American Institute of Mathematical Sciences

Advanced
2009, 2009(Special): 761-770. doi: 10.3934/proc.2009.2009.761

Some classes of surfaces in $\mathbb{R}^3$ and $\M_3$ arising from soliton theory and a variational principle

Suleyman Tek 1,

1. 

Department of Computer Science, University of Arkansas at Little Rock, 2801 S. University Ave., Little Rock, AR 72204, United States

Received  July 2008 Revised  February 2009 Published  September 2009

In this paper, modified Korteweg-de Vries (mKdV) and Harry Dym (HD) surfaces are considered which are arisen from using soliton surface technique and a variational principle. Some of these surfaces belong to Willmore-like and Weingarten surfaces, and surfaces that solve the generalized shape equation classes. Moreover, parameterized form of these surfaces are found for given solutions of the mKdV and HD equations.
Keywords: Soliton surfaces, Weingarten surfaces, integrable equations, Willmore surfaces, shape equa- tion.
Mathematics Subject Classification: Primary: 53A05, 53A35, 53C42, 35Q53, 35Q58; Secondary: 53C80, 35Q80.
Citation: Suleyman Tek. Some classes of surfaces in $\mathbb{R}^3$ and $\M_3$ arising from soliton theory and a variational principle. Conference Publications, 2009, 2009 (Special) : 761-770. doi: 10.3934/proc.2009.2009.761
[1]

Leo T. Butler. A note on integrable mechanical systems on surfaces. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 1873-1878. doi: 10.3934/dcds.2014.34.1873
[2]

Martin Bauer, Philipp Harms, Peter W. Michor. Sobolev metrics on shape space of surfaces. Journal of Geometric Mechanics, 2011, 3 (4) : 389-438. doi: 10.3934/jgm.2011.3.389
[3]

Siran Li, Jiahong Wu, Kun Zhao. On the degenerate boussinesq equations on surfaces. Journal of Geometric Mechanics, 2020, 12 (1) : 107-140. doi: 10.3934/jgm.2020006
[4]

Sergiu Klainerman, Igor Rodnianski. On emerging scarred surfaces for the Einstein vacuum equations. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1007-1031. doi: 10.3934/dcds.2010.28.1007
[5]

Alfonso Artigue. Expansive flows of surfaces. Discrete & Continuous Dynamical Systems, 2013, 33 (2) : 505-525. doi: 10.3934/dcds.2013.33.505
[6]

Kariane Calta, John Smillie. Algebraically periodic translation surfaces. Journal of Modern Dynamics, 2008, 2 (2) : 209-248. doi: 10.3934/jmd.2008.2.209
[7]

Eduard Duryev, Charles Fougeron, Selim Ghazouani. Dilation surfaces and their Veech groups. Journal of Modern Dynamics, 2019, 14: 121-151. doi: 10.3934/jmd.2019005
[8]

Anton Petrunin. Correction to: Metric minimizing surfaces. Electronic Research Announcements, 2018, 25: 96-96. doi: 10.3934/era.2018.25.010
[9]

Anton Petrunin. Metric minimizing surfaces. Electronic Research Announcements, 1999, 5: 47-54.

[10]

Yong Lin, Gábor Lippner, Dan Mangoubi, Shing-Tung Yau. Nodal geometry of graphs on surfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1291-1298. doi: 10.3934/dcds.2010.28.1291
[11]

Gabriella Tarantello. Analytical, geometrical and topological aspects of a class of mean field equations on surfaces. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 931-973. doi: 10.3934/dcds.2010.28.931
[12]

Alexandre Girouard, Iosif Polterovich. Upper bounds for Steklov eigenvalues on surfaces. Electronic Research Announcements, 2012, 19: 77-85. doi: 10.3934/era.2012.19.77
[13]

Seung Won Kim, P. Christopher Staecker. Dynamics of random selfmaps of surfaces with boundary. Discrete & Continuous Dynamical Systems, 2014, 34 (2) : 599-611. doi: 10.3934/dcds.2014.34.599
[14]

Roberto Paroni, Podio-Guidugli Paolo, Brian Seguin. On the nonlocal curvatures of surfaces with or without boundary. Communications on Pure & Applied Analysis, 2018, 17 (2) : 709-727. doi: 10.3934/cpaa.2018037
[15]

José Ginés Espín Buendía, Daniel Peralta-salas, Gabriel Soler López. Existence of minimal flows on nonorientable surfaces. Discrete & Continuous Dynamical Systems, 2017, 37 (8) : 4191-4211. doi: 10.3934/dcds.2017178
[16]

Gareth Ainsworth. The attenuated magnetic ray transform on surfaces. Inverse Problems & Imaging, 2013, 7 (1) : 27-46. doi: 10.3934/ipi.2013.7.27
[17]

François Béguin. Smale diffeomorphisms of surfaces: a classification algorithm. Discrete & Continuous Dynamical Systems, 2004, 11 (2&3) : 261-310. doi: 10.3934/dcds.2004.11.261
[18]

Robert S. Strichartz. Average error for spectral asymptotics on surfaces. Communications on Pure & Applied Analysis, 2016, 15 (1) : 9-39. doi: 10.3934/cpaa.2016.15.9
[19]

Gareth Ainsworth. The magnetic ray transform on Anosov surfaces. Discrete & Continuous Dynamical Systems, 2015, 35 (5) : 1801-1816. doi: 10.3934/dcds.2015.35.1801
[20]

Benjamin Dozier. Equidistribution of saddle connections on translation surfaces. Journal of Modern Dynamics, 2019, 14: 87-120. doi: 10.3934/jmd.2019004

 Impact Factor: 

Tools

Metrics

  • PDF downloads (49)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences