Excitation of Large-Amplitude Localized Nonlinear Waves by the Interaction of Kinks of the Sine-Gordon Equation with Attracting Impurity

Abstract

The generation and evolution of localized waves on an impurity in the scattering of a kink of the sine-Gordon equation are studied. It is shown that the problem can be considered as excitation of oscillations of a harmonic oscillator by a short external impulse. The external impulse is modeled by the scattering of a kink on an impurity. The influence of the modes of motion of a kink on the excitation energy of localized waves is numerically and analytically studied. The method of collective coordinate for the analytical study is used. The value of this energy is determined by the ratio of the impurity parameters and the initial kink velocity. It is shown that the dependence of the energy (and amplitude) of the generated localized waves

Changing the Dynamic Parameters of Localized Breather and Soliton Waves in the Sine-Gordon Model with Extended Impurity, External Force, and Decay in the Autoresonance Mode

“…For some applications, it is necessary to account for dissipation and external force in the system (e. g., [1][2][3][10][11][12][13][14]). Parameters of SGE are also often set as functions of coordinates and time [15][16][17][18][19][20]. In some cases (for small perturbations) the structure, static and dynamic properties of localized waves in such SGE models may be studied analytically [1][2][3][16][17][18][19][20][21][22][23][24][25].…”

“…Parameters of SGE are also often set as functions of coordinates and time [15][16][17][18][19][20]. In some cases (for small perturbations) the structure, static and dynamic properties of localized waves in such SGE models may be studied analytically [1][2][3][16][17][18][19][20][21][22][23][24][25]. However, it is more often necessary to use numerical methods [1-3, 13-15, 22, 23].…”

“…Many articles were devoted to spatial modulation (heterogeneity) of periodic potential, or the presence of an impurity in the system (e. g., [3,[13][14][15][16][18][19][20][21][22][23][24][25]). Impurities acting on a kink as the attracting potential lead to the excitation of different impurity modes -localized oscillating state on an impurity.…”