Radiationless Traveling Waves in Saturable Nonlinear Schrödinger Lattices

Abstract

The long-standing problem of moving discrete solitary waves in nonlinear Schrödinger lattices is revisited. The context is photorefractive crystal lattices with saturable nonlinearity whose grand-canonical energy barrier vanishes for isolated coupling strength values. Genuinely localized traveling waves are computed as a function of the system parameters for the first time. The relevant solutions exist only for finite velocities. DOI: 10.1103/PhysRevLett.97.124101 PACS numbers: 42.65.Tg, 05.45.Yv, 63.20.Pw Recently, the topic of discrete solitons (intrinsic localized modes) in photorefractive materials has received significant attention; see [1] for a review. This interest was initialized by the experimental realization of twodimensional periodic lattices in photorefractive crystals [2] in which solitons were observed [3,4]. Further work has revealed a wealth of additional coherent structures such as dipoles, quadrupoles, soliton trains, vector, necklace, and ring solitons; see, e.g., [5][6][7][8]. Since photorefractive materials feature the so-called saturable nonlinearity, these results for periodic lattices have spawned a parallel interest in genuinely discrete saturable nonlinear lattices [9][10][11]. One particularly intriguing result of these studies is that the stability properties of the localized modes are substantially different from their regular discrete nonlinear Schrödinger (DNLS) analogs. In DNLS, it is well known [12 -14] that site-centered localized modes are always stable, while intersite-centered modes are unstable (and are stabilized only in the continuum limit where these two branches degenerate into the well-known continuum sech-soliton of the integrable cubic NLS). For the photorefractive nonlinearity (i.e., the so-called Vinetskii-Kukhtarev model originally proposed in Ref. [15] and revisited in Refs. [9][10][11]), depending on the coupling strength, the intersite-centered modes may have lower energy than their on site counterparts. Hence, one should expect that the ensuing sign reversal of the so-called Peierls-Nabarro (PN) energy barrier E E IS ÿ E OS (where the subscripts denote intersite and on site, respectively) should cause an exchange between the linear stability properties of the two modes.A related, even more fundamental, question in nonlinear lattice models of DNLS type is whether exponentially localized self-supporting excitations that move with a constant wave speed can exist, the so-called moving discrete solitons. For continuum models, this question is in some sense trivial since the equations posed in a moving frame remain of the same fundamental type. Yet for lattices, passing to a moving frame leads to so-called advance-delay equations that are notoriously hard to analyze. One recent (negative) result in this direction [16] for the so-called Salerno model shows that, starting from the integrable Ablowitz-Ladik equation, mobile discrete solitary waves acquire nonvanishing tails as soon as parameters deviate from the integrable limit. Hence, exponentially localized fu...

Dark solitons in discrete lattices: Saturable versus cubic nonlinearities

“…G (and, in particular, its differences between on-site and inter-site configurations) was shown in [36] to be the proper indicator of the location of the exchange of stability/transparency windows in the focusing version of the model. Therefore, in Fig.…”

“…This result is a clear indication that, in contrast to the saturable self-focusing nonlinearity case, in the self-defocusing regime the saturable lattice possesses no transparency points. It remains as an interesting open question whether such a stability alternation could be achievable in other defocusing lattice models (such as, e.g., possibly a cubic-quintic equation), especially because these transparency windows have been associated with important travelling solution properties of these lattices [36]. We also briefly compare the growth rates of the instabilities in the different models.…”

Dark breathers in granular crystals

“…In that light, a computational study of such states along the lines of earlier similar studies in the realm of the discrete NLS equation [32][33][34] would be of particular interest. The potential emergence of dark breathers in inhomogeneous chains, such as dimers [19,20] or trimers [35] or in defect chains [18], may be another issue to computationally consider in detail, since bright breathers have been identified as relatively robust nonlinear states in such settings [19][20][21].…”

Unifying perspective: Solitary traveling waves as discrete breathers in Hamiltonian lattices and energy criteria for their stability

“…While the latter will capture the lattice instabilities, it may also feature instabilities absent on the lattice, which are a by-product of this PDE's ability to resolve scales smaller than h. Hence, a more systematic connection between the spectra of the two problems (and of the instabilities that each may feature) is of paramount importance. Observe also that while this work dealt with families of STWs parameterized by velocity, in some cases such entities occur for isolated velocity values [23,24], potentially being members of a wider family encompassing waveforms with nonvanishing tails. It would be interesting to explore whether our considerations can be extended to such cases.…”