A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

Ablowitz, Mark J.; Ma, Yi-Ping; Rumanov, Igor

doi:10.1137/16m1099960

Cited by 8 publications

(11 citation statements)

References 56 publications

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“…Yet, our quantitative analysis illustrates that dispersion dominates already for rather weak couplings i.e., > 0.25. Whether a discrete analogue of self-similar dynamics arises for this interval (corresponding to the continuum observations of [18]) is an interesting open question for future study.…”

Section: Numerical Results: Existence Stability and Dynamicsmentioning

confidence: 97%

2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

D'Ambroise¹,

Kevrekidis²

2019

Phys. Scr.

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We derive stationary solutions to the two-dimensional hyperbolic discrete nonlinear Schrödinger (HDNLS) equation by starting from the anti-continuum limit and extending solutions to include nearest-neighbor interactions in the coupling parameter. We use pseudo-arclength continuation to capture the relevant branches of solutions and explore their corresponding stability and dynamical properties (i.e., their fate when unstable). We focus on nine primary types of solutions: single site, double site in-and out-of-phase, squares with four sites in-phase and out-of phase in each of the vertical and horizontal directions, four sites out-of-phase arranged in a line horizontally, and two additional solutions having respectively six and eight nonzero sites. The chosen configurations are found to merge into four distinct bifurcation events. We unveil the nature of the bifurcation phenomena and identify the critical points associated with these states and also explore the consequences of the termination of the branches on the dynamical phenomenology of the model.

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Section: Numerical Results: Existence Stability and Dynamicsmentioning

confidence: 97%

2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

D'Ambroise¹,

Kevrekidis²

2019

Phys. Scr.

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“…Nevertheless, intermediate dynamics can exhibit rich and interesting structures; see, for example, the numerics in Ref. 16 for the hyperbolic case. It might be possible to investigate approximate intermediate solutions with the help of the 2d NLS Whitham system.…”

Section: Discussionmentioning

confidence: 99%

“…Curved propagating DSW fronts are often observed, see, for example, Refs. 7, 15, 16. In this paper, the complete Whitham system for the 2d NLS equation, both elliptic and hyperbolic, is constructed.…”

Section: Introductionmentioning

confidence: 99%

On the Whitham system for the (2+1)‐dimensional nonlinear Schrödinger equation

2022

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Whitham modulation equations are derived for the nonlinear Schrödinger equation in the plane ((2+1)dimensional nonlinear Schrödinger [2d NLS]) with small dispersion. The modulation equations are obtained in terms of both physical and Riemann-type variables; the latter yields equations of hydrodynamic type. The complete 2d NLS Whitham system consists of six dynamical equations in evolutionary form and two constraints. As an application, we determine the linear stability of one-dimensional traveling waves. In both the elliptic and hyperbolic cases, the traveling waves are found to be unstable. This result is consistent with previous investigations of stability by other methods and is supported by direct numerical calculations.

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“…Nevertheless intermediate dynamics can exhibit rich and interesting structures; see e.g. the numerics in [7] for the hyperbolic case. It might be possible to investigate approximate intermediate solutions with the help of the 2d NLS Whitham system.…”

Section: Discussionmentioning

confidence: 99%

“…Curved propagating DSW fronts are often observed, see e.g. [35,38,7]. In this paper, the complete Whitham system for the 2d NLS equation, both elliptic and hyperbolic, is constructed.…”

Section: Introductionmentioning

confidence: 99%

On the Whitham system for the radial nonlinear Schrödinger equation

2019

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Dispersive shock waves (DSWs) of the defocusing radial nonlinear Schrödinger (rNLS) equation in two spatial dimensions are studied. This equation arises naturally in Bose-Einstein condensates, water waves, and nonlinear optics. A unified nonlinear WKB approach, equally applicable to integrable or nonintegrable partial differential equations, is used to find the rNLS Whitham modulation equation system in both physical and hydrodynamic type variables. The description of DSWs obtained via Whitham theory is compared with direct rNLS numerics; the results demonstrate very good quantitative agreement. On the other hand, as expected, comparison with the corresponding DSW solutions of the one-dimensional NLS equation exhibits significant qualitative and quantitative differences. K E Y W O R D S nonlinear waves, nonlinear optics, water waves and fluid dynamics The applicability of the 2d NLS equation for Bose-Einstein condensate (BEC) wave functions in the appropriate geometries with the third coordinate axis being the axis of symmetry is well known and Stud Appl Math. 2019;142:269-313. wileyonlinelibrary.com/journal/sapm Company 269 270 ABLOWITZ ET AL. can be derived from many-body quantum dynamics, see, eg, Ref. 1. This equation can also describe deep water waves with sufficiently high surface tension. 2 A motivation to study the rNLS equation comes from the BEC experiments and the analysis in Ref. 3 where it was found numerically that the rNLS equation provides a good approximation to the dynamics.In the experiments of Ref. 3, an initial hump of the BEC density in the form of a ring was created by the laser beam effectively pushing the particles away from the center. Then, the BEC expanded radially. An additional radial parabolic potential that is absent in Equation 1 caused the BEC in experiments of Ref. 3 to move away from the center. Relative to this motion, however, the BEC expanded radially toward the inside that is what we consider analytically and numerically here.The dimensionless parameter characterizing dispersion (see Ref.3) was very small in these experiments, ≈ 0.012. With such a small value of Whitham theory, which is a nonlinear WKB-expansion in , is expected to be relevant. As pictures of the experiments show, in the expansion, large oscillations of the BEC density were created, which implies a dispersive shock wave (DSW) started from the initial density jump. The oscillations formed a number of concentric rings, ie, the DSW propagation was indeed radial with high degree of accuracy. Similar concentric rings forming a radial DSW were observed in nonlinear optics experiments. 4 However, in Ref.3, only the one-dimensional NLS equation was treated analytically. Whitham modulation theory for the (1 + 1)-dimensional NLS (1d NLS) equation (see Refs. 5 and 6) was applied to this problem. While qualitative agreement with experiments and direct simulations was found, and the solution/experiments exhibit character of a DSW, the analytical results did not always correspond closely (see, eg, figure 24 of Ref. 3). N...

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A Universal Asymptotic Regime in the Hyperbolic Nonlinear Schrödinger Equation

Cited by 8 publications

References 56 publications

2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

2D solutions of the hyperbolic discrete nonlinear Schrödinger equation

On the Whitham system for the (2+1)‐dimensional nonlinear Schrödinger equation

On the Whitham system for the radial nonlinear Schrödinger equation

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