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Spatial patterns described by the extended Fisher-Kolmogorov (EFK) equation: Kinks. (English) Zbl 0826.34056

The authors study the following problem with the Fisher-Kolmogorov equation: (1) -γu (iv) +u '' +u-u 3 =0 for -<x<, u(-)=-1 and u(+)=+1, u(0)=0. A main results is:

Theorem A. For each 0<γ1/8, there exists a unique odd monotone solution u(x,γ) of problem (1). If γ>1/8, there exist no odd monotone solutions of problem (1).

In the case 0<γ1/8, 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.

34E15Asymptotic singular perturbations, general theory (ODE)
34B15Nonlinear boundary value problems for ODE
34C15Nonlinear oscillations, coupled oscillators (ODE)
34C25Periodic solutions of ODE
35B35Stability of solutions of PDE