We introduce novel classes of higher-order spatial optical solitons in analogy with Laguerre-Gaussian and Hermite-Gaussian linear eigenmodes. We reveal that stable higher-order optical solitons can exist in nonlocal nonlinear media in the various forms of soliton necklaces and soliton matrices. Modulational instability can lead to nontrivial transformations between energetically close solitons ...

The quantized vortex state is investigated in a Bose-Einstein condensate, confined in a multiply connected geometry formed by a Laguerre-Gaussian optical trap. Solving the Gross-Pitaevskii equation variationally, we show that the criterion for vortex stability is that the interatomic interaction strength must exceed a critical interaction strength. The time evolution of a freely expanding Lague...

Recently there has been some renewed interest in Laguerre differential geometry [1], [2], [3]. This geometry was to a large extent developed by Blaschke and his school and a large amount of material about it can be found in [4]. Consider the set of oriented spheres in Euclidean space. If we add to this set the space of oriented planes, i.e. the spheres of infinite radii, then we obtain the spac...

Big q-Jacobi polynomials {Pn(·; a, b, c; q)}∞n=0 are classically defined for 0 < a < q −1, 0 < b < q−1 and c < 0. For the family of little q-Jacobi polynomials {pn(·; a, b|q)}∞n=0, classical considerations restrict the parameters imposing 0 < a < q−1 and b < q−1. In this work we extend both families in such a way that wider sets of parameters are allowed, and we establish orthogonality conditio...

Infinitely many explicit solutions of certain second-order differential equations with an apparent singularity of characteristic exponent −2 are constructed by adjusting the parameter of the multi-indexed Laguerre polynomials.