In this article, a singularity structure analysis of a (2+1)-dimensional generalized Korteweg-de Vries equation studied originally by Boiti et al., admitting a weak Lax pair, is carried out and it is proven that the system satisfies the Painleve property. Its bilinear form is constructed in a natural way from the P analysis and then it is used to generate "multidromion" solutions (exponentially decaying solutions in all directions). The same analysis can be extended to construct the multidromion solutions of the generalized Nizhnik-Novikov-Veselov (NNV) equation from which the NNV equation follows as a special case.

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Abstract. In this paper, we investigate the (2+1) dimensional long wave-short wave resonance interaction (LSRI) equation and show that it possess the Painlevé property. We then solve the LSRI equation using Painlevé truncation approach through which we are able to construct solution in terms of three arbitrary functions. Utilizing the arbitrary functions present in the solution, we have generated a wide class of elliptic function periodic wave solutions and exponentially localized solutions such as dromions, multidromions, instantons, multi-instantons and bounded solitary wave solutions.

We study the emergence of Faraday waves in cigar-shaped collisionally inhomogeneous BoseEinstein condensates subject to periodic modulation of the radial confinement. Considering a Gaussian-shaped radially inhomogeneous scattering length, we show through extensive numerical simulations and detailed variational treatment that the spatial period of the emerging Faraday waves increases as the inhomogeneity of the scattering length gets weaker, and that it saturates once the width of the radial inhomogeneity reaches the radial width of the condensate. In the regime of strongly inhomogeneous scattering lengths, the radial profile of the condensate is akin to that of a hollow cylinder, while in the weakly inhomogeneous case the condensate is cigar-shaped and has a Thomas-Fermi radial density profile. Finally, we show that when the frequency of the modulation is close to the radial frequency of the trap, the condensate exhibits resonant waves which are accompanied by a clear excitation of collective modes, while for frequencies close to twice that of the radial frequency of the trap, the observed Faraday waves set in forcefully and quickly destabilize condensates with weakly inhomogeneous two-body interactions.