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Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA


Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania


A Special Type of Invariant Solutions and its Connection with Dispersion Relations

Discontinuity, Nonlinearity, and Complexity 2(4) (2013) 321--331 | DOI:10.5890/DNC.2013.11.002

Nail H. Ibragimov$^{1}$; Ranis N. Ibragimov$^{2}$

$^{1}$ Laboratory “Group analysis of mathematical models in natural and engineering sciences”, Ufa State Aviation Technical University, 12, K. Marx Str., 450000 Ufa, Russia Research centre ALGA, Department of Mathematics and Science, Blekinge Institute of Technology,

SE-371 79 Karlskrona, Sweden

$^{2}$ Department of Mathematics University of Texas at Brownsville, TX 78520, USA

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The concept of dispersion relations is widely used in physics and ap- plied mathematics in investigating wave type solutions of differential equations. On the other hand, Lie group analysis provides another useful method for constructing exact solutions of linear and nonlinear differential equations via the concept of invariant solutions. We show in the present paper that for certain types of differential equations there is a remarkable connection between these two concepts. Namely, the idea of dispersion relations leads to a special type of invariant solutions.


NHI acknowledges a financial support of the Government of Russian Federation through Resolution No.220, Agreement No. 11.G34.31.0042.


  1. [1]  Ibragimov, N.H. and Ibragimov, R.N. (2011) Group analysis of nonlinear internal waves in oceans. I: Lagrangian, conservation laws, invariant solutions. Archives of ALGA 6,19-44 arXiv: math-ph, 1108-1877v1, 1-28.
  2. [2]  Ibragimov, N.H. and Ibragimov, R.N., (2012), "Rotationally symmetric internal gravity waves", International Journal of Non-Linear Mechanics, 47, 46-52,
  3. [3]  Ibragimov, N.H and Ibragimov, R.N. (2011), Applications of Lie Group Analysis in Geophysical Fluid Dynamics, Higher Education Press & World Scientific Publishers, Beijing& Singapore.
  4. [4]  Ibragimov, N.H. and Ibragimov, R.N. (2011), Integration by quadratures of the nonlinear Euler quations modeling atmospheric flows in a thin rotating spherical shell, Physics Letters A, 375, 3858-3865.
  5. [5]  Ibragimov, R.N. and Ibragimov, N.H.(2010), Effects of rotation on self-resonant internal gravity waves in the ocean, Ocean Modelling, 31, 80-87.
  6. [6]  Thorpe, S.A.(1997), On the interactions of internal waves reflecting from slopes, Journal of Physical Oceanography, 27, 2072-2078.