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  • 1.
    Hedberg, Claes
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Rudenko, Oleg
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering.
    Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium2017In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 90, no 3, p. 2083-2091Article in journal (Refereed)
    Abstract [en]

    One of the most important sections of nonlinear wave theory is related to the collisions of single pulses. These often exhibit corpuscular properties. For example, it is well known that solitons described by the Korteweg–de Vries equation and a few other conservative model equations exhibit properties of elastic particles, while shock waves described by dissipative models like Burgers’ equation stick together as absolutely inelastic particles when colliding. The interactions of single pulses in media with modular nonlinearity considered here reveal new physical features that are still poorly understood. There is an analogy between the single pulses collision and the interaction of clots of chemical reactants, such as fuel and oxidant, where the smaller component disappears and the larger one decreases after a reaction. At equal “masses” both clots can be annihilated. In this work various interactions of two and three pulses are considered. The conditions for which a complete annihilation of the pulses occurs are indicated. © 2017 The Author(s)

  • 2. Hedberg, Claes
    et al.
    Rudenko, Oleg
    Interaction between low and high-frequency Modes in a Nonlinear System: Gas-Filled Cylinder Covered by a movable Piston2003In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 32, no 4, p. 405-416Article in journal (Refereed)
    Abstract [en]

    A simple mechanical system containing a low-frequency vibration mode and set of high-frequency acoustic modes is considered. The frequency response is calculated. Nonlinear behaviour and interaction between modes is described by system of functional equations. Two types of nonlinearities are taken into account. The first one is caused by the finite displacement of a movable boundary, and the second one is the volume nonlinearity of gas. New mathematical models based on nonlinear equations are suggested. Some examples of nonlinear phenomena are discussed on the base of derived solutions.

  • 3. Ibragimov, Nail H.
    Laplace type invariants for parabolic equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 125-133Article in journal (Refereed)
    Abstract [en]

    The Laplace invariants pertain to linear hyperbolic differential equations with two independent variables. They were discovered by Laplace in 1773 and used in his integration theory of hyperbolic equations. Cotton extended the Laplace invariants to elliptic equations in 1900. Cotton's invariants can be obtained from the Laplace invariants merely by the complex change of variables relating the elliptic and hyperbolic equations. To the best of my knowledge, the invariants for parabolic equations were not found thus far. The purpose of this paper is to fill this gap by considering what will be called Laplace type invariants (or seminvariants), i.e. the quantities that remain unaltered under the linear transformation of the dependent variable. Laplace type invariants are calculated here for all hyperbolic, elliptic and parabolic equations using the unified infinitesimal method. A new invariant is found for parabolic equations.

  • 4. Ibragimov, Nail H.
    et al.
    Kovalev, Vladimir
    Pustovalov, V.V.
    Symmetries of integro-differential equations: A survey of methods illustrated by the Benny equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 135-153Article in journal (Refereed)
    Abstract [en]

    Classical Lie group theory provides a universal tool for calculating symmetry groups for systems of differential equations. However Lie's method is not as much effective in the case of integral or integro-differential equations as well as in the case of infinite systems of differential equations. This paper is aimed to survey the modern approaches to symmetries of integro-differential equations. As an illustration, an infinite symmetry Lie algebra is calculated for a system of integro-differential equations, namely the well-known Benny equations. The crucial idea is to look for symmetry generators in the form of canonical Lie-Backlund operators.

  • 5. Ibragimov, Nail H.
    et al.
    Magri, F.
    Geometric proof of Lie's linearization theorem2004In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 36, no 1, p. 41-46Article in journal (Refereed)
    Abstract [en]

    S. Lie found in 1883 the general form of all second-order ordinary differential equations transformable to the linear equation by a change of variables and proved that their solution reduces to integration of a linear third-order ordinary differential equation. He showed that the linearizable equations are at most cubic in the first-order derivative and described a general procedure for constructing linearizing transformations by using an over-determined system of four equations. We present here a simple geometric proof of the theorem, known as Lie's linearization test, stating that the compatibility of Lie's four auxiliary equations furnishes a necessary and sufficient condition for linearization.

  • 6. Ibragimov, Nail H.
    et al.
    Svirshchevskii, SR
    Lie-Backlund symmetries of submaximal order of ordinary differential equations2002In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, p. 155-166Article in journal (Refereed)
    Abstract [en]

    It is well known that the maximal order of Lie-Backlund symmetries for any nth-order ordinary differential equation is equal to n-1, and that the whole set of such symmetries forms an infinite-dimensional Lie algebra. Symmetries of the order pless than or equal ton - 2 span a linear subspace (but not a subalgebra) in this algebra. We call them symmetries of submaximal order. The purpose of the article is to prove that for n less than or equal to 4 this subspace is finite-dimensional and it's dimension cannot be greater than 35 for n=4, 10 for n=3 and 3 for n=2. In the case n=3 this statement follows immediately from Lie's result on contact symmetries of third-order ordinary differential equations. The maximal values of dimensions are reached, e.g., on the simplest equations y((n))=0.

  • 7.
    Rudenko, Oleg
    et al.
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering. Blekinge Institute of Technology, School of Engineering, Department of Mechanical Engineering.
    Hedberg, Claes
    Blekinge Institute of Technology, Faculty of Engineering, Department of Mechanical Engineering. Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Science. Blekinge Institute of Technology, School of Engineering, Department of Mechanical Engineering. Blekinge Institute of Technology, School of Engineering, Department of Mathematics and Natural Sciences. Blekinge Institute of Technology, Department of Telecommunications and Mathematics.
    A new equation and exact solutions describing focal fields in media with modular nonlinearity2017In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 89, no 3, p. 1905-1913Article in journal (Refereed)
    Abstract [en]

    Brand-new equations which can be regarded as modifications of Khokhlov–Zabolotskaya–Kuznetsov or Ostrovsky–Vakhnenko equations are suggested. These equations are quite general in that they describe the nonlinear wave dynamics in media with modular nonlinearity. Such media exist among composites, meta-materials, inhomogeneous and multiphase systems. These new models are interesting because of two reasons: (1) the equations admit exact analytic solutions and (2) the solutions describe real physical phenomena. The equations model nonlinear focusing of wave beams. It is shown that inside the focal zone a stationary waveform exists. Steady-state profiles are constructed by the matching of functions describing the positive and negative branches of exact solutions of an equation of Klein–Gordon type. Such profiles have been observed many times during experiments and numerical studies. The non-stationary waves can contain singularities of two types: discontinuity of the wave and of its derivative. These singularities are eliminated by introducing dissipative terms into the equations—thereby increasing their order. © 2017 The Author(s)

  • 8. Rudenko, Oleg
    et al.
    Hedberg, Claes
    Nonlinear Dynamics of Grains in a Liquid-Saturated Soil2003In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 35, no 2, p. 187-200Article in journal (Refereed)
    Abstract [en]

    A new kind of nonlinearity of inertial type caused by accelerated motion of interacting particles is described. The model deals with an ensemble of grains immersed into a vibrating fluid. First, the nonlinear vibration of two connected grains is studied. The temporal behaviours of displacement and velocity, as well as spectrum of vibration, are analysed. Numerical simulations are performed. Then an infinite chain of grains is considered and the corresponding differential-difference equation is derived. For the continuum limit the inhomogeneous nonlinear wave equation is solved and temporal profiles are calculated. A new resonant phenomenon is described and the resonant curves are constructed.

  • 9.
    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.
    The quadratically cubic Burgers equation: an exactly solvable nonlinear model for shocks, pulses and periodic waves2016In: Nonlinear dynamics, ISSN 0924-090X, E-ISSN 1573-269X, Vol. 85, no 2, p. 767-776Article in journal (Refereed)
    Abstract [en]

    A modified equation of Burgers type with a quadratically cubic (QC) nonlinear term was recently pointed out as a new exactly solvable model of mathematical physics. However, its derivation, analytical solution, computer modeling, as well as its physical applications and analysis of corresponding nonlinear wave phenomena have not been published up to now. The physical meaning and generality of this QC nonlinearity are illustrated here by several examples and experimental results. The QC equation can be linearized and it describes the experimentally observed phenomena. Some of its exact solutions are given. It is shown that in a QC medium not only shocks of compression can be stable, but shocks of rarefaction as well. The formation of stationary waves with finite width of shock front resulting from the competition between nonlinearity and dissipation is traced. Single-pulse propagation is studied by computer modeling. The nonlinear evolutions of N- and S-waves in a dissipative QC medium are described, and the transformation of a harmonic wave to a sawtooth-shaped wave with periodically recurring trapezoidal teeth is analyzed. © 2016 The Author(s)

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