This article offers a comprehensive survey of results obtained for solitons and complex nonlinear wave patterns supported by nonlinear lattices (NLs), which represent a spatially periodic modulation of the local strength and sign of the nonlinearity, and their combinations with linear lattices. A majority of the results obtained, thus far, in this field and reviewed in this article are theoretical. Nevertheless, relevant experimental settings are also surveyed, with emphasis on perspectives for implementation of the theoretical predictions in the experiment. Physical systems discussed in the review belong to the realms of nonlinear optics (including artificial optical media, such as photonic crystals, and plasmonics) and Bose-Einstein condensation. The solitons are considered in one, two, and three dimensions. Basic properties of the solitons presented in the review are their existence, stability, and mobility. Although the field is still far from completion, general conclusions can be drawn. In particular, a novel fundamental property of one-dimensional solitons, which does not occur in the absence of NLs, is a finite threshold value of the soliton norm, necessary for their existence. In multidimensional settings, the stability of solitons supported by the spatial modulation of the nonlinearity is a truly challenging problem, for theoretical and experimental studies alike. In both the one-dimensional and two-dimensional cases, the mechanism that creates solitons in NLs in principle is different from its counterpart in linear lattices, as the solitons are created directly, rather than bifurcating from Bloch modes of linear lattices.

Solitons in nonlinear lattices

Nonlinear dynamics of the Frenkel-Kontorova model

Computational methods for the dynamics of the nonlinear Schrödinger/Gross-Pitaevskii equations

Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method

Dispersion-managed solitons in fibre systems and lasers

Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme

We present results concerning kink-impurity interactions in the sine-Gordon (SG) model. In particular, we demonstrate that the SG kink may be totally reflected by an attractive impurity if its initial velocity is in some resonance windows. This effect can be predicted by a suitable collective-coordinate method, and the resonance structures can be explained with an energy exchange between the kink translational mode and the impurity mode. We also study the scattering of the kink by an excited impurity and show that such a scattering strongly depends on the amplitude and phase of the impurity mode. In particular, resonance phenomena are also observed in the scattering. In addition, we consider the interactions of the kink with an isotopic (heavy-mass) impurity. We demonstrate that if the impurity mass is not too large in comparison with the standard mass in the SG model, the kink can pass the impurity almost freely at any initial velocity. However, if the impurity mass is large enough, a higher-velocity kink will be reflected while a lower-velocity kink will pass. We explain this effect analytically and show that the impurity mode plays an important role in the scattering. In all the cases considered we find good agreements between the collective-coordinate analysis and the direct numerical simulations.

Resonant kink-impurity interactions in the sine-Gordon model.

We describe a new type of soliton-impurity interaction and demonstrate that the soliton can be totally reflected by an attractive impurity if its initial velocity lies in certain resonance ``windows.'' This effect has an analogy with the resonance phenomena in kink-antikink collisions [Campbell, Schonfeld, and Wingate, Physica (Amsterdam) 9D, 1 (1983)], and it can be explained by a resonant energy exchange between the soliton and the impurity mode. Taking the sine-Gordon and ${\mathrm{\ensuremath{\varphi}}}^{4}$ models as examples, we find a number of resonance windows by numerical simulations and develop a collective-coordinate approach to describe the effect analytically.

Resonant soliton-impurity interactions.

Two energy conserving numerical schemes for the Sine-Gordon equation

We study kink-impurity interactions in the ${\mathrm{\ensuremath{\varphi}}}^{4}$ model, extending our previous results [Zhang Fei, Y. Kivshar, and L. V\'azquez, Phys. Rev. A 45, 6019 (1992)] to the case where the kink has an internal degree of freedom which may be excited due to scattering. We demonstrate numerically that the kink may be reflected by an attractive impurity when its velocity lies in some ``windows,'' and this effect is due to a resonant energy exchange between the kink translational mode, its internal mode, and the impurity mode. We observe additional features in the resonant interactions, e.g., quasiresonances at some intermediate velocities and ``three-bounce'' resonances. We also develop an analytical approach taking into account three dynamical variables and show that such a collective-coordinate model may explain qualitatively the resonance structures observed in numerical simulations.

Zhang Fei

Papers

Numerical simulation of nonlinear Schro¨dinger systems: a new conservative scheme

Resonant kink-impurity interactions in the sine-Gordon model.

Resonant soliton-impurity interactions.

Two energy conserving numerical schemes for the Sine-Gordon equation

Resonant kink-impurity interactions in the φ 4 model