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Euler’s Figures and Extreme Waves
Published in Shamil U. Galiev, Evolution of Extreme Waves and Resonances, 2020
The sine-Gordon equation is a nonlinear hyperbolic partial differential equation involving the d’Alembert operator and the sine of the unknown function. The equation and several solution techniques were known in the nineteenth century in the course of study of various problems of differential geometry. (We stress that the very particular case of this equation was derived by Kirchhoff [9]. This version was used by him to model the elastica forms.) The equation grew greatly in importance in the 1970s, when it was realized that it led to solitons (so-called “kink” and “antikink”). The sine-Gordon equation appears in a number of physical applications, including applications in relativistic field theory, Josephson junctions, or mechanical transmission lines. For our research, the existence of multivalued solutions similar to Euler’s elastica figures is important, which began to discuss actively at the last time.
Propagation and Energy Transfer
Published in James D. Taylor, Introduction to Ultra-Wideband Radar Systems, 2020
Robert Roussel-Dupré, Terence W. Barrett
Why are solitons so special? Well, one reason is that the sine-Gordon soliton equation (which we will define below) is the unique equation invariant under the infinitesimal Lie transformations x → (1 − ε)x, t → (1 + ε)t, and the finite scale transformations x → a−1x, t → at.141 In other words, the sine-Gordon equation is invariant under certain precisely defined matched compensatory transformations. In laboratory coordinates it is also the unique equation which is Lorentz invariant. Certain solitons are solutions to the sine-Gordon equation. Equations with soliton solutions are examples of infinite dimensional completely integrable Hamiltonian systems with an infinity of polynomial conserved densities.
Solutions of local fractional sine-Gordon equations
Published in Waves in Random and Complex Media, 2019
H. Karayer, D. Demirhan, F. Buyukkilic
Since the noises just appear in the solutions of space fractional sine-Gordon equation, these noises are related to fractal structure of space. They occur due to structural property of relevant system when the fractional order tends to zero. It can be seen that this effect does not appear for greater values of . For a specific system one can talk about the origin of these noises. For instance, when solitary waves obtained from solution of sine-Gordon equation are studied in integer calculus, solitons propagate in a non-viscous medium. In order to give a more realistic picture for displacements of the solitary waves, we consider that the solitary waves move along a viscous medium where space is fractal and time is discrete. Thus it can be interpreted that the noises are associated with viscosity of the medium. Hereby, the effect of viscosity in the medium becomes prominent when approaches to zero.
Generalized Lavrentiev-type regularization method for the Cauchy problem of a semi-linear elliptic equation
Published in Inverse Problems in Science and Engineering, 2019
This problem has important applications in several areas of mathematical physics. For instance investigating the problem here we take , defined in space , then its eigenvalues are , the eigenfunctions are , and for , the inner product , the norm . If , it is the Helmholtz equation that arises in many engineering applications, such as acoustic, hydrodynamic and electromagnetic waves [1,2]. On the other research for the Cauchy problem of Helmholtz equation, we can see [3,4], etc. If we consider the linear case (the source term f does not rely on u), it becomes the Poisson equation that has been studied extensively in both pure and applied aspects. If taking , it can be deduced the classical nonlinear elliptic sine-Gordon equation, which is an integrable PDE. The nonlinear sine-Gordon equation appears in the theory of Josephson effects, superconductors, spin waves in ferromagnets, see [5,6]. This equation is also of considerable interest from a purely mathematical point of view. Indeed, it came into sight in the differential geometry and attracted a lot of attention because of the collisional behaviours of solitons that arise from this equation. In addition, as an example of a nonlinear integrable PDE, it is a time-independent PDE of elliptic type, and therefore it differs from most other nonlinear integrable models that describe a temporal evolution process. On the works for the general sine-Gordon model, we can see [7], etc. Motivated by the above reasons, in the present paper we consider the problem (1), and the assignment is to determine from it.
Solitons in the presence of a small, slowly varying perturbation
Published in Applicable Analysis, 2020
The perturbed sine-Gordon equation is a Hamiltonian evolution equation with Hamiltonian given by and the symplectic form given by where In first order formulation (1) can be written as a system: The unperturbed sine-Gordon equation (F=0), admits soliton solutions , where Here the functions are defined by where and satisfies with boundary conditions as (see [1,3]). The states form the two-dimensional solitary manifold as stressed in [1]. The perturbed sine-Gordon equation (1) arises in various physical phenomena, whereby the physical meaning of the variable θ and of the perturbation F is specific in each situation. Fluxon motion in a one-dimensional long Josephson junction can be described by (1), where the perturbation denotes the bias current density and θ the quantum phase difference between superconducting electrodes (see [4]). Another application of (1) is the description of a charge-density-wave system driven by an external electric field, where the perturbation denotes the electric field applied to the system and θ the phase of the charge-density-wave (see [5]). The perturbed sine-Gordon equation (1) finds also application in the dynamics of atoms adsorbed on a crystal surface (adatoms). Thereby the perturbation denotes an external electric driving force and θ the displacement of adatoms (see [6]). T. H. R. Skyrme [7] proposed the equation to model elementary particles. It has been shown that in many situations the solitons behave like particles driven by external forces, whereby the forces are expressed by the perturbation in (1) (see [8]). Further relevant results and applications of the perturbed Sine-Gordon equation can be found in [9–13].