Study with the several resources on Docsity
Earn points by helping other students or get them with a premium plan
Prepare for your exams
Study with the several resources on Docsity
Earn points to download
Earn points by helping other students or get them with a premium plan
This is the Exam Paper of Mathematical Tripos which includes Solitons and Instantons, Nonlinear Schr¨Odinger Equation, Stationary Soliton Solutions, Differential Equation, Effective Lagrangian, Harmonic Potential, Derrick Scaling Argument, Abelian Higgs Model etc. Key important points are: Nonlinear Patterns, Simple Bifurcation, Earth’s Magnetic Field, Axisymmetric Dipole Magnetic Field, Induced Velocity, Equatorial Dipole Magnetic Field, General Set of Equations, Amplitudes and Phases
Typology: Exams
1 / 3
This page cannot be seen from the preview
Don't miss anything!
Monday, 8 June, 2009 1:30 pm to 3:30 pm
Attempt no more than TWO questions. There are THREE questions in total. The questions carry equal weight.
Cover sheet None Treasury Tag Script paper
You may not start to read the questions printed on the subsequent pages until instructed to do so by the Invigilator.
2
1 A model of the earth’s magnetic field is represented by the interaction of three modes, arising in a simple bifurcation, with the following symmetries:
(a) an axisymmetric dipole magnetic field; (b) an equatorial dipole magnetic field which changes sign under a rotation about the axis of π; (c) an induced velocity field with the same symmetry as the equatorial dipole.
It is assumed that the earth is symmetric with respect to longitude, and that the system is invariant under a change of sign of all magnetic fields, leaving the velocity field unchanged. Finally, it is known that the equations describing the magnetic field evolution are linear in the magnetic field.
Justify the statement that the axisymmetric field can be represented by a real variable B, while the other fields can be represented by complex variables A, V.
Write down the most general set of equations that respect the above conditions, and show that when these are truncated at cubic order they take the form, where the ci, di are complex in general, and the μi, ωi and e are real:
B˙ = μ 1 B + (c 1 A∗V + c.c.) − e|V |^2 B, A^ ˙ = (μ 2 + iω 2 )A + c 2 V B − d 2 |V |^2 A − d 3 V 2 A∗, V^ ˙ = (μ 3 + iω 3 )V + c 3 AB − d 4 V |V |^2 − d 5 V |A|^2 − d 6 V ∗A^2 − d 7 V B^2 ,
and show that c 1 and |c 3 | can be taken as unity without loss of generality.
Now take d 2 = d 3 = d 5 = d 6 = d 7 = 0, and d 4 , c 2 real. Derive equations (which need not be solved) governing the amplitudes and phases of the variables for solutions of the equations where the moduli of A, V, B are constant. Find explicitly a relation between the parameters for which such a solution with A, B 6 = 0 bifurcates from one with A, B = 0, V 6 = 0.
Part III, Paper 70
