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

Abstract

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 ${\varphi }^4$ 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{ö}dinger models. We also study stability of the obtained solutions. As an interesting aside, we derive the list of solutions to the continuum ${\varphi }^4$ equation that fill the entire two-dimensional space of parameters obtained as the continuum limit of the corresponding space of the discrete models.

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

References in corpus (1)

• Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

Cited by in corpus (5)

• Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential
• Comparative Study of Different Discretizations of the ${\varphi }^4$ Model
• High-speed kinks in a generalized discrete ${\varphi }^4$ model
• Exact static solutions to a translationally invariant discrete ${\varphi }^4$ model
• Exact Moving and Stationary Solutions of a Generalized Discrete Nonlinear Schrodinger Equation