Successive phase transitions and kink solutions in $\phi^{8}$,   $\phi^{10}$, and $\phi^{12}$ field theories
  We obtain exact solutions for kinks in $\phi^{8}$, $\phi^{10}$ and $\phi^{12}$ field theories with degenerate minima, which can describe a second-order phase transition followed by a first-order one, a succession of two first-order phase transitions and a second-order phase transition followed by two first-order phase transitions, respectively. Such phase transitions are known to occur in ferroelastic and ferroelectric crystals and in meson physics. In particular, we find that the higher-order field theories have kink solutions with algebraically-decaying tails and also asymmetric cases with mixed exponential-algebraic tail decay, unlike the lower-order $\phi^4$ and $\phi^6$ theories. Additionally, we construct distinct kinks with equal energies in all three field theories considered, and we show the co-existence of up to three distinct kinks (for a $\phi^{12}$ potential with six degenerate minima). We also summarize phonon dispersion relations for these systems, showing that the higher-order field theories have specific cases in which only nonlinear phonons are allowed. For the $\phi^{10}$ field theory, which is a quasi-exactly solvable (QES) model akin to $\phi^6$, we are also able to obtain three analytical solutions for the classical free energy as well as the probability distribution function in the thermodynamic limit. 
Avinash Khare, Ivan C. Christov, Avadh Saxena
math-ph cond-mat.mtrl-sci hep-th math.MP