Comparative study of different discretizations of theϕ4model

Abstract

We examine various recently proposed discretizations of the well-known φ 4 field theory. We compare and contrast the properties of their fundamental solutions including the nature of their kink-type solitary waves and the spectral properties of the linearization around such waves. We study these features as a function of the lattice spacing h, as one deviates from the continuum limit of h → 0. We then proceed to a more "stringent" comparison of the models, by discussing the scattering properties of a kink-antikink pair for the different discretizations. These collisions are well-known to possess properties that quite sensitively depend on the initial speed even at the continuum limit. We examine how typical model behaviors are modified in the presence (and as a function) of discreteness and attempt to extract qualitative trends and issue pertinent warnings about some of the surprising resulting properties.

Formation of martensite crystals in the limiting case of a supersonic growth rate

“…It should be kept in mind that when coordinating the modes (especially in the short-wave range) providing for the selection of a specific variant from the spectrum of possible polytypic MT, an explicit account of the lattice discreteness and an accurate description of the frequency spectrum become essential during the construction of non-linear CWP models. Here the accumulated experience (see, e.g., [33][34][35][36]) in studying nonlinear waves in discrete media is in full measure probable to be in demand, both within an analytical description and during dynamic modeling.…”

Translationally invariant kink solutions of discrete ϕ4 models

“…x ), were constructed and investigated by a large number of authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17]. To begin with, we would like to note that here we mean nonintegrable chains (such as the Toda chain) [18]).…”

Soliton collision in discrete PT-symmetric systems without Peierls-Nabarro potential

“…Известно, что в большинстве случаев дискрет-ные системы обладают потенциалом Пайерлса-Набарро (пПН), что снижает подвижность сигналов в таких си-стемах. В работах [6][7][8][9][10] было показано, что, в некоторых случаях, можно модифицировать дискретную модель та-ким образом, что континуальный предел остается неиз-менным, но для нее становится возможным построить точные решения. В таких моделях отсутствует пПН, и такая модификация системы позволяет изучить дина-мику взаимодействия солитонов более точно, посколь-ку, располагая точным решением, легко задать началь-ные условия, описывающие движущиеся солитоны, и, кроме того, солитоны в таких системах двигаются, не излучая энергию.…”