1306.3765 (E. A. Levchenko et al.)

Pattern formation in terms of semiclassically limited distribution on
lower-dimensional manifolds for nonlocal
Fisher--Kolmogorov--Petrovskii--Piskunov equation    [PDF]

E. A. Levchenko, A. V. Shapovalov, A. Yu Trifonov

We have investigated the pattern formation in systems described by the nonlocal Fisher--Kolmogorov--Petrovskii--Piskunov equation for the cases where the dimension of the pattern concentration area is less than that of independent variables space. We have obtained a system of integro-differential equations which describe the dynamics of the concentration area and the semiclassically limited distribution of a pattern in the class of trajectory concentrated functions. Also, asymptotic large-time solutions have been obtained that describe the semiclassically limited distribution of a quasi-steady-state pattern on the concentration manifold. The approach is illustrated by an example for which the analytical solution is in good agreement with the prediction of a numerical simulation.

View original: http://arxiv.org/abs/1306.3765