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)
On the conservation laws and invariant solutions of the mKdV equation. (English) Zbl 1180.35471
Summary: We consider modified Korteweg-de Vries (mKdV) equation. By using the nonlocal conservation theorem method and the partial Lagrangian approach, conservation laws for the mKdV equation are presented. It is observed that only nonlocal conservation theorem method lead to the nontrivial and infinite conservation laws. In addition, invariant solution is obtained by utilizing the relationship between conservation laws and Lie-point symmetries of the equation.
MSC:
35Q53KdV-like (Korteweg-de Vries) equations
37K05Hamiltonian structures, symmetries, variational principles, conservation laws
35B06Symmetries, invariants, etc. (PDE)
References:
[1]Noether, E.: Invariante variationsprobleme, Nachr. konig. Gesell. wiss. Gottingen math.-phys. Kl. heft 2, No. 3, 235-257 (1918)
[2]Bessel-Hagen, E.: Uber die erhaltungssatze der elektrodynamik, Math. ann. 84, 258-276 (1921) · Zbl 48.0877.02 · doi:10.1007/BF01459410 · doi:http://jfm.sub.uni-goettingen.de/JFM/digit.php?an=JFM+48.0877.02
[3]P.S. Laplace, Celestial Mechanics, New York, 1966 (English translation)
[4]Anco, S. C.; Bluman, G. W.: Direct construction method for conservation laws of partial differential equations. Part II: General treatment, European J. Appl. math. 9, 567-585 (2002) · Zbl 1034.35071 · doi:10.1017/S0956792501004661
[5]Steudel, H.: Uber die zuordnung zwischen invarianzeigenschaften und erhaltungssatzen, Z. naturforsch 17A, 129-132 (1962)
[6]Olver, P. J.: Application of Lie groups to differential equations, (1993)
[7]Kara, A. H.; Mahomed, F. M.: Relationship between symmetries and conservation laws, Int. J. Theor. phys. 39, 23-40 (2000) · Zbl 0962.35009 · doi:10.1023/A:1003686831523
[8]Kara, A. H.; Mahomed, F. M.: Noether-type symmetries and conservation laws via partial Lagrangians, Nonlinear dynam. 45, 367-383 (2006) · Zbl 1121.70014 · doi:10.1007/s11071-005-9013-9
[9]Kolev, B.: Poisson brackets in hydrodynamics, Discrete contin. Dyn. syst. 19, 555-574 (2007) · Zbl 1139.53040
[10]Ibragimov, N. H.: A new conservation theorem, J. math. Anal. appl. 333, 311-328 (2007) · Zbl 1160.35008 · doi:10.1016/j.jmaa.2006.10.078
[11]Benjamin, T. J.; Feir, J. E.: Disintegration of wave trains on deep water, J. fluid mech. 27, 417-437 (1967) · Zbl 0144.47101 · doi:10.1017/S002211206700045X
[12]Constantin, A.; Strauss, W.: Stability properties of steady water waves with vorticity, Comm. pure appl. Math. 60, 911-950 (2007) · Zbl 1125.35081 · doi:10.1002/cpa.20165
[13]Naz, R.; Mahomed, F. M.; Mason, D. P.: Comparison of different approaches to conservation laws for some partial differential equations in fluid mechanics, Appl. math. Comput. 205, 212-230 (2008) · Zbl 1153.76051 · doi:10.1016/j.amc.2008.06.042
[14]Atherton, R. W.; Homsy, G. M.: On the existence and formulation of variational principles for nonlinear differential equations, Stud. appl. Math. 54, 31-60 (1975) · Zbl 0322.49019
[15]Sjöberg, A.: Double reduction of pdes from the association of symmetries with conservation laws with applications, Appl. math. Comput. 184, 608-616 (2007) · Zbl 1116.35004 · doi:10.1016/j.amc.2006.06.059
[16]Olver, P. J.: Evolution equations possessing infinitely many symmetries, J. math. Phys. 18, 1212-1215 (1977) · Zbl 0348.35024 · doi:10.1063/1.523393