zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Uniqueness of the stationary wave for the extended Fisher-Kolmogorov equation. (English) Zbl 0965.34039

Summary: The extended Fisher-Kolmogorov equation

u t =-βu xxxx +u xx +u-u 3 ,β>0,

models a binary system near the Lifshitz critical point and is known to exhibit a stationary heteroclinic solution joining the equilibria ±1. For the classical case, β=0, the heteroclinic is u(x)=tanh(x/2) and is unique up to the obvious symmetries.

The author proves the conjecture that the uniqueness persists all the way to β=1/8, where the onset of spatial chaos associated with the loss of monotonicity of the stationary wave is known to occur. The method used are non-perturbative and employ a global cross-section to the Hamiltonian flow of the stationary fourth-order equation on the energy level of ±1. He also proves uniform a priori bounds on all bounded stationary solutions, valid for any β>0.

34C37Homoclinic and heteroclinic solutions of ODE
34C28Complex behavior, chaotic systems (ODE)
34F05ODE with randomness