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A family of higher-order rational lumps on non-zero constant background of Davey-Stewartson (DS) II equation are investigated. These solutions have multiple peaks whose heights and trajectories are approximately given by asymptotical analysis. It is found that the heights are time-dependent and for large time they approach the same constant height value of the first-order fundamental lump. The resulting trajectories are considered and it is found that the scattering angle can assume arbitrary values in the interval of $(\frac{\pi}{2}, \pi)$ which is markedly distinct from the necessary orthogonal scattering for the higher-order lumps on zero background. Additionally, it is illustrated that the higher-order lumps containing multi-peaked $n$-lumps can be regarded as a nonlinear superposition of $n$ first-order ones as $|t|\rightarrow\infty$., Comment: 17 pages,10 figures

A prototypical example of a rogue wave structure in a two-dimensional model is presented in the context of the Davey-Stewartson~II (DS~II) equation arising in water waves. The analytical methodology involves a Taylor expansion of an eigenfunctionof the model's Lax pair which is used to form a hierarchy of infinitely many new eigenfunctions. These are used for the construction of two-dimensional (2D) rogue waves (RWs) of the DS~II equation by the even-fold Darboux transformation (DT). The obtained 2D RWs, which are localized in both space and time, can be viewed as a 2D analogue of the Peregrine soliton and are thus natural candidates to describe oceanic RW phenomena,as well as ones in 2D fluid systems and water tanks., Comment: 6 pages, 3 color figures

We study the extended Jaynes-Cummings-Hubbard model on triangular cavity lattices and zigzag ladders. By using density-matrix renormalization group methods, we observe various types of solids with different density patterns and find evidence for light supersolids, which exist in extended regions of the phase diagram of the zigzag ladder. Furthermore, we observe strong pair correlations in the supersolid phase due to the interplay between the atoms in the cavities and atom-photon interaction. By means of cluster mean-field simulations and a scaling of the cluster size extending our analysis to two-dimensional triangular lattices, we present evidence for the emergence of a light supersolid in this case also., Comment: 11 pages, 16 figures

By the photoassociation method, the trimer superfluid phase disappears in the one dimensional state-dependent optical lattice if the ratio of the three-body interaction $W$ to the trimer tunneling $J$is kept at $W/J=12$ [Phys Rev A. {\bf 90}, 033622(2014)]. To search for a trimer superfluid and trimer supersolid, we load the cold atom into two-dimensional lattices, whose coordinate number $z$ and kinetic energy $-zJ$ are respectively larger and lower than those of a one dimensional lattice. Herein, we study the Bose-Hubbard model, which has an additional trimer tunneling term, a three-body interaction and a next-nearest repulsion. The on-site trimer and trimer superfluid exist if the on-site two-body repulsion and three-body repulsion are smaller than some thresholds. With atom-tunneling terms, the phase transitions from trimer superfluid phase to both atom superfluid and atom supersolid phases are first order. With $W/J=12$, in a one dimensional lattice, the trimer superfluid phase does not exist at all. In contrast, the trimer superfluid phase, exists in the lower density regions $0 \textless \rho \textless2$ on the square lattice if $J$ is not very large. The trimer superfluid phase emerges in a wider range $0 \textless \rho \textless3$ in the triangular lattice, or in the cubic lattice ($z=6$). When the three-body interaction is turned on, a trimer supersolid phase emerges due to the classical degeneracy between the quasi trimer solid and the trimer solid being broken by the quantum fluctuation. The phase transitions from the trimer supersolid phase to quasi trimer solid are first order and the phase transition from the trimer supersolid phase to trimer solid is continuous. Our results, obtained by mean-field and quantum Monte Carlo methods, will be helpful in realizing the trimer superfluid and supersolid by cold atom experiments., Comment: 9 pages, 9 figures

The the mixed Chen-Lee-Liu derivative nonlinear Schr\"odinger equation (CLL-NLS) can be considered as simplest model to approximate the dynamics of weakly nonlinear and dispersive waves, taking into account the self-steepnening effect (SSE). The latter effect arises as a higher-order correction of the nonlinear Schr\"ordinger equation (NLS), which is known to describe the dynamics of pulses in nonlinear fiber optics, and constiutes a fundamental part of the generalized NLS. Similar effects are decribed within the framework of the modified NLS, also referred to as the Dysthe equation, in hydrodynamics. In this work, we derive fundamental and higher-order solutions of the CLL-NLS by applying the Darboux transformation (DT). Exact expressions of non-vanishing boundary solitons, breathers and a hierarchy of rogue wave solutions are presented. In addition, we discuss the localization characters of such rogue waves, by characterizing their length and width. In particular, we describe how the localization properties of first-order NLS rogue waves can be modified by taking into account the SSE, presented in the CLL-NLS. This is illustrated by use of an analytical and a graphical method. The results may motivate similar analytical studies, extending the family of the reported rogue wave solutions as well as possible experiments in several nonlinear dispersive media, confirming these theoretical results., Comment: We have made remarkable changes of the early version, and so this version has 29 pages including 12 figures

We present determinant expressions for vector rogue wave solutions of the Manakov system, a two-component coupled nonlinear Schr\"odinger equation. As special case, we generate a family of exact and non-symmetric rogue wave solutions of the nonlinear Schr\"odinger equation up to third-order, localized in both space and time. The derived non-symmetric doubly-localized second-order solution is generated experimentally in a water wave flume for deep-water conditions. Experimental results, confirming the characteristic non-symmetric pattern of the solution, are in very good agreement with theory as well as with numerical simulations, based on the modified nonlinear Schr\"odinger equation, known to model accurately the dynamics of weakly nonlinear wave packets in deep-water., Comment: 15 pages, 7 figures, accepted by Proceedings of the Royal Society A

In this paper, we provide a simple method to generate higher order position solutions and rogue wave solutions for the derivative nonlinear Schr\"odinger equation. The formulae of these higher order solutions are given in terms of determinants. The dynamics and structures of solutions generated by this method are studied.

We construct higher order rogue wave solutions for the Gerdjikov-Ivanov equation explicitly in term of determinant expression. Dynamics of both soliton and non-soliton solutions is discussed. A family of solutions with distinct structures are presented, which are new to the Gerdjikov-Ivanov equation.