Exact static solutions for discrete $\phi^4$ models free of the Peierls-Nabarro barrier: Discretized first integral approach

摘要

We propose a generalization of the discrete Klein-Gordon models free of thePeierls-Nabarro barrier derived in Nonlinearity {\bf 12}, 1373 (1999) and Phys.Rev. E {\bf 72}, 035602(R) (2005), such that they support not only kinks but aone-parameter set of exact static solutions. These solutions can be obtainediteratively from a two-point nonlinear map whose role is played by thediscretized first integral of the static Klein-Gordon field, as suggested in J.Phys. A {\bf 38}, 7617 (2005). We then discuss some discrete $\phi^4$ modelsfree of the Peierls-Nabarro barrier and identify for them the full space ofavailable static solutions, including those derived recently in Phys. Rev. E{\bf 72} 036605 (2005) but not limited to them. These findings are alsorelevant to standing wave solutions of discrete nonlinear Schr{\"o}dingermodels. We also study stability of the obtained solutions. As an interestingaside, we derive the list of solutions to the continuum $\phi^4$ equation thatfill the entire two-dimensional space of parameters obtained as the continuumlimit of the corresponding space of the discrete models.