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)
Application of simplest equations of Bernoulli and Riccati kind for obtaining exact traveling-wave solutions for a class of pdes with polynomial nonlinearity. (English) Zbl 1222.35062

Summary: We search for traveling-wave solutions of equations of the form

p=1 N 1 α p p Q t p + q=1 N 2 β q q Q x q + m=1 M μ m Q m =0

where α p , β q and μ m are parameters. We obtain such solutions by the method of the simplest equation for the cases when the simplest equation is the equation of Bernoulli or the equation of Riccati. We modify the methodology of the simplest equation of Kudryashov as follows. Kudryashov uses the first step of the test for the PainlevĂ© property in order to determine the size of the solution of the studied PDE. We divide the PDEs under consideration into two parts: part A, which contains the derivatives, and part B, which contains the rest of the equation. The application of the ansatz or the extended ansatz of Kudryashov transforms part A and part B into two polynomials. We balance the highest powers of the polynomials for the parts A and B and thus obtain a balance equation which depends on the kind of the simplest equation used. The balance equations are connected to nonlinear algebraic systems of relationships among the parameters of the equations and the parameters of the solution. On the basis of these systems, we obtain numerous solutions of the class of equations considered.

MSC:
35G05General theory of linear higher-order PDE
35C07Traveling wave solutions of PDE