Exact static solutions for discrete phi4 models free of the Peierls-Nabarro barrier: discretized first-integral...

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-Nabarro barrier derived in Spreight [Nonlinearity 12, 1373 (1999)] and Barashenkov [Phys. Rev. E 72, 035602(R) (2005)], such that they support not only kinks but a one-parameter set of exact static solutions. These solutions can be obtained iteratively from a two-point nonlinear map whose role is played by the discretized first integral of the static Klein-Gordon field, as suggested by Dmitriev [J. Phys. A 38, 7617 (2005)]. We then discuss some discrete phi4 models free of the Peierls-Nabarro barrier and identify for them the full space of available static solutions, including those derived recently by Cooper [Phys. Rev. E 72, 036605 (2005)] but not limited to them. These findings are also relevant to standing wave solutions of discrete nonlinear Schrödinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum phi4 equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.