  We describe exact kink soliton solutions to nonlinear partial differential
equations in the generic form u_{t} + P(u) u_{x} + \nu u_{xx} + \delta u_{xxx}
= A(u), with polynomial functions P(u) and A(u) of u=u(x,t), whose generality
allows the identification with a number of relevant equations in physics. We
emphasize the study of chirality of the solutions, and its relation with
diffusion, dispersion, and nonlinear effects, as well as its dependence on the
parity of the polynomials $P(u)$ and $A(u)$ with respect to the discrete
symmetry $u\to-u$. We analyze two types of kink soliton solutions, which are
also solutions to 1+1 dimensional phi^{4} and phi^{6} field theories.
