A detailed numerical study of the long time behaviour of dispersive shock waves in solutions to the Kadomtsev–Petviashvili (KP) I equation is presented. It is shown that modulated lump solutions emerge from the dispersive shock waves. For the description of dispersive shock waves, Whitham modulation equations for KP are obtained. It is shown that the modulation equations near the soliton line are hyperbolic for the KPII equation while they are elliptic for the KPI equation leading to a focusing effect and the formation of lumps. Such a behaviour is similar to the appearance of breathers for the focusing nonlinear Schrödinger equation in the semiclassical limit.

VL - 474 UR - https://royalsocietypublishing.org/doi/abs/10.1098/rspa.2017.0458 ER - TY - JOUR T1 - NURBS-SEM: A hybrid spectral element method on NURBS maps for the solution of elliptic PDEs on surfaces JF - COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING Y1 - 2018 A1 - Giuseppe Pitton A1 - Luca Heltai VL - 338 UR - https://arxiv.org/abs/1804.08271 ER - TY - RPRT T1 - Non-linear Schrödinger system for the dynamics of a binary condensate: theory and 2D numerics Y1 - 2016 A1 - Alessandro Michelangeli A1 - Giuseppe Pitton AB - We present a comprehensive discussion of the mathematical framework for binary Bose-Einstein condensates and for the rigorous derivation of their effective dynamics, governed by a system of coupled non-linear Gross-Pitaevskii equations. We also develop in the 2D case a systematic numerical study of the Gross-Pitaevskii systems in a wide range of relevant regimes of population ratios and intra-species and inter-species interactions. Our numerical method is based on a Fourier collocation scheme in space combined with a fourth order integrating factor scheme in time. UR - http://urania.sissa.it/xmlui/handle/1963/35266 U1 - 35572 U2 - Mathematics U4 - 1 ER -