# Two-dimensional optical lattice solitons.

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Nikolaos K Efremidis, Jul 08, 2015 Available from:-
- "Thus it is of interest to develop a fundamental understanding of the effect of inhomogeneities in a medium on the dynamics of nonlinear dispersive waves and, in particular, on the dynamics of solitons. See, for example, [24] for an experimental investigation of solitons in periodic structures. "

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**ABSTRACT:**We consider a class of nonlinear Schrodinger / Gross-Pitaevskii (NLS/GP) equations with periodic potentials, having an even symmetry. We construct "solitons", centered about any point of symmetry of the potential. For focusing (attractive) nonlinearities, these solutions bifurcate from the zero state at the lowest band edge frequency, into the semi-infinite spectral gap. Our results extend to bifurcations into finite spectral gaps, for focusing or defocusing (repulsive) nonlinearities under more restrictive hypotheses. Soliton nonlinear bound states with frequencies near a band edge are well-approximated by a slowly decaying solution of a homogenized NLS/GP equation, with constant homogenized effective mass tensor and effective nonlinear coupling coefficient, modulated by a Bloch state. For the critical NLS equation with a periodic potential, e.g. the cubic two dimensional NLS/GP with a periodic potential, our results imply that the limiting soliton power, as the spectral band edge frequency is approached, is equal to a constant \zeta_* times the minimal mass soliton of the translation invariant critical NLS equation. \zeta_* is expressible in terms of the band edge Bloch eigenfunction and the determinant of the effective mass tensor; and 0SIAM Journal on Multiscale Modeling and Simulation 01/2010; 8(4):1055-1101. DOI:10.1137/090769417 · 1.80 Impact Factor -
- "Example 3. The next example is the 2D NLS equation with a periodic potential iU t þ DU þ jU j 2 U À V ðx; yÞU ¼ 0; ð4:7Þ where D ¼ o xx þ o yy , and V ðx; yÞ ¼ 1:5ðcos 2 x þ cos 2 yÞ. This equation arises in Bose–Einstein condensates trapped in optical lattices and light propagation in periodic non-linear media [21] [22] [23]. This equation admits vortex solitons of the form U ðx; y; tÞ ¼ uðx; yÞe Àilt , where uðx; yÞ is a complex function with a unit topological charge [21]. "

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**ABSTRACT:**Three iteration methods are proposed for the computation of eigenvalues and eigenfunctions in the linear stability of solitary waves. These methods are based on iterating certain time evolution equations associated with the linear stability eigenvalue problem. The first method uses the fourth-order Runge–Kutta method to iterate the pre-conditioned linear sta-bility operator, and it usually converges to the most unstable eigenvalue of the solitary wave. The second method uses the Euler method to iterate the ''square" of the pre-conditioned linear stability operator. This method is shown to converge to any discrete eigenvalue in the stability spectrum. The third method is obtained by incorporating the mode elimination tech-nique into the second method, which speeds up the convergence considerably. These methods are applied to various exam-ples of physical interest, and are found to be efficient, easy to implement, and low in memory usage.Journal of Computational Physics 05/2008; 227(14). DOI:10.1016/j.jcp.2008.03.039 · 2.49 Impact Factor -
- "which has recently attracted much interest due to its application to optical lattices and Bose- Einstein condensates [3] [35] [36]. This equation admits a family of nodeless solitary waves. "

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**ABSTRACT:**Two accelerated imaginary-time evolution methods are proposed for the computation of solitary waves in arbitrary spatial dimensions. For the first method (with traditional power normalization), the convergence conditions as well as conditions for optimal accelerations are derived. In addition, it is shown that for nodeless solitary waves, this method converges if and only if the solitary wave is linearly stable. The second method is similar to the first method except that it uses a novel amplitude normalization. The performance of these methods is illustrated on various examples. It is found that while the first method is competitive with the Petviashvili method, the second method delivers much better performance than the first method and the Petviashvili method.Studies in Applied Mathematics 12/2007; DOI:10.1111/j.1467-9590.2008.00398.x · 1.15 Impact Factor