Generalized Variant of the Variational Ginzburg–Landau Equation
—The version of the Ginzburg–Landau variational equation occurring in condensed matter physics which is called the ψ6-model is studied. As in the case of the ψ4-model, the corresponding equation is studied together with periodic boundary conditions. Conditions for the existence of single-mode equilibrium states are obtained for a periodic boundary value problem. An answer is given to the question of their stability in the sense of Lyapunov’s definition. In the case of a change in stability by single-mode equilibrium states, local bifurcations near these solutions are studied. It is shown that subcritical (rigid) bifurcations are characteristic of the problem under consideration. Bifurcation analysis involves the methods of the theory of dynamical systems with infinite-dimensional phase space and, first of all, the method of integral manifolds and the method of Poincaré normal forms. It is shown that two-dimensional invariant manifolds formed by spatially inhomogeneous solutions other than single-mode ones bifurcate from single-mode equilibrium states. For such solutions, asymptotic formulas are obtained.