, while for the group velocity C(k) = we have |C(k)| ≤ 1. = dw dk (b) Establish that the Fourier transform solution is given by 1 u(x,t) = 2/7/7 √∞ = ∞ (A(k)e' (k+t√k²+1) + B(k)e³(k=−t√k²+1)) dk. 2πT Further, show that, in view of the fact that u(x, 0) and ut(x, 0) are real we indeed have 1 u(x,t) = 2πT ·S∞ (A(k)ei(kz+t√k²+1)) dk + (), where () denotes the complex conjugate of the first integral. (c) Show that as t∞ with x/t fixed, |x/t| < 1, we have 1 u(x, t) ~ t -x A √√2π (t²x²)3/4 ei√t²−x²+i¹ + (*). +2 x2

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter4: Eigenvalues And Eigenvectors

Section4.6: Applications And The Perron-frobenius Theorem

Problem 69EQ: Let x=x(t) be a twice-differentiable function and consider the second order differential equation...

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