Spatially chaotic solutions to parabolic equations and preservation of homotopies (Q1385877)

Many natural phenomena involve dynamics of complex spatial forms that are stably reproduced during long time intervals and eventually stiffen if energy dissipates. This generates spatial structures, which are frequently characterized by a chaotic irregular arrangement of elements. This paper describes a general and stable mechanism that can explain the originating and preserving of such forms when their dynamics are described by multicomponent parabolic equations. The main idea is to describe the complexity of a solution in terms of its homotopic type. If the energy of the solutions is not very large, then their values belong to the space formed by the level sets of the nonlinear potential that is included in the equation and has a nontrivial topological structure. In this work, we systematically use the Jacobi variational functional, which allows us to obtain more accurate results, in particular, to describe the soliton dynamics. We consider a parabolic system of the form \[ \partial_tu= a\partial^2_x u-\nabla F(u), \] where \(u\in \mathbb{R}^d\), \(u= u(x, t)\), \(x\in \mathbb{R}\), \(t\geq 0\), \(a>0\), and \(F\) is twice continuously differentiable on \(\mathbb{R}^d\) and positive everywhere except for zero, \(F(u)\geq 0\), \(F(0)= 0\), \(F(u)>0\), for \(u\neq 0\).