Kinks in higher-order polynomial models

We consider a class of Lorentz-invariant field-theoretic models with one real scalar field in (1+1)-dimensional spacetime with self-action in the form of even-degree polynomials having two degenerate minima. Kink solutions are obtained, their masses are found, and asymptotic properties are studied. Some features of the Sturm-Liouville problem defining the spectrum of soliton excitations in a linear approximation are shown. In particular, the form of the quantum mechanical potential and the zero mode can be found for such solitons in parametric form.