Applicable Analysis: An International Journal

Bambusi, Dario; Paleari, Simone; Penati, Tiziano

doi: 10.1080/00036811003627518pmid: N/A

We construct small amplitude breathers in one-dimensional (1D) and two-dimensional (2D) Klein–Gordon (KG) infinite lattices. We also show that the breathers are well-approximated by the ground state of the nonlinear Schrödinger equation. The result is obtained by exploiting the relation between the KG lattice and the discrete nonlinear Schrödinger model. The proof is based on a Lyapunov–Schmidt decomposition and continuum approximation techniques introduced in [Bambusi and Penati, Continuous approximation of ground states in DNLS lattices, Nonlinearity 23 (2010), pp. 143–157], actually using its main result as an important lemma.

Belmonte-Beitia, Juan; Pelinovsky, Dmitry

doi: 10.1080/00036810903330538pmid: N/A

We address the Gross–Pitaevskii equation with a periodic linear potential and a periodic sign-varying nonlinearity coefficient. Contrary to the claims in the previous works, we show that the intersite cubic nonlinear terms in the discrete nonlinear Schrödinger (DNLS) equation appear beyond the applicability of assumptions of the tight-binding approximation. Instead of these terms, for an even linear potential and an odd nonlinearity coefficient, the DNLS equation and other reduced equations have the quintic nonlinear term, which correctly describes bifurcation of gap solitons in the semi-infinite gap.

Cuevas, J.; Karachalios, N.I.; Palmero, F.

doi: 10.1080/00036810903277135pmid: N/A

We discuss the existence of breathers and of energy thresholds for their formation in DNLS lattices with linear and nonlinear impurities. In the case of linear impurities, we present some new results concerning important differences between the attractive and repulsive impurity which is interplaying with a power nonlinearity. These differences concern the coexistence or the existence of staggered and unstaggered breather profile patterns. We also distinguish between the excitation threshold (the positive minimum of the power observed when the dimension of the lattice is greater or equal to some critical value) and explicit analytical lower bounds on the power (predicting the smallest value of the power a discrete breather one-parameter family), which are valid for any dimension. Extended numerical studies in one-, two- and three-dimensional lattices justify that the theoretical bounds can be considered as thresholds for the existence of the frequency parameterized families. The discussion reviews and extends the issue of the excitation threshold in lattices with nonlinear impurities while lower bounds, with respect to the kinetic energy, are also discussed for travelling waves in FPU periodic lattices.

Fečkan, Michal; Rothos, Vassilis M.

doi: 10.1080/00036810903208130pmid: N/A

Existence and bifurcation results are derived for quasi periodic travelling waves of discrete nonlinear Schrödinger equations with nonlocal interactions and with polynomial-type potentials. Variational tools are used. Several concrete nonlocal interactions are studied as well.

Giannoulis, Johannes

doi: 10.1080/00036811003649124pmid: N/A

We investigate the macroscopic dynamics of sets of an arbitrary finite number of weakly amplitude-modulated pulses in a multidimensional lattice of particles. The latter are assumed to exhibit scalar displacement under pairwise nonlinear interaction potentials of arbitrary range and are embedded in a nonlinear background field. By an appropriate multiscale ansatz, we derive formally the explicit evolution equations for the macroscopic amplitudes up to an arbitrarily high-order of the scaling parameter, thereby deducing the resonance and nonresonance conditions on the fixed wave vectors and frequencies of the pulses, which are required for that. The derived equations are justified rigorously in time intervals of macroscopic length. Finally, for sets of up to three pulses we present a complete list of all possible interactions and discuss their ramifications for the corresponding, explicitly given macroscopic systems.

James, Guillaume; Levitt, Antoine; Ferreira, Cynthia

doi: 10.1080/00036810903437788pmid: N/A

We study the existence of discrete breathers (time-periodic and spatially localized oscillations) in a chain of coupled nonlinear oscillators modelling the breathing of DNA. We consider a modification of the Peyrard–Bishop model introduced by Peyrard et al. [Nonlinear analysis of the dynamics of DNA breathing, J. Biol. Phys. 35 (2009), 73–89], in which the reclosing of base pairs is hindered by an energy barrier. Using a new kind of continuation from infinity, we prove for weak couplings the existence of large amplitude and low frequency breathers oscillating around a localized equilibrium, for breather frequencies lying outside resonance zones. These results are completed by numerical continuation. For resonant frequencies (with one multiple belonging to the phonon band) we numerically obtain discrete breathers superposed on a small oscillatory tail.

Kopylova, E.A.

doi: 10.1080/00036810903277176pmid: N/A

The long-time asymptotics is analysed for finite energy solutions of the 1D discrete Klein–Gordon equation coupled to a nonlinear oscillator. The coupled system is invariant with respect to the phase rotation group U(1). For initial states close to a solitary wave, the solution converges to a sum of another solitary wave and dispersive wave which is a solution to the free Klein–Gordon equation. The proofs develop the strategy of Buslaev–Perelman: the linearization of the dynamics on the solitary manifold, the symplectic orthogonal projection, method of majorants, etc.

Mielke, Alexander; Patz, Carsten

doi: 10.1080/00036810903517605pmid: N/A

We derive dispersive stability results for oscillator chains like the FPU chain or the discrete Klein–Gordon chain. If the nonlinearity is of degree higher than four, then small localized initial data decay exactly as in the linear case. For this, we provide sharp decay estimates for the linearized problem using oscillatory integrals and avoiding the nonoptimal interpolation between different ℓ p spaces.

N'Guérékata, Gaston M.; Pankov, Alexander

doi: 10.1080/00036810902889591pmid: N/A

We consider the initial value problem for the discrete non-linear Schrödinger equation in weighted l 2-spaces. Under quite general assumptions we prove that the problem is globally well-posed. The proof is based on simple general facts on abstract evolution equations.

Schneider, Guido

doi: 10.1080/00036810903277150pmid: N/A

We prove that the evolution of a slowly varying envelope of small amplitude of an underlying oscillating wave packet in the Fermi–Pasta–Ulam (FPU) system can be described approximately by the nonlinear Schrödinger equation. In contrast to other lattice equations for which this question has been addressed in the existing literature, the FPU system possesses a nontrivial quadratic resonance due to the curves of eigenvalues which vanish at the wave number k = 0. The proof of the error estimates is based on normal form transforms and a wave number-dependent scaling of the error function.