# 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 explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation. (English) Zbl 1258.35050

Summary: We study explicit nonlinear wave solutions and their bifurcations of the generalized Camassa-Holm equation

${u}_{t}+2k{u}_{x}-{u}_{xxt}+3{u}^{2}{u}_{x}=2{u}_{x}{u}_{xx}+u{u}_{xxx}·$

Not only are the precise expressions of the explicit nonlinear wave solutions obtained, but some interesting bifurcation phenomena are revealed.

Firstly, it is verified that $k=3/8$ is a bifurcation parametric value for several types of explicit nonlinear wave solutions.

When $k<3/8$, there are five types of explicit nonlinear wave solutions, which are

(i) hyperbolic peakon wave solution,

(ii) fractional peakon wave solution,

(iii) fractional singular wave solution,

(iv) hyperbolic singular wave solution,

(v) hyperbolic smooth solitary wave solution.

When $k=3/8$, there are two types of explicit nonlinear wave solutions, which are fractional peakon wave solution and fractional singular wave solution.

When $k>3/8$, there is not any type of explicit nonlinear wave solutions.

Secondly, it is shown that there are some bifurcation wave speed values such that the peakon wave and the anti-peakon wave appear alternately.

Thirdly, it is displayed that there are other bifurcation wave speed values such that the hyperbolic peakon wave solution becomes the fractional peakon wave solution, and the hyperbolic singular wave solution becomes the fractional singular wave solution.

##### MSC:
 35C08 Soliton solutions of PDE 35C07 Traveling wave solutions of PDE 35K55 Nonlinear parabolic equations 34A05 Methods of solution of ODE 34B15 Nonlinear boundary value problems for ODE 34C37 Homoclinic and heteroclinic solutions of ODE 34C23 Bifurcation (ODE)