Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Dynamical Transition Theory of Hexagonal Pattern Formations   The main goal of this paper is to understand the formation of hexagonal
patterns from the dynamical transition theory point of view. We consider the
transitions from a steady state of an abstract nonlinear dissipative system. To
shed light on the formation of mixed mode patterns such as the hexagonal
pattern, we consider the case where the linearized operator of the system has
two critical real eigenvalues, at a critical value $\lambda_c$ of a control
parameter $\lambda$ with associated eigenmodes having a roll and rectangular
pattern. By using center manifold reduction, we obtain the reduced equations of
the system near the critical transition value $\lambda_c$. By a through
analysis of these equations, we fully characterize all possible transition
scenarios when the coefficients of the quadratic part of the reduced equations
do not vanish. We consider three problems, two variants of the 2D
Swift-Hohenberg equation and the 3D surface tension driven convection, to
demonstrate that all the main theoretical results we obtain here are indeed
realizable.

10
The categories of this paper are: q-bio
31
The categories of this paper are: ao-sci
35
The categories of this paper are: mtrl-th
19
The categories of this paper are: eess
15
The categories of this paper are: nucl-ex
5
The categories of this paper are: gr-qc
12
The categories of this paper are: nucl-th
1
The categories of this paper are: cs
4
The categories of this paper are: cond-mat
11
The categories of this paper are: hep-lat
34
The categories of this paper are: chem-ph
21
The categories of this paper are: adap-org
18
The categories of this paper are: econ
26
The categories of this paper are: bayes-an
3
The categories of this paper are: physics
0
The categories of this paper are: hep-ph