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The work of A.K.N.S. which is based on the sl(2, R) valued soliton connection is extended to obtain new integrable coupled nonlinear partial differential equations. This is achieved by assuming the soliton connection having values in a simple Lie, Kac-Moody, Lie superalgebras. Extensions of some of the integrable nonlinear partial differential equations are given explicitly. In particular the coupled NLS equations on various homogeneous spaces and the coupled modified KdV integro-differential equations are obtained on symmetric spaces.

This work presents our investigations of mezomorphic properties of two polymorphic liquid crystals, namely, 4-nonyloxy-4-butoxyphenyl benzoate and N-(-4-heptyloxybenzylidene-4-butylaniline) in a wide temperature range, particularly, in the phase transition regions. By means of an original experimental method, the heterophase regions and also the phase transition temperatures have been determined for these materials with high accuracy. These phase transition intervals have been analyzed using a mean field model.

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Using mean-field theory with P2θ2 coupling term in the free energy expansion, we calculate the polarization as a function of concentration for the AC* phase transition in a binary mixture of C7 and 10.O.4 liquid crystalline material. Our calculated values of polarization are in good agreement with the experimental data.