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  • 1.
    Ibragimov, Nail H.
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Ibragimov, Ranis N.
    Galiakberova, L. R.
    Symmetries and Conservation Laws of a Spectral Nonlinear Model for Atmospheric Baroclinic Jets2014In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 9, no 5, p. 111-118Article in journal (Refereed)
    Abstract [en]

    In this paper, we shall obtain the symmetries of the mathematical model describing spontaneous relaxation of eastward jets into a meandering state and use these symmetries for constructing the conservation laws. The basic eastward jet is a spectral parameter of the model, which is in geostrophic equilibrium with the basic density structure and which guarantees the existence of nontrivial conservation laws.

  • 2. Ibragimov, Nail
    et al.
    Ibragimov, Ranis
    Applications of Lie Group Analysis to Mathematical Modelling in Natural Sciences2012In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 7, no 2, p. 52-65Article in journal (Refereed)
    Abstract [en]

    Today engineering and science researchers routinely confront problems in mathematical modeling involving solutions techniques for differential equations. Sometimes these solutions can be obtained analytically by numerous traditional ad hoc methods appropriate for integrating particular types of equations. More often, however, the solutions cannot be obtained by these methods, in spite of the fact that, e.g. over 400 types of integrable second-order ordinary differential equations were summarized in voluminous catalogues. On the other hand, many mathematical models formulated in terms of nonlinear differential equations can successfully be treated and solved by Lie group methods. Lie group analysis is especially valuable in investigating nonlinear differential equations, for its algorithms act here as reliably as for linear cases. The aim of this article is, from the one hand, to provide the wide audience of researchers with the comprehensive introduction to Lie's group analysis and, from the other hand, is to illustrate the advantages of application of Lie group analysis to group theoretical modeling of internal gravity waves in stratified fluids.

  • 3.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mathematics and Natural Sciences.
    Ibragimov, Ranis N.
    Univ Wisconsin Parkside, USA.
    Kovalev, Vladimir F.
    Russian Acad Sci, RUS.
    INVARIANT SOLUTIONS AND SHOCK ATMOSPHERIC WAVES IN A THIN CIRCULAR LAYER2018In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 13, no 2, article id UNSP 19Article in journal (Refereed)
    Abstract [en]

    The objective of this paper is to investigate the nonlinear mathematical model describing equatorial waves from Lie group analysis point of view in order to understand the nature of shallow water model theory, which is associated to planetary equatorial waves. Such waves correspond to the Cauchy-Poisson free boundary problem on the nonstationary motion of a perfect incompressible fluid circulating around a solid circle of a large radius.

  • 4.
    Ibragimov, Nail
    et al.
    Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences.
    Ibragimov, R.N.
    Bifurcation of Nonlinear Conservation Laws from the Classical Energy Conservation Law for Internal Gravity Waves in Cylindrical Wave Field2013In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 8, no 5, p. 119-130Article in journal (Refereed)
    Abstract [en]

    New conservation laws bifurcating from the classical form of conservation laws are constructed to the nonlinear Boussinesq model describing internal Kelvin waves propagating in a cylindrical wave field of an uniformly stratified water affected by the earth's rotation. The obtained conservation laws are different from the well known energy conservation law for internal waves and they are associated with symmetries of the Boussinesq model. Particularly, it is shown that application of Lie group analysis provide three infinite sets of nontrivial integral conservation laws depending on two arbitrary functions, namely a(t, theta),b(t, r) and an arbitrary function c(t, theta, r) which is given implicitly as a nontrivial solution of a partial differential equation involving a(t, theta) and b(t,r).

  • 5.
    Rudenko, Oleg
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Hedberg, Claes
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering. Blekinge Inst Technol, S-37179 Karlskrona, Sweden..
    SINGLE SHOCK AND PERIODIC SAWTOOTH-SHAPED WAVES IN MEDIA WITH NON-ANALYTIC NONLINEARITIES2018In: Mathematical Modelling of Natural Phenomena, ISSN 0973-5348, E-ISSN 1760-6101, Vol. 13, no 2, article id UNSP 18Article in journal (Refereed)
    Abstract [en]

    The review of new mathematical models containing non-analytic nonlinearities is given. These equations have been proposed recently, over the past few years. The models describe strongly nonlinear waves of the first type, according to the classification introduced earlier by the authors. These models are interesting because of two reasons: (i) equations admit exact analytic solutions, and (ii) solutions describe the real physical phenomena. Among these models are modular and quadratically cubic equations of Hopf, Burgers, Korteveg-de Vries, Khokhlov-Zabolotskaya and Ostrovsky-Vakhnenko type. Media with non-analytic nonlinearities exist among composites, meta-materials, inhomogeneous and multiphase systems. Some physical phenomena manifested in the propagation of waves in such media are described on the qualitative level of severity.

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