### Bifurcation theory methods in problem about crystallization of liquid phase state in statistical crystal theory

#### B. V. Loginov^{1}, O. V. Makeev^{2}

Annotation | Applications of the results [16---21] are briefly presented (p. 2) to the problem on crystallization of liquid phase state in statistical crystal theory governed by nonlinear integral equation of Hammerstein type with integrals on the whole space $\mathbb{R}^3$ and kernels depending on modulus of arguments difference. All solutions have primary type and respectively allow only symmetry of symmorphic spatial crystallographic groups. Basic attention is paid (p. 3) to crystallization problem with composite lattices governed by systems of nonlinear Hammerstein type integral equations. Here vectorial zero-subspace arise and respectively vectorial bifurcation with higher orders of degeneracy. By this an approach is indicated to bifurcation problems allowing non-symorphic crystallographic group symmetries. As concrete example it is considered the construction of the bifurcation equation for crystallization problem with the group $C_{2h}^5$ symmetry of monoclinic syngony governed by the system of four nonlinear integral equations. The constructed branching equation inherits the indicated symmetry. Bifurcating solutions are written out. It is considered only case of composite lattice consisting of the identical sublattices that corresponds to one bifurcation parameter. Here n.5 of the article Proc. MVMS v. 11, No. 1 is presented in which more complicated case of different sublattices is briefly considered. The most general situation will be subject of future investigations. |
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Keywords | symmetry breaking problems; statistical theory of crystal; nonlinear integral equations of Hammerstein type; bifurcation and symmetry |

^{1}Ulyanovsk state technical University, Ulyanovsk; loginov@ulstu.ru.^{2}Ulyanovsk state technical University, Ulyanovsk; o.makeev@ulstu.ru.

**Citation**: B. V. Loginov, O. V. Makeev, "[Bifurcation theory methods in problem about crystallization of liquid phase state in statistical crystal theory]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,11:2 (2009) 194–196 (In Russian)