Whitham modulation theory for the two-dimensional Benjamin-Ono equation

Ablowitz, Mark J.; Biondini, Gino; Qiao, Weihong

doi:10.1103/physreve.96.032225

Cited by 22 publications

(63 citation statements)

References 32 publications

(69 reference statements)

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“…15 This spawned further development of Whitham theory for the KP equation, the (2 + 1)-dimensional Benjamin-Ono equation, and more generally for equations of KP type. [16][17][18] See also earlier work 19,20 about plane dark solitons and oblique DSWs of (2 + 1)dimensional NLS equation in supersonic fluid (BEC) flow around obstacles. The present study provides another example of Whitham theory applied to (2 + 1)-dimensional PDEs and their reductions; as indicated below, our rNLS results are quite different from those for the corresponding 1d NLS system.…”

mentioning

confidence: 85%

“…and = ( ) and = ( ) are the first and second complete elliptic integrals, respectively. Equation 17 itself turns out to lead to Equation C.9, obtained here as a secularity condition, see Appendix C. Instead, we use the combination of Equations 17 and 12, 2 ⋅ (17) + 2 ⋅ (12), to get what would be the (hydrodynamic) momentum conservation law in the 1d NLS case (ie, without 1∕ terms),…”

Section: Whitham System For the Rnls Equationmentioning

confidence: 99%

“…The general nonlinear WKB approach and singular perturbation theory which we use will be an indispensable tool in this research. It has already proven to be extremely useful for (2 + 1)-dimensional PDEs of KP-type, [16][17][18] which are important mathematical models in many applications.…”

Section: Might Be Helpful In Such Investigationsmentioning

confidence: 99%

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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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confidence: 85%

Section: Whitham System For the Rnls Equationmentioning

confidence: 99%

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On the Whitham system for the radial nonlinear Schrödinger equation

2019

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“…All other Wannier modes are obtained through integer shifts, see Eq. (18). Take expansion (21) and sub-stitute it into the full (M(r) = 0) system (6) to obtain…”

Section: Derivation Of Tight-binding Modelmentioning

confidence: 99%

“…The latter, while very useful for Schrödinger operators cf. [18,19], have not been found to be effective in this topological class of problems. The difficulty with using a direct Wannier mode expansion is that the MO system is a Chern insulator and as a result the decay rate of the Wannier functions is slow [20].…”

Section: Introductionmentioning

confidence: 99%

Discrete approximation of topologically protected modes in magneto-optical media

Ablowitz

Cole

2020

Phys. Rev. A

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Topologically protected waves in the linearly polarized Maxwell's equations with gyrotropic/magneto-optic media were studied a decade ago both computationally and experimentally. This paper develops a robust tight-binding model for this system that makes careful use of Wannier function representations. The model provides very good approximations to the underlying band structure. When solved on a semi-infinite strip, it produces exponentially localized edge modes whose corresponding eigenvalues span the frequency band gaps. A set of coupled differential equations are derived which allows one to find how the electromagnetic field propagates unidirectionally, without backscatter from defects. Furthermore, the discrete model predicts topologically protected edge modes with nontrivial Chern number which are consistent with direct simulation.arXiv:2002.06718v1 [physics.optics]

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Evolution of initial discontinuity for the defocusing complex modified KdV equation

Kong

Wang

et al. 2019

Nonlinear Dyn

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Whitham modulation theory for the two-dimensional Benjamin-Ono equation

Cited by 22 publications

References 32 publications

On the Whitham system for the radial nonlinear Schrödinger equation

On the Whitham system for the radial nonlinear Schrödinger equation

Discrete approximation of topologically protected modes in magneto-optical media

Evolution of initial discontinuity for the defocusing complex modified KdV equation

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