| 1. |
- Hedberg, Claes, et al.
(author)
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Collisions, mutual losses and annihilation of pulses in a modular nonlinear medium
- 2017
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In: Nonlinear dynamics. - : Springer Netherlands. - 0924-090X .- 1573-269X. ; 90:3, s. 2083-2091
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Journal article (peer-reviewed)abstract
- 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)
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| 2. |
- Khamitova, Raisa
(author)
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Symmetries and conservation laws
- 2009
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Doctoral thesis (other academic/artistic)abstract
- Conservation laws play an important role in science. The aim of this thesis is to provide an overview and develop new methods for constructing conservation laws using Lie group theory. The derivation of conservation laws for invariant variational problems is based on Noether’s theorem. It is shown that the use of Lie-Bäcklund transformation groups allows one to reduce the number of basic conserved quantities for differential equations obtained by Noether’s theorem and construct a basis of conservation laws. Several examples on constructing a basis for some well-known equations are provided.Moreover, this approach allows one to obtain new conservation laws even for equations without Lagrangians. A formal Lagrangian can be introduced and used for computing nonlocal conservation laws. For self-adjoint or quasi-self-adjoint equations nonlocal conservation laws can be transformed into local conservation laws.One of the fields of applications of this approach is electromagnetic theory, namely, nonlocal conservation laws are obtained for the generalized Maxwell-Dirac equations. The theory is also applied to the nonlinear magma equation and its nonlocal conservation laws are computed.
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| 3. |
- Mikhailov, S. G., et al.
(author)
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A simple nonlinear element model
- 2017
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In: Acoustical Physics. - : MAIK NAUKA/INTERPERIODICA/SPRINGER. - 1063-7710 .- 1562-6865. ; 63:3, s. 270-274
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Journal article (peer-reviewed)abstract
- We study experimentally the behavior of a nonlinear element, a light plate pressed to the opening in the cavity of an acoustic resonator. Measurements of field oscillations inside and outside the cavity have shown that for large amplitudes, they become essentially anharmonic. The time dependences of displacement of the plate with increasing amplitude of the exciting voltage demonstrates a gradual change in the shape of vibrations from harmonic to half-period oscillation. A constant component appears in the cavity: rarefaction or outflow of the medium through the orifice. We construct a theory for nonlinear oscillations of a plate taking into account its different elastic reactions to compression and rarefaction with allowance for monopole radiation by the small-wave-size plate or radiation of a plane wave by the plate. We calculate the amplitudes of the harmonics and solve the problem of low-frequency stationary noise acting on the plate. We obtain expressions for the correlation function and mean power at the output given a normal random process at the input.
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| 4. |
- Nedic, Mitja, et al.
(author)
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Herglotz functions and applications in electromagnetics
- 2020
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In: Advances in Mathematical Methods for Electromagnetics. - : Institution of Engineering and Technology. - 9781785613845 - 9781785613852 ; , s. 491-514
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Book chapter (other academic/artistic)abstract
- Herglotz functions inevitably appear in pure mathematics, mathematical physics, and engineering with a wide range of applications. In particular, they are the pertinent functions to model passive systems, and thus appear in modeling of electromagnetic phenomena in circuits, antennas, materials, and scattering. In this chapter, we review the basic theory of Herglotz functions and its applications to determine sum rules and physical bounds for passive systems.
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| 5. |
- Rudenko, Oleg
(author)
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Equation admitting linearization and describing waves in dissipative media with modular, quadratic, and quadratically cubic nonlinearities
- 2016
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In: Doklady Mathematics. - : Maik Nauka/Interperiodica. - 1064-5624. ; 94:3, s. 703-707
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Journal article (peer-reviewed)abstract
- A second-order partial differential equation admitting exact linearization is discussed. It contains terms with nonlinearities of three types—modular, quadratic, and quadratically cubic—which can be present jointly or separately. The model describes nonlinear phenomena, some of which have been studied, while others call for further consideration. As an example, individual manifestations of modular nonlinearity are discussed. They lead to the formation of singularities of two types, namely, discontinuities in a function and discontinuities in its derivative, which are eliminated by dissipative smoothing. The dynamics of shock fronts is studied. The collision of two single pulses of different polarity is described. The process reveals new properties other than those of elastic collisions of conservative solitons and inelastic collisions of dissipative shock waves.
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| 6. |
- Rudenko, Oleg
(author)
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Modular solutions
- 2016
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In: Doklady Mathematics. - : Maik Nauka/Interperiodica. - 1064-5624. ; 94:3, s. 708-711
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Journal article (peer-reviewed)abstract
- Solutions to a partial differential equation of the third order containing the modular nonlinearity are studied. The model describes, in particular, elastic waves in media with weak high-frequency dispersion and with different response to tensile and compressive stresses. This equation is linear for solutions preserving their sign. Nonlinear phenomena only manifest themselves to alternating solutions. Stationary solutions in the form of solitary waves or solitons are found. It is shown how the linear periodic wave becomes nonlinear after exceeding a certain critical value of the amplitude, and how it transforms into a soliton with further increase in the amplitude.
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| 7. |
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