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 Weierstrass elliptic function expansion method and its applications in nonlinear wave equations. (English) Zbl 1142.35603
Summary: Based on the close relationship between the Weierstrass elliptic function Weierstrass 𝔭(ξ;g 2 ,g 3 ) and nonlinear ordinary differential equation, a Weierstrass elliptic function expansion method is developed in terms of the Weierstrass elliptic function instead of many Jacobi elliptic functions. The mechanism is constructive and can be carried out in computer with the aid of computer algebra (Maple). Many important nonlinear wave equations arising from nonlinear science are chosen to illustrate this technique such as the new integrable Davey-Stewartson-type equation, the (2+1)-dimensional modified KdV equation, the generalized Hirota equation in 2+1 dimensions, the Generalized KdV equation, the (2+1)-dimensional modified Novikov-Veselov equations, (2+1)-dimensional generalized system of modified KdV equation, the coupled Klein-Gordon equation, and the (2+1)-dimensional generalization of coupled nonlinear Schrödinger equation. As a consequence, some new doubly periodic solutions are obtained in terms of the Weierstrass elliptic function. Moreover solitary wave solutions and singular solitary wave solutions are also given as simple limits of doubly periodic solutions. These solutions may be useful to explain some physical phenomena. The algorithm is also applied to other many nonlinear wave equations. Moreover we also present the general form of the method.
MSC:
35Q55NLS-like (nonlinear Schrödinger) equations
33E05Elliptic functions and integrals
35C05Solutions of PDE in closed form
35Q53KdV-like (Korteweg-de Vries) equations
37K10Completely integrable systems, integrability tests, bi-Hamiltonian structures, hierarchies
Software:
Maple