Collisional-inhomogeneity-induced generation of matter-wave dark solitons

We propose an experimentally relevant protocol for the controlled generation of matter-wave dark solitons in atomic Bose-Einstein condensates (BECs). In particular, using direct numerical simulations, we show that by switching-on a spatially inhomogeneous (step-like) change of the s-wave scattering length, it is possible to generate a controllable number of dark solitons in a quasi-one-dimensional BEC. A similar phenomenology is also found in the two-dimensional setting of "disk-shaped" BECs but, as the solitons are subject to the snaking instability, they decay into vortex structures. A detailed investigation of how the parameters involved affect the emergence and evolution of solitons and vortices is provided.Comment: 8 pages, 5 Figures, Physics Letters A (in press

On the properties of a nonlocal nonlinear schrödinger model and its soliton solutions

Nonlinear waves are normally described by means of certain nonlinear evolution equations. However, finding physically relevant exact solutions of these equations is, in general, particularly difficult. One of the most known nonlinear evolution equation is the nonlinear Schrödinger (NLS), a universal equation appearing in optics, Bose-Einstein condensates, water waves, plasmas, and many other disciplines. In optics, the NLS system is used to model a unique balance between the critical effects that govern propagation in dispersive nonlinear media, namely dispersion/diffraction and nonlinearity. This balance leads to the formation of solitons, namely robust localized waveforms that maintain their shape even when they interact. However, for several physically relevant contexts the standard NLS equation turns out to be an oversimplified description. This occurs in the case of nonlocal media, such as nematic liquid crystals, plasmas, and optical media exhibiting thermal nonlinearities. Here, we study the properties and soliton solutions of such a nonlocal NLS system, composed by a paraxial wave equation for the electric field envelope and a diffusion-type equation for the medium’s refractive index. The study of this problem is particularly interesting since remarkable properties of the traditional NLS—associated with complete integrability—are lost in the nonlocal case. Nevertheless, we identify cases where derivation of exact solutions is possible while, in other cases, we resort to multiscale expansions methods. The latter, allows us to reduce this systems to a known integrable equation with known solutions, which in turn, can be used to approximate the solutions of the initial system. By doing so, a plethora of solutions can be found; solitary waves solutions, planar or ring-shaped, and of dark or anti-dark type, are predicted to occur. © Springer International Publishing AG, part of Springer Nature 2018

Dynamical oscillations in nonlinear optical media

The spatial dynamics of pulses in Kerr media with parabolic index profile are examined. It is found that when diffraction and graded-index have opposite signs propagating pulses exhibit an oscillatory pattern, similar to a breathing behavior. Furthermore, if the pulse and the index profile are not aligned the pulse oscillates around the index origin with frequency that depends on the values of the diffraction and index of refraction. These oscillations are not observed when diffraction and graded-index share the same sign. © 2009 Elsevier B.V. All rights reserved

Dynamics of a Higher-Order Ginzburg–Landau-Type Equation

We study possible dynamical scenarios associated with a higher-order Ginzburg–Landau-type equation. In particular, first we discuss and prove the existence of a limit set (attractor), capturing the long-time dynamics of the system. Then, we examine conditions for finite-time collapse of the solutions of the model at hand, and find that the collapse dynamics is chiefly controlled by the linear/nonlinear gain/loss strengths. Finally, considering the model as a perturbed nonlinear Schrödinger equation, we employ perturbation theory for solitons to analyze the influence of gain/loss and other higher-order effects on the dynamics of bright and dark solitons. © 2021, Springer Nature Switzerland AG

Collisional-inhomogeneity-induced generation of matter-wave dark solitons

We propose an experimentally relevant protocol for the controlled generation of matter-wave dark solitons in atomic Bose-Einstein condensates (BECs). In particular, using direct numerical simulations, we show that by switching-on a spatially inhomogeneous (step-like) change of the s-wave scattering length, it is possible to generate a controllable number of dark solitons in a quasi-one-dimensional BEC. A similar phenomenology is also found in the two-dimensional setting of "disk-shaped" BECs but, as the solitons are subject to the snaking instability, they decay into vortex structures. A detailed investigation of how the parameters involved affect the emergence and evolution of solitons and vortices is provided. © 2010 Elsevier B.V. All rights reserved

Higher-dimensional extended shallow water equations and resonant soliton radiation

The higher order corrections to the equations that describe nonlinear wave motion in shallow water are derived from the water wave equations. In particular, the extended cylindrical Korteweg-de Vries and Kadomtsev-Petviashvili equations—which include higher order nonlinear, dispersive, and nonlocal terms—are derived from the Euler system in (2+1) dimensions, using asymptotic expansions. It is thus found that the nonlocal terms are inherent only to the higher dimensional problem, both in cylindrical and Cartesian geometry. Asymptotic theory is used to study the resonant radiation generated by solitary waves governed by the extended equations, with an excellent comparison obtained between the theoretical predictions for the resonant radiation amplitude and the numerical solutions. In addition, resonant dispersive shock waves (undular bores) governed by the extended equations are studied. It is shown that the asymptotic theory, applicable for solitary waves, also provides an accurate estimate of the resonant radiation amplitude of the undular bore. ©2021 American Physical Societ

Traveling waves of the regularized short pulse equation

The properties of the so-called regularized short pulse equation (RSPE) are explored with a particular focus on the traveling wave solutions of this model. We theoretically analyze and numerically evolve two sets of such solutions. First, using a fixed point iteration scheme, we numerically integrate the equation to find solitary waves. It is found that these solutions are well approximated by a finite sum of hyperbolic secants powers. The dependence of the solitons parameters (height, width, etc) to the parameters of the equation is also investigated. Second, by developing a multiple scale reduction of the RSPE to the nonlinear Schrödinger equation, we are able to construct (both standing and traveling) envelope wave breather type solutions of the former, based on the solitary wave structures of the latter. Both the regular and the breathing traveling wave solutions identified are found to be robust and should thus be amenable to observations in the form of few optical cycle pulses. © 2014 IOP Publishing Ltd