Creation of Spin Waves in ³HeB

Abstract

This article is largely concerned with the following problems: the solution of the double sine-Gordon equations [1,2,3]

$${u_{xx}} - {u_{tt}} = _ + ^ - \,\left( {\sin \,u + \frac{1}{2}\lambda \sin \frac{1}{2}u} \right)$$

(1)

for boundary conditions u,u_x,u_xx, etc. → 0, as |x| → ∞, and initial data u(x,0) = 0; u_t(x,0) = a, |x| ≤ ℓ, u_t(x,0) = 0, |x| > ℓ. Two cases are directly relevant to the spin waves in the superfluid phases of He below 2.6 mK: these are the -ve sign and λ = 1, and the +ve sign and λ = 0. The latter is an initial value problem for the sine-Gordon equation and can be solved by an inverse scattering method [4]. The double sine-Gordon equations which arise for λ ≠ 0 are not soluble by any of the techniques presently available for solving nonlinear evolution equations [2,3,5]: there are, for example, apparently only three conservation laws [2] and the systems are not ‘integrable’. Evidently only singular perturbation theory [6] and numerical integration are available to solve this problem. Despite success with perturbation theories for the case of the positive sign [6], we do not yet know how to handle similar perturbation theory for the negative sign: u_xx − u_tt = − sin u is unstable and its multisoliton solutions are unstable. This par therefore confines its report to the results of numerical work.