Nonlinear Sciences – Pattern Formation and Solitons
Scientific paper
2005-06-08
Nonlinearity 19 (2006) 217-235
Nonlinear Sciences
Pattern Formation and Solitons
To be published in Nonlinearity. 22 pages, 5 figures. Extensive clarifications to the text have been made
Scientific paper
10.1088/0951-7715/19/1/011
In recent years, three exceptional discretizations of the phi^4 theory have been discovered [J.M. Speight and R.S. Ward, Nonlinearity 7, 475 (1994); C.M. Bender and A. Tovbis, J. Math. Phys. 38, 3700 (1997); P.G. Kevrekidis, Physica D 183, 68 (2003)] which support translationally invariant kinks, i.e. families of stationary kinks centred at arbitrary points between the lattice sites. It has been suggested that the translationally invariant stationary kinks may persist as 'sliding kinks', i.e. discrete kinks travelling at nonzero velocities without experiencing any radiation damping. The purpose of this study is to check whether this is indeed the case. By computing the Stokes constants in beyond-all-order asymptotic expansions, we prove that the three exceptional discretizations do not support sliding kinks for most values of the velocity - just like the standard, one-site, discretization. There are, however, isolated values of velocity for which radiationless kink propagation becomes possible. There is one such value for the discretization of Speight and Ward and three 'sliding velocities' for the model of Kevrekedis.
Barashenkov Igor V.
Oxtoby O. F.
Pelinovsky Dmitry E.
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