Exact static solutions for discrete ϕ4\phi^4 models free of the Peierls-Nabarro barrier: Discretized first integral approach

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.Comment: Accepted for publication in PRE; the M/S has been revised in line with the referee repor

Quantum interference in the classically forbidden region: a parametric oscillator

We study tunneling between period two states of a parametrically modulated oscillator. The tunneling matrix element is shown to oscillate with the varying frequency of the modulating field. The effect is due to spatial oscillations of the wave function and the related interference in the classically forbidden region. The oscillations emerge already in the ground state of the oscillator Hamiltonian in the rotating frame, which is quartic in the momentum.Comment: Submitted to PR

Discrete Nonlinear Schrodinger Equations Free of the Peierls-Nabarro Potential

We derive a class of discrete nonlinear Schr{\"o}dinger (DNLS) equations for general polynomial nonlinearity whose stationary solutions can be found from a reduced two-point algebraic problem. It is demonstrated that the derived class of discretizations contains subclasses conserving classical norm or a modified norm and classical momentum. These equations are interesting from the physical standpoint since they support stationary discrete solitons free of the Peierls-Nabarro potential. As a consequence, even in highly-discrete regimes, solitons are not trapped by the lattice and they can be accelerated by even weak external fields. Focusing on the cubic nonlinearity we then consider a small perturbation around stationary soliton solutions and, solving corresponding eigenvalue problem, we (i) demonstrate that solitons are stable; (ii) show that they have two additional zero-frequency modes responsible for their effective translational invariance; (iii) derive semi-analytical solutions for discrete solitons moving at slow speed. To highlight the unusual properties of solitons in the new discrete models we compare them with that of the classical DNLS equation giving several numerical examples.Comment: Misprints noticed in the journal publication are corrected [in Eq. (1) and Eq. (34)