Plane boundary-value problems for the sine-Helmholtz equation in the theory of elasticity of liquid crystals in non-uniform magnetic fields

It is shown that, with certain limitations, the equations of orientational equilibrium of a nematic liquid crystal in the two-dimensional domain, placed in a non-uniform magnetic field, reduce to the non-linear sine-Hehmholtz equation in the plane of conjugate magnetic potentials mu and eta. The part played by the conformal mappings mu,eta-->x,y is investigated. Plane boundary-value problems for this equation in the one- and two-dimensional regions are considered. Criteria of stability of the two-dimensional solutions in open and closed volumes are established. The explicit forms of the solutions, which are expressed in terms of periodic elliptic functions and quasi-periodic theta functions of two arguments, are analysed. The inverse problem is solved. Solutions of the two-dimensional kink and the Jacobi delta-function type in a closed volume are obtained. (C) 1996 Elsevier Science Ltd.