Periodic problem for an evolution equation with a quadratic and a cubic nonlinearity

Komarov, M. V.

doi:10.1134/s0012266111120032

Cited by 1 publication

(2 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This solution is spatially homogeneous and monotonically depending on time. By analogy with previous studies [28][29][30][31][32], we have assumed that the patterns above can be described as large time perturbations of this exact solution. The large time asymptotics for the SLD ρ(t, s) are constructed explicitly, to within O(1/T 2 ), in the class of functions which tend to the above exact solution as T → ∞.…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

Pattern formation in terms of semiclassically limited distribution on lower dimensional manifolds for the nonlocal Fisher–Kolmogorov–Petrovskii–Piskunov equation

Levchenko¹,

Шаповалов²,

Trifonov³

2013

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

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.

show abstract

Section: Discussionmentioning

confidence: 99%

“…The solutions of equation ( 5.3) tend to a steady-state solution at large times [28][29][30][31][32]. This property has led us to seek solutions to equation (5.3) in a class of functions closely similar to (5.16).…”

Section: Large-time Asymptoticsmentioning

confidence: 99%