Symmetry breaking and bifurcating solutions in the classical complex ϕ⁶field theory

The authors consider the equation of motion for a complex classical field in a (3+1)-dimensional ϕ 6 model. The resultant complex non-linear Klein-Gordon equation is solved using an ansatz in which the envelope satisfies a scalar non-linear Klein-Gordon equation and the carrier satisfies either a wave equation or Laplace's equation with additional constraints. They use the results of the symmetry reduction method to exactly solve both the carrier equations and the envelope equation. The latter has recently been analysed and here they only briefly discuss the types of the appropriate solutions. However, they present a detailed discussion on one particular solution which is physically important. It bifurcates both in real space and in phase space. Possible physical applications have been outlined in the last section and they include superfluidity, superconductivity, liquid crystals and helicoidal metamagnets.