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Using these expansion coef- ficients ak inside Eq. Since the conical soliton solution scales only linearly in Lr , we conclude that the domain-wall solution scaling FIG.

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To this end, we estimate energetics costly and conical soliton texture is the preferred state. Lr a more detailed analysis is required. To compute a conical soliton energy in a cell of a finite 1. Vortex line width w requires an extension of the solution to a finite geometry. The energy tion. Its energy can be simply estimated. Examining of such a state is clearly given by Fig.

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Putting these two contributions together, we find Lr 2. The difficulty has to do with the fail- 2 0 0 a 55 ure of a periodic soliton solution [obtained by picking the scaling with the area of the cell. Conical soliton surface vortex ditions. However, a good approximate symmetric solution, il- lustrated in Fig. In this notation, Eq. Smalyukh, V. Gurarie and S.

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Choi for p discussions and comments on the paper. Thus the periodic conical Foundation through Grants No. DMR and No. The analogy of the E-L equation [Eq. Naively, corresponding to the particle starting with a vanishing this maybe done by choosing the value of k such that velocity and below the potential maximum.

The subse- the period matches the cell thickness, w. Prost, The physics of liquid crystals, J. Kosterlitz and D.

Thouless, J. C 6, 2nd ed. Oxford, New York, Halperin and D. Nelson, Phys. Smalyukh, R. Pratibha, N. Madhusudana, and O. Nelson and B. Halperin, Phys. B 19, D. Lavrentovich, Eur.

E 16, Young, ibid. Saupe, Mol. Cardy and S. Ostlund, Phys. B 25, Chaikin and T.

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In Ref. Mermin, Rev. Meyer, Philos. Chandrasekhar, G. Nair, D. Rao, S. Prasad, satisfied exactly. Praefcke, and D.