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)
Instability of solitary waves for a generalized Benney-Luke equation. (English) Zbl 1155.35081

Motivated by the fact that the generalized Benney-Luke equation is a formally valid approximation for describing two-way water wave propagation, the authors study the linear instability of its solitary wave solutions.

The method employed starts with the use of Fourier transforms which provide a description of solutions in terms of a Fourier multiplier. It is noted that, in small amplitude, the one-dimensional Benney-Like equation is asymptotically related to the generalized KdV equation. An operator generalization of Rouché’s theorem due to Gohberg and Sigal is used to establish the existence of an unstable eigenvalue. This method hence avoids the use of Evans functions.

The last part of the paper validates numerically the above results for the generalized Benney-Luke equation through a finite difference numerical scheme which combines an explicit predictor and an implicit corrector step to compute solutions.

35Q51Soliton-like equations
35Q53KdV-like (Korteweg-de Vries) equations
35Q35PDEs in connection with fluid mechanics
35B35Stability of solutions of PDE
65M06Finite difference methods (IVP of PDE)