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

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

2006-03-31

arxiv.org/abs/nlin/0603074v2

Nonlinear Sciences

Pattern Formation and Solitons

Accepted for publication in PRE; the M/S has been revised in line with the referee report

Scientific paper

10.1103/PhysRevE.74.046609

We propose a generalization of the discrete Klein-Gordon models free of the Peierls-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 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 in J. Phys. A {\bf 38}, 7617 (2005). We then discuss some discrete $\phi^4$ models free of the Peierls-Nabarro barrier and identify for them the full space of available static solutions, including those derived recently in Phys. Rev. E {\bf 72} 036605 (2005) but not limited to them. These findings are also relevant to standing wave solutions of discrete nonlinear Schr{\"o}dinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum $\phi^4$ equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.

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