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Solitons in a modified discrete nonlinear Schrödinger equation

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Solitons in a modified discrete nonlinear Schrödinger equation

Examination of the time evolution of discrete solitons in the limit of strongly localized modes, suggests ways to manage the Peierls-Nabarro barrier, facilitating in this way a degree of soliton steering. The long-time propagation of an initially localized excitation shows that, at long evolution times, nonlinear effects become negligible and as a result, the propagation becomes ballistic. The qualitative similarity of the results for the mDNLS to the ones obtained for the standard DNLS, suggests that this kind of discrete soliton is an robust entity capable of transporting an excitation across a generic discrete medium that models several systems of interest.

In the semiclassical approach to the coupled electron-phonon problem, the electronic degrees of freedom are coupled to the vibrational ones, where the latter are pictured as classical oscillators. A further approximation assumes that these oscillators are enslaved to the electron, thus reducing the number of equations which now contain only electronic coordinates.

Identical equation appears in other systems such as coupled waveguide arrays in optics 8 , 10 , 11 , BECs in coupled magneto-optical traps 13 , 14 and biomolecules 15 — 19 , to name a few. On the other hand, when the oscillators are taken as of the Debye, or acoustic type, one arrives to the less-known Modified Discrete Nonlinear mDNLS equation:. Hereafter, we will dispense with the physical origin of the mDNLS and proceed to focus on its selftrapping and transport properties, and show that its soliton phenomenology is similar to the one found previously in DNLS.

This is very important since it supports the idea that a discrete soliton is a robust excitation of the system, regardless of the precise nature of the underlying vibrational degrees of freedom. Our system is a Hamiltonian one since from Eq. The mDNLS was first found in earlier studies of polaron formation, from the coupled electron-phonon equations in the adiabatic limit 20 , The mDNLS has been used to study certain recurrences that occur in the coupled electron-phonon problem 21 , The selftrapping properties of the mDNLS were also examined in ref.

In this work we will focus on the different families of nonlinear bulk and surface modes and their stability properties, transport exponents and on the propagation of mDNLS solitons, looking for possible means of propagation control. The modes shown here have different power content. Thus, the linear stability of the nonlinear modes is determined by the eigenvalue spectra of the matrices AB and BA. A convenient parameter to quantify the stability of a mode is the instability gain G, defined as.


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When G is zero, the mode is stable; otherwise it is unstable. This parameter is nothing else but the largest growth rate of the mode and is given by the imaginary part of the square root of the complex eigenvalue of A B or B A. We note that, while for the bulk modes, there are at least two modes with no threshold power, for the surface modes they all require a minimum power threshold nonlinearity to exist. The only stable lowest-order bulk mode is the odd one, which is stable all the way down to the linear band. For the surface modes we observe that they all require a minimum power threshold to exist.

Power content versus eigenvalue for the nonlinear modes of Fig. Continuous dashed curves denote stable unstable modes.

Introduction

We consider here the propagation of an approximate soliton solution using Eq. As an initial condition we will use the form Open in a separate window which is a discretization of the exact continuous NLS one-soliton solution. Parameter k represents the initial momentum of the pulse and n c is the position of the soliton center. This ansatz is reasonable for wide solitons where the discrete character of the lattice is of no consequence. In the case of large kick, the soliton propagates across the lattice and bounces elastically from the ends of the chain. For low values of k , the soliton propagates some short distance and gets selftrapped eventually around some lattice site.

It should be noted that similar results are obtained for more generic spatial profiles that are localized in space and endowed with an initial kick.

1. Introduction

One example of this is using the profile corresponding to the fundamental stationary mode. One of the major problems for achieving controllable steering of discrete solitons is the existence of an effective periodic potential, known as the Peierls-Nabarro PN potential, that appears as a result of lattice discreteness.


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While in the continuous case, the presence of translational invariance favors soliton propagation, in a discrete system a minimum impulse is needed to effect soliton motion. The magnitude of the PN potential can be roughly estimated as A 4 , where A is the soliton amplitude We can shed some light into this problem by the use of strongly localized modes SLMs 24 — We consider a stationary odd SLM in the form.

Now we assume that the odd and even SLM are different states of the same soliton. This implies that both SLMs possess the same norm power. Therefore, Open in a separate window. The even Hamiltonian becomes. We see that, to a first approximation, the barrier could be tuned by an appropriate choice of the amplitude, momentum and nonlinearity parameter. If our objective is not to effect a free propagation, but to deliver the soliton at a given location where it will remain due to selftrapping , like in a multiport switching, one could in principle, resort to an engineering of the couplings 27 , 28 to bring the soliton from a given position to any desired site.

When dealing with the dynamical evolution of the mDLNS equation, it is natural to ask under which circumstance the system will create discrete solitons instead of radiation. A clue about this comes from examining the linear stability of an initially uniform nonlinear profile. When this profile becomes unstable, the profile will tend to fragment and the largest fragments could serve as seeds for discrete solitons. Usually this depends on the strength of nonlinearity.

In the case of large kick, the soliton propagates across the lattice and bounces elastically from the ends of the chain. For low values of k , the soliton propagates some short distance and gets selftrapped eventually around some lattice site. It should be noted that similar results are obtained for more generic spatial profiles that are localized in space and endowed with an initial kick.

One example of this is using the profile corresponding to the fundamental stationary mode. One of the major problems for achieving controllable steering of discrete solitons is the existence of an effective periodic potential, known as the Peierls-Nabarro PN potential, that appears as a result of lattice discreteness. While in the continuous case, the presence of translational invariance favors soliton propagation, in a discrete system a minimum impulse is needed to effect soliton motion.

The magnitude of the PN potential can be roughly estimated as A 4 , where A is the soliton amplitude We can shed some light into this problem by the use of strongly localized modes SLMs 24 — We consider a stationary odd SLM in the form. Now we assume that the odd and even SLM are different states of the same soliton. This implies that both SLMs possess the same norm power. Therefore, Open in a separate window.

Nonlinear Schrodinger systems: continuous and discrete - Scholarpedia

The even Hamiltonian becomes. We see that, to a first approximation, the barrier could be tuned by an appropriate choice of the amplitude, momentum and nonlinearity parameter.

If our objective is not to effect a free propagation, but to deliver the soliton at a given location where it will remain due to selftrapping , like in a multiport switching, one could in principle, resort to an engineering of the couplings 27 , 28 to bring the soliton from a given position to any desired site. When dealing with the dynamical evolution of the mDLNS equation, it is natural to ask under which circumstance the system will create discrete solitons instead of radiation.

Linear and Nonlinear Systems (With Examples)/Linear vs Nonlinear Systems/Linearity and Superposition

A clue about this comes from examining the linear stability of an initially uniform nonlinear profile. When this profile becomes unstable, the profile will tend to fragment and the largest fragments could serve as seeds for discrete solitons. Usually this depends on the strength of nonlinearity. After inserting this solution into Eq.

We insert this solution into Eqs 6 and 7 and obtain. This procedure gives the same information as a semi-analytical method Results are shown in Fig. For positive nonlinearity parameter the gain is positive signaling instability of the uniform profile and thus, adequate conditions for the creation of discrete solitons. For negative nonlinearity strength, the gain is identically zero.

This results are in agreement with those obtained for the DNLS in the limit of uniform initial profile Left: Instability gain versus nonlinearity strength. However, since in our case the nonlinearity parameter is proportional to the square of the electron-phonon interaction, it is always positive and we can conclude that the system is modulationally unstable and thus, prone to generating discrete solitons.

Finally, let us look at the transport properties of the mDNLS system. The typical thing to do is to examine the mean square displacement of an initially localized initial condition, at long evolution times. This can be explained noting that as time evolves, the profile expands and brings the nonlinearity terms down. In other words, at long times the nonlinearity contribution in Eq. Mean square displacement versus time for several nonlinearity parameter values.

We have computed the linear stability of the lowest-order modes and have also computed the modulational stability of the uniform solution. We conclude that the fundamental bulk mode is stable with a stability curve extending all the way from the high nonlinearity region down to the linear band. In general, higher modes need a minimum nonlinearity to exist and can posses alternate stability as a function of power content.

The surface modes, on the other hand, all need a minimum nonlinearity threshold to exist, with a stable fundamental mode. The existence of this threshold has been also observed in the DNLS case 30 , We have also estimated the dynamical barrier for the motion of a localized excitation across the lattice and obtained an approximate expression in terms of the amplitude, initial momentum and nonlinearity.

The modulation stability of the special uniform solution was computed, concluding that the system is modulationally unstable. This means that the system favors the creation of nonlinear localized excitations solitons. Finally, we computed the asymptotic transport exponent, by examining the mean square displacement of an initially localized excitation. We found that, at long times, and as a result of norm conservation, nonlinear effects becomes smaller and smaller and, as a result the propagation exponent becomes the ballistic one in the limit of an infinite time.

It should be mentioned that qualitatively similar results have been found previously for the complementary DNLS case. This is interesting and points out to the robustness of the phenomenology found here concerning the modes stability, the discrete soliton propagation, and the asymptotic propagation exponent.

It also reinforces the idea of this type of discrete soliton as a robust entity capable of transporting an excitation across a generic discrete medium and thus, useful in several different physical systems.