The authors study the following problem with the Fisher-Kolmogorov equation: (1) for , and , . A main results is:
Theorem A. For each , there exists a unique odd monotone solution of problem (1). If , there exist no odd monotone solutions of problem (1).
In the case , for the solutions ensured by Theorem A, some qualitative properties also are established (Theorem B). In the context of phase transitions, such solutions are known as kinks separating regions of different phases.