# zbMATH — the first resource for mathematics

##### Examples
 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.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 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)
The long-wave limit for the water wave problem. I: The case of zero surface tension. (English) Zbl 1034.76011
The paper revisits the classical problem of the description of long small-amplitude weakly nonlinear weakly dispersive surface waves in a long channel, neglecting the capillary effects. The analysis starts with a system of equations for two-dimensional irrotational inviscid flow. It is proved that, on a relatively short time scale, $\sim 1/ϵ$, a general initial perturbation, in the form of a localized pulse, splits into two pulses which travel in opposite directions. At a longer time scale, $\sim {ϵ}^{-3}$, each of the two pulses splits into an array of solitons obeying Korteweg-de Vries (KdV) equation. At the latter time scale, it is also proved that collisions between solitons can be described asymptotically correctly by KdV equation. The proofs are based on estimates of the difference between the solutions to the full water-wave system of equations and the asymptotic KdV equation. A novelty in comparison with previous works analyzing the rigorous correspondence between the full system and the approximation based on the KdV equation is that the class of functions admitted in the analysis does not exclude solitons and soliton trains.

##### MSC:
 76B25 Solitary waves (inviscid fluids) 35Q51 Soliton-like equations 76M45 Asymptotic methods, singular perturbations (fluid mechanics)