Stability of multiple pulses in discrete systems

Abstract

The stability of multiple-pulse solutions to the discrete nonlinear Schrödinger equation is considered. A bound state of widely separated single pulses is rigorously shown to be unstable, unless the phase shift Delta phi between adjacent pulses satisfies Delta phi=pi. This instability is accounted for by positive real eigenvalues in the linearized system. The analysis leading to the instability result does not, however, determine the linear stability of those multiple pulses for which Delta phi=pi between adjacent pulses. A direct variational approach for a two-pulse predicts that it is linearly stable if Delta phi=pi, and if the separation between the individual pulses satisfies a certain condition. The variational approach can easily be generalized to study the stability of N pulses for any N>or=3. The analysis is supplemented with a detailed numerical stability analysis.

Instabilities and bifurcations of nonlinear impurity modes

“…(3.5) in [29], as well as in references therein. Finally, it is worth mentioning that the series in Eq.…”

Two-dimensional paradigm for symmetry breaking: The nonlinear Schrödinger equation with a four-well potential

“…[24,25] for the corresponding 1D and 3D stability results). Gross features of these findings are that, whenever two adjacent sites are in-phase, a real eigenvalue pair is expected to emerge due to their interaction, while whenever such sites are out-ofphase, the relevant eigenvalue is expected to be imaginary [26], but with negative Krein signature [22], which implies a potential for a Hamiltonian-Hopf bifurcation. It should also be noted that, in the limit of the infinite lattice, it is naturally expected that the asymmetries observed herein in many of the branches will disappear (i.e., the amplitudes in different wells will be equal) -see also a relevant discussion in Ref.…”

Asymptotic calculation of discrete non-linear wave interactions

“…(12)- (13) are retrieved, but with the RHS expression multiplied by cos(φ). This is rather natural as in-phase ALNLS solitons are expected to attract each other, while out-of-phase ones are expected to experience mutual repulsion [19,20,21]. …”

“…We should mention here that for continuum systems, there exists a large variety of methods for computing such interactions. These range from the perturbation theoretical works of [18], to the variational methods of [19,20,21], the Fredholm-alternative based technique of [22] or the more rigorous calculations of [23] based on Lin's method. In a recent publication [24], we took an alternative route to these methods, by implementing an asymptotic calculation using the approach proposed by Manton [25] (which, in turn, was generalizing the earlier work of [26]).…”

“…One of the complications is that, aside from the tail-tail interaction of the waves, each wave may also reside on a periodic substrate potential, usually termed the Peierls-Nabarro barrier [12]. As a result, except for the (typically exponential in the wave separation) tail-induced potential, a local potential may arise due to discreteness, resulting in a washboard potential structure which can be captured by variational techniques [20,21].…”