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...

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We investigate traveling solitary waves in the one-dimensional (1D) Salerno model, which interpolates between the cubic discrete nonlinear Schrödinger (DNLS) equation and the integrable Ablowitz-Ladik (AL) model. In a traveling frame the model becomes an advance-delay differential equation to which we analyze the existence of homoclinic orbits to the rest state. The method of beyond all orders asymptotics is used to compute the so-called Stokes constant that measures the splitting of the stable and unstable manifolds. Through computing zeros of the Stokes constant, we identify a number of solution families that may bifurcate for parameter values between the DNLS and AL limits of the Salerno model. Using a pseudospectral method, we numerically continue these solution families and show that their parameters approach the curves of the zero level of the Stokes constant as the soliton amplitude approaches zero. An interesting topological structure of solutions occurs in parameter space. As the AL limit is approached, solution sheets of single-hump solutions undergo folds and become double-hump solitons. Numerical simulation suggests that the single-humps are stable and interact almost inelastically.

We address the existence of traveling single-humped localized solutions in the spatially discrete nonlinear Schrödinger (NLS) equation. A mathematical technique is developed for analysis of persistence of these solutions from a certain limit in which the dispersion relation of linear waves contains a triple zero. The technique is based on using the Implicit Function Theorem for solution of an appropriate differential advance-delay equation in exponentially weighted spaces. The resulting Melnikov calculation relies on a number of assumptions on the spectrum of the linearization around the pulse, which are checked numerically. We apply the technique to the so-called Salerno model and the translationally invariant discrete NLS equation with a cubic nonlinearity. We show that the traveling solutions terminate in the Salerno model whereas they generally persist in the translationally invariant NLS lattice as a one-parameter family of solutions. These results are found to be in a close correspondence with numerical approximations of traveling solutions with zero radiation tails. Analysis of persistence also predicts the spectral stability of the one-parameter family of traveling solutions under time evolution of the discrete NLS equation.

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