Solitons in laser physics.

Spin waves in 3He A are governed by the sine-Gordon equation. Its kink and breather solutions are elementary nonlinear excitations which generalize the usual concept of spin waves in a linearised theory.2 Spin waves in 3He B are governed by a non-integrable "double sine-Gordon equation". This has two kink solutions with jumps 2δ ≡ 4 cos-1 (-¼) and 4π - 2δ. As noted previously kinks always bump in collision; but 2δ kink-antikink pairs convert to - (4π - 2δ) antikink-kink pairs; 4π - 2δ kink-antikink pairs bump below a velocity threshold V ≲ 0.92 in dimensionless units and convert to - 2δ antikink-kink pairs above this threshold. A new result is that below V = 0.36 4π - 2δ kink-antikink pairs bind into long lived breather-like states. If c-number theory is applicable in the form derived we expect the breathers in both 3He A and 3He B to play some role in the NMR response. But if the systems are quantised no breathers can be excited in 3He A and no breather-like states in 3He B.We investigate two c-number initial value problems for the creation of spin waves, one in 3He A, one in 3He B. The critical fields for homogeneous NMR in both systems are shown to remain critical fields for spatially inhomogeneous systems. For the chosen boundary and initial conditions only breathers are created below first threshold: these are standing waves in 3He A but move outwards in 3He B. For the chosen boundary conditions the ringing observed by Wheatley would have to be attributed to these breathers. We conclude from this on the one hand that the systems scarcely can be quantised and on the other that the actual boundary conditions could prove important to the associated physics. At first threshold ut = 3/2 3He B spin waves move outwards as single 4π - 2δ kinks, as 4π - 2δ kink-antikink pairs, or as combinations of these with outgoing breathers. In contrast it is above the single threshold at ut = 2 in 3He A that single 2π kinks first move out and any remaining breathers stand; more 2π kinks move out further above threshold. 3He B has a second threshold above which combinations of 4π - 2δ kinks and antikinks and 2δ kinks move outwards. The different behaviors of spin waves in the two systems 3He A and 3He B may permit a magnetic experiment which could help to confirm the different symmetries of the order parameters assigned to these different phases.

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