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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) $$-\gamma u^{(iv)}+ u''+ u- u^3= 0$$ for $$- \infty< x< \infty$$, $$u(- \infty)= - 1$$ and $$u(+ \infty)= +1$$, $$u(0)= 0$$. A main results is:
Theorem A. For each $$0< \gamma\leq 1/8$$, there exists a unique odd monotone solution $$u(x, \gamma)$$ of problem (1). If $$\gamma> 1/8$$, there exist no odd monotone solutions of problem (1).
In the case $$0< \gamma\leq 1/8$$, 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.

MSC:
 34E15 Singular perturbations, general theory for ordinary differential equations 34B15 Nonlinear boundary value problems for ordinary differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 35B35 Stability in context of PDEs