By Yuejin Z., Guodong X.

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In one respect, this process is incomplete: if a more elaborate (and more suspect) form of perturbation theory is used, the energy denominators in the perturbation series are no longer constants, but are themselves functions of the energy shift to be calculated. The anisotropy constants would thus be functions of energy. This is not acceptable, and it suggests (since energy and time are conjugate variables) that the elimination of L should be carried out in the framework of time-dependent perturbation theory.

Seeking solutions of the form φ(x−vs t) reduces the number of independent variables from MODELS OF SMALL MOTIONS 29 four to three. Further specializing to solutions depending on only one spatial variable along the velocity vs , turns it into an ordinary nonlinear diﬀerential equation q with a well-known solution, proportional to 1/cosh D (x − vs t) , sometimes 2 2 called a soliton. Here, D = ∂ ωk0 /∂k0 , where the k0 -derivative is in the chosen direction. Of course, a solution assumed to depend only on the particular combination (x − vs t) is not necessarily stable.

However, mode-mode coupling can yield an eﬀective attraction among the constituents of the packet that will tend to keep the pulse together, counteracting the eﬀect of dispersion. A highly condensed summary of Slavin et al. (1994) on the subject is presented here: Conﬁning themselves to interaction energy involving the product of four spin waves, these authors start with the equations of motion within the manifold 2ωk = ωk+κ +ωk−κ to conserve energy and momentum (point Q, Fig. 28) Combine these modes to form a narrow wavepacket Ψ = on k0 .

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