Collective coordinates, asymptotics and domain wall dynamics in ferromagnets

Duration: 51 mins 32 secs

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Description:	Robbins, J Wednesday 30th January 2019 - 17:00 to 18:00


Created:	2019-02-05 09:56
Collection:	The mathematical design of new materials
Publisher:	Isaac Newton Institute
Copyright:	Robbins, J
Language:	eng (English)


Abstract:	The method of collective coordinates is a simple and widely used variational procedure for finding approximate solutions to many- or infinite-dimensional, possibly damped and driven, Hamiltonian systems. The approximate solutions are typically characterised by a small number of time-dependent parameters, which are understood to describe a small number of activated modes. The simplicity of the method comes at a price, however, as it does not allow a determination of how good (or bad) the approximation is. In certain regimes, asymptotic expansions can provide the requisite estimates, though they require more work. This is illustrated for the problem of the motion of domain walls in ferromagnets. Domain walls are interfaces between differently oriented magnetic domains, and the dynamics of these interfaces under applied magnetic fields and currents is a problem of current physical and technological interest. We also describe behaviour in a high-field regime, beyond the well-known Walker breakdown, where one of the domains becomes unstable. A new type of dynamics emerges that appears to be beyond the reach of a collective coordinate description. It can be described using front propagation theory, but rigorous results (akin to a KPP analysis) appear to be challenging. This is joint work with Arseni Goussev, Valeriy Slastikov, and Sergiy Vasylkevych.

Abstract:

The method of collective coordinates is a simple and widely used variational procedure for finding approximate solutions to many- or infinite-dimensional, possibly damped and driven, Hamiltonian systems. The approximate solutions are typically characterised by a small number of time-dependent parameters, which are understood to describe a small number of activated modes. The simplicity of the method comes at a price, however, as it does not allow a determination of how good (or bad) the approximation is. In certain regimes, asymptotic expansions can provide the requisite estimates, though they require more work. This is illustrated for the problem of the motion of domain walls in ferromagnets. Domain walls are interfaces between differently oriented magnetic domains, and the dynamics of these interfaces under applied magnetic fields and currents is a problem of current physical and technological interest. We also describe behaviour in a high-field regime, beyond the well-known Walker breakdown, where one of the domains becomes unstable. A new type of dynamics emerges that appears to be beyond the reach of a collective coordinate description. It can be described using front propagation theory, but rigorous results (akin to a KPP analysis) appear to be challenging. This is joint work with Arseni Goussev, Valeriy Slastikov, and Sergiy Vasylkevych.

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