Light Meets Water in Nonlocal Media: Surface Tension Analogue in Optics

Abstract

Shallow water wave phenomena find their analogue in optics through a nonlocal nonlinear Schrödinger (NLS) model in (2 + 1)-dimensions. We identify an analogue of surface tension in optics, namely a single parameter depending on the degree of nonlocality, which changes the sign of dispersion, much like surface tension does in the shallow water wave problem. Using multiscale expansions, we reduce the NLS model to a KadomtsevPetviashvilli (KP) equation, which is of the KPII (KPI) type, for strong (weak) nonlocality. We demonstrate the emergence of robust optical antidark solitons forming Y-, X-and H-shaped wave patterns, which are approximated by colliding KPII line solitons, similar to those observed in shallow waters. 05.45.Yv, 42.65.Tg, 47.11.St, 47.35.Fg Many physically different contexts can be brought together through common modeling and mathematical description. A common (and rather unlike) example is water waves and nonlinear optics. Two models are inextricably linked with both subjects: the universal Korteweg-de Vries (KdV) and nonlinear Schrödinger (NLS) equations [1]. Furthermore, these models can be reduced from one to the other [2], thus suggesting that phenomena occurring in water waves may also exist in optics. Here, using such reductions for a nonlocal NLS, we find that surface tension -which causes fluids to minimize the area they occupy-has a direct analogue in optics.Key to our findings are solitons, i.e., robust localized waves that play a key role in numerous studies in physics [3], applied mathematics [4] and engineering [5]. A unique property of solitons is that they feature a particle-like character, i.e., they interact elastically, preserving their shapes and velocities after colliding with each other. Such elastic collisions, as well as pertinent emerging wave patterns, can sometimes be observed even in everyday life. A predominant example is the one pertaining to flat beaches: in such shallow water wave settings, two line solitons merging at proper angles give rise to patterns of X-, H-, or Y-shaped waves, as well as other more complicated nonlinear waveforms [6]. All these shallow water wave structures are actually exact analytical multi-dimensional line soliton solutions of the Kadomtsev-Petviashvilli (KP) equation (which generalizes KdV to two dimensions (2D) [1,4]) of the KPII type -a key model in the theory of shallow water waves with weak surface tension [1]. The relative equation with strong surface tension is referred to as KPI.Here we show that such patterns can also be observed in a quite different physical setting, i.e., the one related to optical beam propagation in media with a spatially nonlocal defocusing nonlinearity. Such media include thermal nonlinear optical media [7,8], partially ionized plasmas [9, 10], nematic liquid crystals [11,12], and dipolar bosonic quantum gases [13]. It is shown that approximate solutions of the nonlocal NLS model satisfy, at proper scales, equations that appear in the context of water waves: a Boussinesq or Benney-Luke (BL) [14], as well...

Solitary waves of the two-dimensional Camassa–Holm—nonlinear Schrödinger equation

“…Note that the present analysis generalizes the results of Ref. [24] (where the 1D case was studied) to the 2D setting; in that regard, it is worth mentioning that similar 2D Boussinesq equations were derived from 2D NLS equations with either a local [27,28] or a nonlocal [15,20] defocusing nonlinearity. Next, using a multiscale expansion method similar to the one employed in the water wave problem [2], we will derive the far-field of the Boussinesq equation, namely a pair of two KP equations for right-and left-going waves.…”

“…To proceed further, and express the KP of Eq. (20) in its standard form, we introduce the rescaling:Ŷ…”

Asymptotic expansions and solitons of the Camassa–Holm – nonlinear Schrödinger equation

“…These models have been used to describe bidirectional shallow water waves in the framework of small-amplitude and long-wavelength approximations; see, e.g., the expositions of [11,19] They were also used in other contexts including ion-acoustic waves in plasmas [18,36], mechanical lattices and electrical transmission lines [37]. It is worth mentioning that an analysis similar to that presented above can also be performed in two-dimensional (2D) settings: indeed, 2D Boussinesq equations were derived from 2D NLS equations with either a local [38] or a nonlocal [33,34] defocusing nonlinearity. Such studies are also relevant to investigations concerning the transverse instability of planar dark solitons [39].…”

“…This approach also allowed for the prediction of the existence of a structure known as antidark soliton, namely a dark soliton with reverse-sign amplitude, having the form of a hump (instead of a dip) on top of the continuous-wave (cw) background density [27][28][29][30][31]. Note that relevant studies employing multiscale expansion methods and predicting the occurrence of antidark solitons were recently extended to settings involving nonlocal nonlinearities [32][33][34].…”

Multiscale expansions avector solitons of a two‐dimensional nonlocal nonlinear Schrödinger system

“…Our approach resembles the one used for single-component nonlocal NLS equations, where similar soliton solutions where found. [32][33][34] We thus predict the existence of weak antidark solitons that are supported by the nonlocality. In addition, however, there is a key element in our case, namely the role of multicomponents.…”