Dynamics and numerics on the Hopf bundle Matlab m-Files Written by Rupert Way,  University of Surrey, UK

Numerical integration along paths in the Hopf bundle       The base of the Hopf bundle is the 2-sphere, with the 1-sphere as fibre, and 3-sphere as total space. The strategy for integrating on the Hopf bundle is to integrate on the 2-sphere and then lift the trajectory to the full space using the natural connection. The generated path is then horizontal, and produces a geometric phase. The four Matlab codes listed below perform this integration and output plots of paths on the 2-sphere and graphs of the generated phase.         * integrate_x_direction.m   * integration_at_L.m   * integration_at_L_prime.m   * lambda_plane_eval_search.m

Numerics on a fibre-bundle view of the CGL equation       Slow space and time modulation of the small-amplitude travelling periodic waves bifurcating from the neutral curve in fluid mechanics, such as two-dimensional plane Poiseuille flow, leads to a complex Ginzburg-Landau model equation, typically of the form shown to the right where A(x,t) is complex valued. A class of solitary wave solutions is the Hocking-Stewartson pulse (HS pulse). The linear stability problem for the HS pulse can be reduced to a complex four-dimensional ODE, and the exterior algebra formulation gives a six-dimensional ODE. A Hopf bundle, with total space the 11-dimensional sphere and 1-sphere fibre, is constructed and the linear stability problem is integrated on this bundle. The Matlab codes for this integration are given below.         * integration_at_L_prime_HSpulse.m   * lambda_plane_eval_search_HSpulse.m

Numerics on a fibre bundle view of the Schrödinger equation       The Schrödinger equation in quantum mechanics, with complex spectral parameter, can be viewed as an 2-dimensional complex ODE. Since the length of an eigenfunction is not important, solutions can be viewed as paths on the 3-sphere, and therefore the eigenvalue problem can be studied on the Hopf bundle. Here, a class of Schrödinger equations with localized potential is studied numerically. The Matlab codes for generating the paths in this case are listed below. Outputs are the paths on the 2-sphere, and the motion of the geometric phase, as illustrated to the right.         * integration_at_L_prime_schrodinger.m   * lambda_plane_eval_search_schrodinger.m

Taylor-Goldstein equation from hydrodynamic stability       The Taylor-Goldstein equation governs the stability of a shear-flow of an inviscid fluid of variable density. A special case of the Taylor-Goldstein equation is the Rayleigh equation. Both of these equations can be viewed as complex 2-dimensional equations, like the Schrödinger equation, although the coefficients in this case can be singular. The equation can again be viewed as a path in the Hopf bundle. Matlab codes for this problem are given below.         * integration_at_L_prime_taylor_gold.m   * lambda_plane_eval_search_taylor_gold.m

Hopf bundle homepage     Mathematics homepage