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

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Authors

  • Nail H. Ibragimov 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, Author
  • Ranis N. Ibragimov Department of Mathematics University of Texas at Brownsville, TX 78520, USA Author

DOI:

https://doi.org/10.5890/DNC.2013.11.002

Abstract

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.

References

[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] Ibragimov, N.H. and Ibragimov, R.N., (2012), "Rotationally symmetric internal gravity waves", International Journal of Non-Linear Mechanics, 47, 46-52,

[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] 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] 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] Thorpe, S.A.(1997), On the interactions of internal waves reflecting from slopes, Journal of Physical Oceanography, 27, 2072-2078.

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PublishedNovember 2013

Usage tracking begins September 1, 2026.

History Published

Issue

Vol. 2 No. 4 (2013): Regular Issue

Section

Research Articles

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How to Cite

Ibragimov, N. H., & Ibragimov, R. N. (2026). A Special Type of Invariant Solutions and its Connection with Dispersion Relations. Discontinuity, Nonlinearity, and Complexity, 2(4), 321-331. https://doi.org/10.5890/DNC.2013.11.002

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