Solution of the Riemann problem for polarization waves in a two-component Bose-Einstein condensate

2017

Phys. Rev. A

Self Cite

We consider the Riemann problem of evolution of initial discontinuities for the photon fluid propagating in a normal dispersion fiber with account of self-steepening effects. The dynamics of light field is described by the nonlinear Schrödinger (NLS) equation with self-steepening term appearing due to retardation of the fiber material response to variations of the electromagnetic signal. It is shown that evolution dynamics in this case is much richer than that for the NLS equation. Complete classification of possible wave structures is given for all possible jump conditions at the discontinuity.

Section: Discussionmentioning

confidence: 99%

Section: Classification Of Wave Patternsmentioning

confidence: 99%

Riemann problem for the photon fluid: Self-steepening effects

Ivanov

2017

Phys. Rev. A

Self Cite

“…Therefore it seems more effective to formulate the principles according to which one can predict the wave pattern evolving from a discontinuity with given parameters. Similar method was used [34,35] in classification of wave patterns evolving from initial discontinuities according to the generalized NLS equation and the Landau-Lifshitz equation for easy-plane magnetics or polarization waves in two-component Bose-Einstein condensate.…”

Section: Classification Of Wave Patternsmentioning

confidence: 99%

“…Here we develop the method which permits one to predict the wave pattern arising from any given data for an initial discontinuity. The method is quite general and it was applied to the generalized NLS equations [34] with Kerr-type cubic nonlinearity added to (1), what is important for nonlinear optics applications, and to the Landau-Lifshitz equation for magnetics with easy-plane anisotropy [35]. Here we develop a similar theory for the equation (1).…”

Section: Introductionmentioning

confidence: 99%

Evolution of initial discontinuities in the DNLS equation theory

2018

J. Phys. Commun.

We present the full classification of wave patterns evolving from an initial step-like discontinuity for arbitrary choice of boundary conditions at the discontinuity location in the DNLS equation theory. In this non-convex dispersive hydrodynamics problem, solutions of the Whitham modulation equations are mapped to parameters of a modulated wave by two-valued functions what makes situation much richer than that for a convex case of the NLS equation type. In particular, new types of simple-wave-like structures, such as trigonometric shocks and combined shocks, appear as building elements of the whole wave pattern observable in the long-time evolution of pulses after the wave-breaking point. The developed theory can find applications to propagation of light pulses in fibers and to the theory of Alfvén dispersive shock waves.

“…The full classification of possible wave structures is established for propagation of long pulses in fibers with account of the steepening effect [5] and for polarization waves in BEC [6,7]. Example of a typical wave pattern is shown in Fig.…”

mentioning

confidence: 99%

Dispersive shock waves in nonlinear and atomic optics

2017

EPJ Web Conf.

Abstract.A brief review is given of dispersive shock waves observed in nonlinear optics and dynamics of Bose-Einstein condensates. The theory of dispersive shock waves is developed on the basis of Whitham modulation theory for various situations taking place in these two fields. In particular, the full classification is established for types of wave structures evolving from initial discontinuities for propagation of long light pulses in fibers with account of steepening effect and for dynamics of the polarization mode in two-component Bose-Einstein condensates.Dispersive shock waves (DSWs) are universal phenomena of nonlinear physics. They were observed in water waves, plasma, solid state physics, nonlinear optics and dynamics of Bose-Einstein condensates (BECs). As examples of DSWs in the last two fields of research, one can mention observation of DSWs in BEC under action of a repulsive potential [1] and formation of shocks in evolution of light pulses in fibers [2]. In this report, we give a brief review of phenomena that have been recently observed and provide introduction into theoretical methods of their description. The main attention is paid to situations when the standard approach based on the Whitham theory for the NLS (Gross-Pitaevskii) equation is not applicable. For example, long distance propagation of light pulses in fibers can depend essentially on effects of retardation of the medium response on the electromagnetic field. Dynamics of polarization mode in BECs is even more complicated. In both these cases the nonlinear dispersionless velocities are not monotonous functions of the pulse parameters that leads to appearance of new types of wave structures evolving after wave breaking of the pulse. Such wave structures include kinks, trigonometric shocks, combined shocks, and their appearance makes the problem of classification of possible wave patterns much more difficult than in standard and well-known situations described by the NLS equation.In this talk, we consider approach to the formulated above problem that is based on the developed earlier Whitham theory for the DNLS equation and Landau-Lifshitz equation for magnetics with easy-plane anisotropy. Generally speaking, the suggested method is applicable to nonlinear wave equations that are completely integrable in framework of the Ablowitz-Kaup-Newell-Segur scheme when the inverse scattering transform can be developed. In this case, the reduced finite-gap integration method yields the periodic solutions parameterized by the zeroes of the resolvent R(ν) of the polynomial P(λ) that defines the solution whereas the zeroes of P(λ) play the role of the Riemann invariants of the