# 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 sine-cosine method for obtaining solutions with compact and noncompact structures. (English) Zbl 1061.35121
Summary: We establish compact and noncompact solutions for nonlinear dispersive equations. A sine-cosine method is used to demonstrate this work. The different physical structures of the focusing branch and the defocusing branch are emphasized. Many models are approached to illustrate the usage of our main results.

##### MSC:
 35Q53 KdV-like (Korteweg-de Vries) equations 35Q51 Soliton-like equations 37K40 Soliton theory, asymptotic behavior of solutions
Full Text:
##### References:
 [1] Wadati, M.: Introduction to solitons. Pramana: journal of physics 57, No. 5--6, 841-847 (2001) [2] Wadati, M.: The exact solution of the modified kortweg--de Vries equation. J. phys. Soc. jpn 32, 1681-1687 (1972) [3] Wadati, M.: The modified kortweg--de Vries equation. J. phys. Soc. jpn 34, 1289-1296 (1973) [4] Rosenau, P.; Hyman, J. M.: Compactons: solitons with finite wavelengths. Phys. rev. Lett 70, No. 5, 564-567 (1993) · Zbl 0952.35502 [5] Rosenau, P.: Compact and noncompact dispersive structures. Phys. lett. A 275, No. 3, 193-203 (2000) · Zbl 1115.35365 [6] Ismail, M. S.; Taha, T.: A numerical study of compactons. Math. comput. Simulat 47, 519-530 (1998) · Zbl 0932.65096 [7] Wazwaz, A. M.: Partial differential equations: methods and applications. (2002) · Zbl 1079.35001 [8] Wazwaz, A. M.: New solitary-wave special solutions with compact support for the nonlinear dispersive $K(m,n)$ equations. Chaos, solitons and fractals 13, No. 2, 321-330 (2002) · Zbl 1028.35131 [9] Wazwaz, A. M.: Exact specific solutions with solitary patterns for the nonlinear dispersive $K(m,n)$ equations. Chaos, solitons and fractals 13, No. 1, 161-170 (2001) · Zbl 1027.35115 [10] Wazwaz, A. M.: General compactons solutions for the focusing branch of the nonlinear dispersive $K(n,n)$ equations in higher dimensional spaces. Appl. math. Comput 133, No. 2--3, 213-227 (2002) · Zbl 1027.35117 [11] Wazwaz, A. M.: General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive $K(n,n)$ equations in higher dimensional spaces. Appl. math. Comput 133, No. 2/3, 229-244 (2002) · Zbl 1027.35118 [12] Wazwaz, A. M.: A study of nonlinear dispersive equations with solitary-wave solutions having compact support. Math. comput. Simulat 56, 269-276 (2001) · Zbl 0999.65109 [13] Wazwaz, A. M.: Compactons dispersive structures for variants of the $K(n,n)$ and the KP equations. Chaos, solitons and fractals 13, No. 5, 1053-1062 (2002) · Zbl 0997.35083 [14] Wazwaz, A. M.: Compactons and solitary patterns structures for variants of the KdV and the KP equations. Appl. math. Comput 139, No. 1, 37-54 (2003) · Zbl 1029.35200 [15] Wazwaz, A. M.: Construction of soliton solutions and periodic solutions of the Boussinesq equation by the modified decomposition method. Chaos, solitons and fractals 12, No. 8, 1549-1556 (2001) · Zbl 1022.35051 [16] Wazwaz, A. M.: A computational approach to soliton solutions of the Kadomtsev--petviashili equation. Appl. math. Comput 123, No. 2, 205-217 (2001) · Zbl 1024.65098 [17] Wazwaz, A. M.: A first course in integral equations. (1997) · Zbl 0924.45001 [18] Adomian, G.: Solving frontier problems of physics: the decomposition method. (1994) · Zbl 0802.65122 [19] Adomian, G.: A review of the decomposition method in applied mathematics. J. math. Anal. appl 135, 501-544 (1998) · Zbl 0671.34053