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IntroductionThe moment quantities associated with the nonlinear Schrödinger equation offer important insights into the evolution dynamics of such dispersive wave partial differential equation (PDE) models. The effective dynamics of the moment quantities are amenable to both analytical and numerical treatments. MethodsIn this paper, we present a data-driven approach associated with the “Sparse Identification of Nonlinear Dynamics” (SINDy) to capture the evolution behaviors of such moment quantities numerically. Results and DiscussionOur method is applied first to some well-known closed systems of ordinary differential equations (ODEs) which describe the evolution dynamics of relevant moment quantities. Our examples are, progressively, of increasing complexity and our findings explore different choices within the SINDy library. We also consider the potential discovery of coordinate transformations that lead to moment system closure. Finally, we extend considerations to settings where a closed analytical form of the moment dynamics is not available. 

Abstract In this paper, we focus on a discrete physical model describing granular crystals, whose equations of motion can be described by a system of differential difference equations. After revisiting earlier continuum approximations, we propose a regularized continuum model variant to approximate the discrete granular crystal model through a suitable partial differential equation. We then compute, both analytically and numerically, its travelling wave and periodic travelling wave solutions, in addition to its conservation laws. Next, using the periodic solutions, we describe quantitatively various features of the dispersive shock wave (DSW) by applying Whitham modulation theory and the DSW fitting method. Finally, we perform several sets of systematic numerical simulations to compare the corresponding DSW results with the theoretical predictions and illustrate that the continuum model provides a good approximation of the underlying discrete one. 

Abstract The nonlinear Schrödinger (NLS) equation in one dimension is considered in the presence of an intensity-dependent dispersion term. We study bright solitary waves with smooth profiles that extend from the limit where the dependence of the dispersion coefficient on the wave intensity is negligible to the limit where the solitary wave becomes singular due to vanishing dispersion coefficient. We analyse and numerically explore the stability for such smooth solitary waves, showing with the help of numerical approximations that the family of solitary waves becomes unstable in an intermediate region between the two limits, while being stable in both limits. This bistability, which has also been observed in other NLS equations with generalized nonlinearity, brings about interesting dynamical transitions from one stable branch to another stable branch, which are explored in direct numerical simulations of the NLS equation with the intensity-dependent dispersion term. 

Abstract In the present work we revisit the problem of the quantum droplet in atomic Bose–Einstein condensates with an eye towards describing its ground state in the large density, so-called Thomas–Fermi (TF) limit. We consider the problem as being separable into 3 distinct regions: an inner one, where the TF approximation is valid, a sharp transition region where the density abruptly drops towards the (vanishing) background value and an outer region which asymptotes to the background value. We analyze the spatial extent of each of these regions, and develop a systematic effective description of the rapid intermediate transition region. Accordingly, we derive a uniformly valid description of the ground state that is found to accurately match our numerical computations. As an additional application of our considerations, we show that this formulation allows for an analytical approximation of excited states such as the (trapped) dark soliton in the large density limit. 

Abstract Solitons are nonlinear solitary waves which maintain their shape over time and through collisions, occurring in a variety of nonlinear media from plasmas to optics. We present an experimental and theoretical study of hydrodynamic phenomena in a two-component atomic Bose-Einstein condensate where a soliton array emerges from the imprinting of a periodic spin pattern by a microwave pulse-based winding technique. We observe the ensuing dynamics which include shape deformations, the emergence of dark-antidark solitons, apparent spatial frequency tripling, and decay and revival of contrast related to soliton collisions. For the densest arrays, we obtain soliton complexes where solitons undergo continued collisions for long evolution times providing an avenue towards the investigation of soliton gases in atomic condensates. 

Abstract Motivated by the work of Jang et al., Nat Commun 6:7370 (2015), where the authors experimentally tweeze cavity solitons in a passive loop of optical fiber, we study the amenability to tweezing of cavity solitons as the properties of a localized tweezer are varied. The system is modeled by the Lugiato-Lefever equation, a variant of the complex Ginzburg-Landau equation. We produce an effective, localized, trapping tweezer potential by assuming a Gaussian phase-modulation of the holding beam. The potential for tweezing is then assessed as the total (temporal) displacement and speed of the tweezer are varied, and corresponding phase diagrams are presented. As the relative speed of the tweezer is increased we find two possible dynamical scenarios: successful tweezing and release of the cavity soliton. We also deploy a non-conservative variational approximation (NCVA) based on a Lagrangian description which reduces the original dissipative partial differential equation to a set of coupled ordinary differential equations for the cavity soliton parameters. We illustrate the ability of the NCVA to accurately predict the separatrix between successful and failed tweezing. This showcases the versatility of the NCVA to provide a low-dimensional description of the experimental realization of the temporal tweezing. 

Abstract Here we revisit the topic of stationary and propagating solitonic excitations in self-repulsive three-dimensional (3D) Bose–Einstein condensates by quantitatively comparing theoretical analysis and associated numerical computations with our experimental results. Motivated by numerous experimental efforts, including our own herein, we use fully 3D numerical simulations to explore the existence, stability, and evolution dynamics of planar dark solitons. This also allows us to examine their instability-induced decay products including solitonic vortices and vortex rings. In the trapped case and with no adjustable parameters, our numerical findings are in correspondence with experimentally observed coherent structures. Without a longitudinal trap, we identify numerically exact traveling solutions and quantify how their transverse destabilization threshold changes as a function of the solitary wave speed. 

Abstract In the seminal work (Weinstein 1999Nonlinearity12673), Weinstein considered the question of the ground states for discrete Schrödinger equations with power law nonlinearities, posed on ${\mathbb{Z}}^d$. More specifically, he constructed the so-called normalised waves, by minimising the Hamiltonian functional, for fixed powerP(i.e.l²mass). This type of variational method allows one to claim, in a straightforward manner, set stability for such waves. In this work, we revisit these questions and build upon Weinstein’s work, as well as the innovative variational methods introduced for this problem in (Laedkeet al1994Phys. Rev. Lett.731055 and Laedkeet al1996Phys. Rev.E544299) in several directions. First, for the normalised waves, we show that they are in fact spectrally stable as solutions of the corresponding discrete nonlinear Schroedinger equation (NLS) evolution equation. Next, we construct the so-called homogeneous waves, by using a different constrained optimisation problem. Importantly, this construction works for all values of the parameters, e.g.l²supercritical problems. We establish a rigorous criterion for stability, which decides the stability on the homogeneous waves, based on the classical Grillakis–Shatah–Strauss/Vakhitov–Kolokolov (GSS/VK) quantity ${\mathrm{\partial }}_{\omega }\parallel {\phi }_{\omega }{\parallel }_{l^2}^2$. In addition, we provide some symmetry results for the solitons. Finally, we complement our results with numerical computations, which showcase the full agreement between the conclusion from the GSS/VK criterion vis-á-vis with the linearised problem. In particular, one observes that it is possible for the stability of the wave to change as the spectral parameterωvaries, in contrast with the corresponding continuous NLS model.